174 results on '"Mathematics - Optimization and Control"'
Search Results
52. Observability Inequalities on Measurable Sets for the Stokes System and Applications
- Author
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Can Zhang, Diego A. Souza, and Felipe W. Chaves-Silva
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Control and Optimization ,Lebesgue measure ,Applied Mathematics ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,Optimization and Control (math.OC) ,FOS: Mathematics ,Applied mathematics ,Shape optimization ,Observability ,0101 mathematics ,Stokes operator ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as an observability inequalities on space-time measurable sets of positive measure for non-stationary Stokes system. Furthermore, we provide their applications in the theory of shape optimization and time optimal control problems.
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- 2020
53. Model Predictive Control, Cost Controllability, and Homogeneity
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Jean-Michel Coron, Karl Worthmann, Lars Grüne, Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Control And GEometry (CaGE ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Mathematisches Institut [Bayreuth], Universität Bayreuth, Technische Universität Ilmenau (TU ), ANR-15-CE23-0007,Finite4SoS,Commande et estimation en temps fini pour les Systèmes de Systèmes(2015), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
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0209 industrial biotechnology ,Control and Optimization ,Homogeneous approximation ,Applied Mathematics ,Homogeneity (statistics) ,010102 general mathematics ,Stability guarantee ,02 engineering and technology ,01 natural sciences ,Controllability ,Model predictive control ,Cost controllability ,020901 industrial engineering & automation ,Exponential stability ,Optimization and Control (math.OC) ,Linearization ,Control theory ,FOS: Mathematics ,Homogeneity ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,0101 mathematics ,Mathematics - Optimization and Control ,Closed loop ,Mathematics - Abstract
International audience; We are concerned with the design of Model Predictive Control (MPC) schemes such that asymptotic stability of the resulting closed loop is guaranteed even if the linearization at the desired set point fails to be stabilizable. Therefore, we propose to construct the stage cost based on the homogeneous approximation and rigorously show that applying MPC yields an asymp-totically stable closed-loop behavior if the homogeneous approximation is asymptotically null controllable. To this end, we verify cost controllability-a condition relating the current state, the stage cost, and the growth of the value function w.r.t. time-for this class of systems in order to provide stability and performance guarantees for the proposed MPC scheme without stabilizing terminal costs or constraints.
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- 2020
54. On the Minimum Pair Approach for Average Cost Markov Decision Processes with Countable Discrete Action Spaces and Strictly Unbounded Costs
- Author
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Huizhen Yu
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Discrete mathematics ,Control and Optimization ,Applied Mathematics ,State (functional analysis) ,Discrete action ,Mathematics::Logic ,90C39, 90C40, 93E20 ,Optimization and Control (math.OC) ,FOS: Mathematics ,Countable set ,Markov decision process ,Mathematics - Optimization and Control ,Average cost ,Mathematics - Abstract
We consider average-cost Markov decision processes (MDPs) with Borel state spaces, countable, discrete action spaces, and strictly unbounded one-stage costs. For the minimum pair approach, we introduce a new majorization condition on the state transition stochastic kernel, in place of the commonly required continuity conditions on the MDP model. We combine this majorization condition with Lusin's theorem to prove the existence of a stationary minimum pair, i.e., a stationary policy paired with an invariant probability measure induced on the state space, with the property that the pair attains the minimum long-run average cost over all policies and initial distributions. We also establish other optimality properties of a stationary minimum pair, and for the stationary policy in such a pair, under additional recurrence or regularity conditions, we prove its pathwise optimality and strong optimality. Our results can be applied to a class of countable action space MDPs in which the dynamics and one-stage costs are discontinuous with respect to the state variable., 25 pages; to appear in SIAM Journal on Control and Optimization. This paper extends the author's earlier results in Section 4 of arXiv:1901.03374v1
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- 2020
55. Finite-Time Stability for Differential Inclusions with Applications to Neural Networks
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Sławomir Plaskacz, Radosław Matusik, Andrzej Nowakowski, and Andrzej Rogowski
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Control and Optimization ,Differential inclusion ,Artificial neural network ,Optimization and Control (math.OC) ,Applied Mathematics ,Gronwall's inequality ,FOS: Mathematics ,Stable equilibrium ,Applied mathematics ,Finite time ,Mathematics - Optimization and Control ,Stability (probability) ,Mathematics - Abstract
The paper investigates sufficient conditions on a differential inclusion which guarantee that the origin is a finite time stable equilibrium, namely a weak local one, a weak global one or a strong local one. The analysis relies on the existence of a Lyapunov function. A new Gronwall type results are used to estimate the settling time. An example of a neural network which is finite-time stable is given
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- 2020
56. A Universal Dynamic Program and Refined Existence Results for Decentralized Stochastic Control
- Author
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Serdar Yüksel
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Stochastic control ,0209 industrial biotechnology ,Mathematical optimization ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,02 engineering and technology ,01 natural sciences ,Decentralised system ,Dynamic programming ,020901 industrial engineering & automation ,Action (philosophy) ,Optimization and Control (math.OC) ,FOS: Mathematics ,0101 mathematics ,Control (linguistics) ,Mathematics - Optimization and Control ,Mathematics - Abstract
For sequential stochastic control problems with standard Borel measurement and control action spaces, we introduce a general (universally applicable) dynamic programming formulation, establish its well-posedness, and provide new existence results for optimal policies. Our dynamic program builds in part on Witsenhausen's standard form, but with a different formulation for the state, action, and transition dynamics. Using recent results on measurability properties of strategic measures in decentralized control, we obtain a standard Borel controlled Markov model. This allows for a well-defined dynamic programming recursion through universal measurability properties of the value functions for each time stage. In addition, new existence results are obtained for optimal policies in decentralized stochastic control. These state that for a static team with independent measurements, it suffices for the cost function to be continuous (only) in the actions for the existence of an optimal policy under mild compactness (or tightness) conditions. These also apply to dynamic teams which admit static reductions with independent measurements through a change of measure transformation. We show through a counterexample that weaker conditions may not lead to existence of an optimal team policy. The paper's existence results generalize those previously reported in the literature. A summary of and comparison with previously reported results and some applications are presented., To appear in SIAM Journal on Control and Optimization
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- 2020
57. A Variational Formula for Risk-Sensitive Control of Diffusions in $\mathbb{R}^d$
- Author
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Ari Arapostathis and Anup Biswas
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0209 industrial biotechnology ,Pure mathematics ,Control and Optimization ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Principal (computer security) ,Mathematics::Analysis of PDEs ,02 engineering and technology ,Risk sensitive ,01 natural sciences ,Elliptic operator ,Mathematics - Analysis of PDEs ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,FOS: Mathematics ,Primary 60J60, Secondary 60J25, 35P15, 60F10, 49G05 ,0101 mathematics ,Control (linguistics) ,Mathematics - Optimization and Control ,Mathematics - Probability ,Eigenvalues and eigenvectors ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We address the variational problem for the generalized principal eigenvalue on $\mathbb{R}^d$ of linear and semilinear elliptic operators associated with nondegenerate diffusions controlled through the drift. We establish the Collatz-Wielandt formula for potentials that vanish at infinity under minimal hypotheses, and also for general potentials under blanket geometric ergodicity assumptions. We also present associated results having the flavor of a refined maximum principle., Comment: 19 pages
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- 2020
58. A Regularity Criterion for a 3D Chemo-Repulsion System and Its Application to a Bilinear Optimal Control Problem
- Author
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Exequiel Mallea-Zepeda, Francisco Guillén-González, María Ángeles Rodríguez-Bellido, Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico, and Universidad de Sevilla. FQM131: Ec.diferenciales,Simulacion Num.y Desarrollo Software
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0209 industrial biotechnology ,optimality conditions ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,bilinear optimal control ,Bilinear interpolation ,02 engineering and technology ,Optimal control ,01 natural sciences ,35K51, 35Q92, 49J20, 49K20 ,Strong solutions ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,chemo-repulsion and production model ,weak solutions ,FOS: Mathematics ,Production (economics) ,Applied mathematics ,strong solutions ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper we study a bilinear optimal control problem associated to a 3D chemo-repulsion model with linear production. We prove the existence of weak solutions and we establish a regularity criterion to get global in time strong solutions. As a consequence, we deduce the existence of a global optimal solution with bilinear control and, using a Lagrange multipliers theorem, we derive first-order optimality conditions for local optimal solutions., 41 pages. arXiv admin note: text overlap with arXiv:1806.10076
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- 2020
59. Error Estimates for Space-Time Discretization of Parabolic Time-Optimal Control Problems with Bang-Bang Controls
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Konstantin Pieper, Boris Vexler, and Lucas Bonifacius
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0209 industrial biotechnology ,Control and Optimization ,Spacetime ,Discretization ,Applied Mathematics ,Space time ,010102 general mathematics ,Numerical Analysis (math.NA) ,02 engineering and technology ,49K20, 49M25, 65M15, 65M60 ,Time optimal ,01 natural sciences ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,A priori and a posteriori ,Mathematics - Numerical Analysis ,0101 mathematics ,Galerkin method ,Control (linguistics) ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper a priori error estimates are derived for full discretization (in space and time) of time-optimal control problems. Various convergence results for the optimal time and the control variable are proved under different assumptions. Especially the case of bang-bang controls is investigated. Numerical examples are provided to illustrate the results.
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- 2019
60. Approximate Public-Signal Correlated Equilibria for Nonzero-Sum Differential Games
- Author
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Yurii Averboukh
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Computer Science::Computer Science and Game Theory ,Correlated equilibrium ,Class (set theory) ,Control and Optimization ,EQUILIBRIUM VALUE ,91A23, 91A10, 49N70, 91A28 ,MathematicsofComputing_NUMERICALANALYSIS ,NONZERO-SUM DIFFERENTIAL GAMES ,Signal ,CONTROL WITH MODEL ,STOCHASTIC GAME ,STOCHASTIC DIFFERENTIAL GAME ,APPROXIMATE EQUILIBRIUMS ,Differential game ,FOS: Mathematics ,Applied mathematics ,Mathematics - Optimization and Control ,CORRELATED EQUILIBRIA ,Mathematics ,CONTINUOUS TIME SYSTEMS ,CONTINUOUS-TIME ,Applied Mathematics ,APPROXIMATE EQUILIBRIUM ,NONZERO-SUM DIFFERENTIAL GAME ,ComputingMilieux_PERSONALCOMPUTING ,Construct (python library) ,STOCHASTIC SYSTEMS ,GAME THEORY ,Optimization and Control (math.OC) ,PUBLIC-SIGNAL CORRELATED STRATEGIES ,Approximate equilibrium ,Differential (mathematics) - Abstract
We construct an approximate public-signal correlated equilibrium for a nonzero-sum differential game in the class of stochastic strategies with memory. The construction is based on a solution of an auxiliary nonzero-sum continuous-time stochastic game. This class of games includes stochastic differential games and continuous-time Markov games. Moreover, we study the limit of approximate equilibrium outcomes in the case when the auxiliary stochastic games tend to the original deterministic one. We show that it lies in the convex hull of the set of equilibrium values provided by deterministic punishment strategies., 35 pages
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- 2019
61. Effects of Parametric Uncertainties in Cascaded Open Quantum Harmonic Oscillators and Robust Generation of Gaussian Invariant States
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Igor G. Vladimirov, Ian R. Petersen, and Matthew R. James
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Quantum Physics ,Control and Optimization ,Applied Mathematics ,Gaussian ,FOS: Physical sciences ,Systems and Control (eess.SY) ,81S22, 81S25, 81P16, 81Q15, 94A17, 93E15, 49L20, 60G15, 93B35, 93B51 ,symbols.namesake ,Stochastic differential equation ,Optimization and Control (math.OC) ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,symbols ,Computer Science - Systems and Control ,Statistical physics ,Invariant (mathematics) ,Quantum Physics (quant-ph) ,Mathematics - Optimization and Control ,Quantum ,Harmonic oscillator ,Parametric statistics ,Mathematics - Abstract
This paper is concerned with the generation of Gaussian invariant states in cascades of open quantum harmonic oscillators governed by linear quantum stochastic differential equations. We carry out infinitesimal perturbation analysis of the covariance matrix for the invariant Gaussian state of such a system and the related purity functional subject to inaccuracies in the energy and coupling matrices of the subsystems. This leads to the problem of balancing the state-space realizations of the component oscillators through symplectic similarity transformations in order to minimize the mean square sensitivity of the purity functional to small random perturbations of the parameters. This results in a quadratic optimization problem with an effective solution in the case of cascaded one-mode oscillators, which is demonstrated by a numerical example. We also discuss a connection of the sensitivity index with classical statistical distances and outline infinitesimal perturbation analysis for translation invariant cascades of identical oscillators. The findings of the paper are applicable to robust state generation in quantum stochastic networks., Comment: 41 pages, 3 figures, 1 table
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- 2019
62. Sufficiency of Deterministic Policies for Atomless Discounted and Uniformly Absorbing MDPs with Multiple Criteria
- Author
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Eugene A. Feinberg and Aleksey B. Piunovskiy
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Computer Science::Machine Learning ,Computer Science::Computer Science and Game Theory ,0209 industrial biotechnology ,Mathematical optimization ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,02 engineering and technology ,State (functional analysis) ,01 natural sciences ,Mathematics::Logic ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,FOS: Mathematics ,Multiple criteria ,Markov decision process ,0101 mathematics ,State distribution ,Mathematics - Optimization and Control ,Mathematics - Abstract
This paper studies Markov decision processes (MDPs) with atomless initial state distributions and atomless transition probabilities. Such MDPs are called atomless. The initial state distribution is considered to be fixed. We show that for discounted MDPs with bounded one-step reward vector-functions, for each policy there exists a deterministic (that is, nonrandomized and stationary) policy with the same performance vector. This fact is proved in the paper for a more general class of uniformly absorbing MDPs with expected total rewards, and then it is extended under certain assumptions to MDPs with unbounded rewards. For problems with multiple criteria and constraints, the results of this paper imply that for atomless MDPs studied in this paper it is sufficient to consider only deterministic policies, while without the atomless assumption it is well-known that randomized policies can outperform deterministic ones. We also provide an example of an MDP demonstrating that if a vector measure is defined on a standard Borel space, then Lyapunov's convexity theorem is a special case of the described results.
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- 2019
63. The Optimal Equilibrium for Time-Inconsistent Stopping Problems---The Discrete-Time Case
- Author
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Zhou Zhou and Yu-Jui Huang
- Subjects
TheoryofComputation_MISCELLANEOUS ,Computer Science::Computer Science and Game Theory ,0209 industrial biotechnology ,Mathematical optimization ,Control and Optimization ,02 engineering and technology ,49K21, 60J05, 91A13, 93E20 ,01 natural sciences ,Subgame perfect equilibrium ,FOS: Economics and business ,020901 industrial engineering & automation ,Fixed-point iteration ,FOS: Mathematics ,Optimal stopping ,Dynamic inconsistency ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Discounting ,Applied Mathematics ,010102 general mathematics ,TheoryofComputation_GENERAL ,Mathematical Finance (q-fin.MF) ,Discrete time and continuous time ,Quantitative Finance - Mathematical Finance ,Optimization and Control (math.OC) - Abstract
We study an infinite-horizon discrete-time optimal stopping problem under non-exponential discounting. A new method, which we call the iterative approach, is developed to find subgame perfect Nash equilibria. When the discount function induces decreasing impatience, we establish the existence of an equilibrium through fixed-point iterations. Moreover, we show that there exists a unique optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency.
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- 2019
64. Extended Mean Field Control Problems: Stochastic Maximum Principle and Transport Perspective
- Author
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René Carmona, Beatrice Acciaio, and Julio Backhoff-Veraguas
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0209 industrial biotechnology ,Control and Optimization ,02 engineering and technology ,01 natural sciences ,020901 industrial engineering & automation ,Perspective (geometry) ,Maximum principle ,Joint probability distribution ,FOS: Mathematics ,Applied mathematics ,QA Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Stochastic control ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Process (computing) ,Function (mathematics) ,State (functional analysis) ,16. Peace & justice ,93E20, 90C08, 60H15, 60H30, 49K45, 60K35 ,Mean field theory ,Optimization and Control (math.OC) ,Mathematics - Probability - Abstract
We study Mean Field stochastic control problems where the cost function and the state dynamics depend upon the joint distribution of the controlled state and the control process. We prove suitable versions of the Pontryagin stochastic maximum principle, both in necessary and in sufficient form, which extend the known conditions to this general framework. Furthermore, we suggest a variational approach to study a weak formulation of these control problems. We show a natural connection between this weak formulation and optimal transport on path space, which inspires a novel discretization scheme., Comment: We changed the title, added an example, and suggest a discretization scheme
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- 2019
65. Contracting Theory with Competitive Interacting Agents
- Author
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Romuald Elie, Dylan Possamaï, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), and Université Paris Dauphine-PSL-Centre National de la Recherche Scientifique (CNRS)
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0209 industrial biotechnology ,General Economics (econ.GN) ,Control and Optimization ,Moral hazard ,02 engineering and technology ,Quantitative Finance - Economics ,01 natural sciences ,Nash equilibrium ,FOS: Economics and business ,Multidimensional quadratic BSDEs ,Competition (economics) ,Microeconomics ,Terminal value ,symbols.namesake ,020901 industrial engineering & automation ,FOS: Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Principal (computer security) ,Principal point ,Reservation ,Resolution (logic) ,relative performance ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Principal multi-agents problems ,Optimization and Control (math.OC) ,symbols ,competition ,Mathematics - Probability - Abstract
In a framework close to the one developed by Holmstr\"om and Milgrom [44], we study the optimal contracting scheme between a Principal and several Agents. Each hired Agent is in charge of one project, and can make efforts towards managing his own project, as well as impact (positively or negatively) the projects of the other Agents. Considering economic Agents in competition with relative performance concerns, we derive the optimal contracts in both first best and moral hazard settings. The enhanced resolution methodology relies heavily on the connection between Nash equilibria and multidimensional quadratic BSDEs. The optimal contracts are linear and each agent is paid a fixed proportion of the terminal value of all the projects of the firm. Besides, each Agent receives his reservation utility, and those with high competitive appetence are assigned less volatile projects, and shall even receive help from the other Agents. From the principal point of view, it is in the firm interest in our model to strongly diversify the competitive appetence of the Agents., Comment: 36 pages
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- 2019
66. Pseudo-Backstepping and Its Application to the Control of Korteweg--de Vries Equation from the Right Endpoint on a Finite Domain
- Author
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Türker Özsarı, Ahmet Batal, and Izmir Institute of Technology. Mathematics
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0209 industrial biotechnology ,Control and Optimization ,Mathematics::Analysis of PDEs ,Boundary controller ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,Domain (mathematical analysis) ,Pseudo-backstepping ,Mathematics - Analysis of PDEs ,020901 industrial engineering & automation ,Korteweg-de Vries equation ,FOS: Mathematics ,93D15, 35Q53, 93C20, 93C10, 93D20, 35A01, 35B45 ,0101 mathematics ,Korteweg–de Vries equation ,Control (linguistics) ,Mathematics - Optimization and Control ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics ,Vries equation ,Feedback stabilization ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Backstepping ,Optimization and Control (math.OC) ,Analysis of PDEs (math.AP) - Abstract
In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries (KdV) equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping controllers that act at the right endpoint of the domain is more challenging than its left endpoint counterpart. The standard application of the backstepping method fails, because corresponding kernel models become overdetermined. In order to deal with this difficulty, we introduce the pseudo-backstepping method, which uses a pseudo-kernel that satisfies all but one desirable boundary condition. Moreover, various norms of the pseudo-kernel can be controlled through a parameter in one of its boundary conditions. We prove that the boundary controllers constructed via this pseudo-kernel still exponentially stabilize the system with the cost of a low exponential rate of decay. We show that a single Dirichlet controller is sufficient for exponential stabilization with a slower rate of decay. We also consider a second order feedback law acting at the right Dirichlet boundary condition. We show that this approach works if the main equation includes only the third order term, while the same problem remains open if the main equation involves the first order and/or the nonlinear term(s). At the end of the paper, we give numerical simulations to illustrate the main result., Comment: 23 pages, 11 figures, 1 table. In this version, an additional reference has been added and discussed within the text. Also some typos were corrected
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- 2019
67. Synchronization of Kuramoto Oscillators: Inverse Taylor Expansions
- Author
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Saber Jafarpour, Francesco Bullo, and Elizabeth Y. Huang
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Inverse ,Systems and Control (eess.SY) ,Dynamical Systems (math.DS) ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Synchronization (computer science) ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,Taylor series ,symbols ,Computer Science - Systems and Control ,Applied mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
Synchronization in networks of coupled oscillators is a widely studied topic with extensive scientific and engineering applications. In this paper, we study the frequency synchronization problem for networks of Kuramoto oscillators with arbitrary topology and heterogeneous edge weights. We propose a novel equivalent transcription for the equilibrium synchronization equation. Using this transcription, we develop a power series expansion to compute the synchronized solution of the Kuramoto model as well as a sufficient condition for the strong convergence of this series expansion. Truncating the power series provides (i) an efficient approximation scheme for computing the synchronized solution, and (ii) a simple-to-check, statistically-correct hierarchy of increasingly accurate synchronization tests. This hierarchy of tests provides a theoretical foundation for and generalizes the best-known approximate synchronization test in the literature. Our numerical experiments illustrate the accuracy and the computational efficiency of the truncated series approximation compared to existing iterative methods and existing synchronization tests.
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- 2019
68. Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach
- Author
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Dan Goreac, PS, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), and Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Control and Optimization ,Open set ,Markov process ,Near-Viability ,symbols.namesake ,FOS: Mathematics ,PDMP ,Viscosity Solutions ,Gene Networks ,Mathematics - Optimization and Control ,Brownian motion ,Mathematics ,Applied Mathematics ,Probability (math.PR) ,Invariance ,Order (ring theory) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Brownian Diffusion ,MSC: 93E20 ,49L25 ,60J60 ,60J75 ,34H05 ,92C42 ,Optimization and Control (math.OC) ,Viscosity (programming) ,symbols ,Piecewise ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Mathematics - Probability - Abstract
International audience; Motivated by the result of invariance of regular-boundary open sets in \cite{CannarsaDaPratoFrankowska2009} and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems). Comments on optimality are provided via examples. Second, we provide a border avoidance (near-viability) counterpart of \cite{CannarsaDaPratoFrankowska2009} for controlled Brownian diffusions and piecewise deterministic switched Markov processes (PDsMP). We equally discuss to which extent Lipschitz-continuity of the driving coefficients is needed. Finally, by applying the theoretical result on PDsMP to Hasty's model of bacteriophage (\cite{hasty_pradines_dolnik_collins_00}, \cite{crudu_debussche_radulescu_09}), we show the necessity of explicit modeling for the environmental cue triggering lysis.
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- 2019
69. Boundary Controllability of Two Vibrating Strings Connected by a Point Mass with Variable Coefficients
- Author
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Jamel Ben Amara and Emna Beldi
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0209 industrial biotechnology ,Control and Optimization ,Point particle ,I.2.7 ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,02 engineering and technology ,Vibrating string ,F.2.2 ,01 natural sciences ,35P, 47A, 93B ,Controllability ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,FOS: Mathematics ,0101 mathematics ,Constant (mathematics) ,Mathematics - Optimization and Control ,Mathematics ,Variable (mathematics) - Abstract
S. Hansen and E. Zuazua [SIAM J. Cont. Optim., 1995] studied the problem of exact controllability of two strings connected by a point mass with constant physical coefficients. In this paper we study the same problem with variable physical coefficients. This system is generated by the following equations $$\rho(x) u_{tt}=(\sigma(x) u_{x})_{x}-q(x)u,~~~~x\in (-1,0)\cup (0,1),~t>0,$$ $$Mu_{tt}(0,t)+\sigma_{1}(0)u_{x}(0^{-},t)-\sigma_{2}(0)u_{x}(0^{+},t)=0,~~~t>0,$$ with Dirichlet boundary condition on the left end and a control acts on the right end. We prove that this system is exactly controllable in an asymmetric space for the control time $T> 2\int_{-1}^{1}(\frac{\rho(x)}{\sigma(x)})^{\frac{1}{2}}dx$. We establish the equivalence between a suitable asymmetric norm of the initial data and the $L^{2}(0,T)$-norm of $u_{x}(1,t)$. Our approach is mainly based on a detailed spectral analysis and the theory of divided differences. More precisely, we prove that the spectral gap tends to zero with a precise asymptotic estimate., Comment: 34pages and 0 figures
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- 2019
70. Approximation of Sweeping Processes and Controllability for a Set-Valued Evolution
- Author
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Marco Mazzola, Khai T. Nguyen, Alberto Bressan, Pennsylvania State University (Penn State), Penn State System, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), North Carolina A&T State University, and University of North Carolina System (UNC)
- Subjects
Discrete mathematics ,Control and Optimization ,Applied Mathematics ,16. Peace & justice ,controllability ,Omega ,Set (abstract data type) ,Controllability ,Quantitative Biology::Quantitative Methods ,Optimization and Control (math.OC) ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,set-valued evolution ,[MATH]Mathematics [math] ,Mathematics - Optimization and Control ,sweeping processes ,Mathematics - Abstract
We consider a controlled evolution problem for a set $\Omega(t)\in\mathbb{R}^d$, originally motivated by a model where a dog controls a flock of sheep. Necessary conditions and sufficient conditions are given, in order that the evolution be completely controllable. Similar techniques are then applied to the approximation of a sweeping process. Under suitable assumptions, we prove that there exists a control function such that the corresponding evolution of the set $\Omega(t)$ is arbitrarily close to the one determined by the sweeping process., Comment: 28 pages, 3 figures
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- 2019
71. An Approximation Scheme for Semilinear Parabolic PDEs with Convex and Coercive Hamiltonians
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Gechun Liang, Shuo Huang, and Thaleia Zariphopoulou
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0209 industrial biotechnology ,Class (set theory) ,Control and Optimization ,Mathematics::Optimization and Control ,Mathematics::Analysis of PDEs ,02 engineering and technology ,01 natural sciences ,Mathematics::Numerical Analysis ,020901 industrial engineering & automation ,FOS: Mathematics ,Applied mathematics ,35K65, 65M12, 93E20 ,Mathematics - Numerical Analysis ,0101 mathematics ,QA ,Mathematics - Optimization and Control ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,Feynman–Kac formula ,Numerical Analysis (math.NA) ,16. Peace & justice ,Parabolic partial differential equation ,Optimization and Control (math.OC) ,Scheme (mathematics) ,Project portfolio management - Abstract
We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac and the Hopf-Lax formulae. We establish the convergence of the scheme and determine the convergence rate, combining Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation., Comment: 24 pages
- Published
- 2020
- Full Text
- View/download PDF
72. CONVERGENCE PROPERTIES OF ADAPTIVE SYSTEMS AND THE DEFINITION OF EXPONENTIAL STABILITY
- Author
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Benjamin Jenkins, Travis E. Gibson, Anuradha M. Annaswamy, and Eugene Lavretsky
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Adaptive control ,Applied Mathematics ,02 engineering and technology ,State (functional analysis) ,Systems and Control (eess.SY) ,010501 environmental sciences ,01 natural sciences ,Article ,020901 industrial engineering & automation ,Exponential stability ,Rate of convergence ,Optimization and Control (math.OC) ,Adaptive system ,Bounded function ,Convergence (routing) ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Computer Science - Systems and Control ,Reference model ,Mathematics - Optimization and Control ,0105 earth and related environmental sciences ,Mathematics - Abstract
The convergence properties of adaptive systems in terms of excitation conditions on the regressor vector are well known. With persistent excitation of the regressor vector in model reference adaptive control the state error and the adaptation error are globally exponentially stable, or equivalently, exponentially stable in the large. When the excitation condition however is imposed on the reference input or the reference model state it is often incorrectly concluded that the persistent excitation in those signals also implies exponential stability in the large. The definition of persistent excitation is revisited so as to address some possible confusion in the adaptive control literature. It is then shown that persistent excitation of the reference model only implies local persistent excitation (weak persistent excitation). Weak persistent excitation of the regressor is still sufficient for uniform asymptotic stability in the large, but not exponential stability in the large. We show that there exists an infinite region in the state-space of adaptive systems where the state rate is bounded. This infinite region with finite rate of convergence is shown to exist not only in classic open-loop reference model adaptive systems, but also in a new class of closed-loop reference model adaptive systems., Comment: 22 pages, 5 figures
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- 2019
73. Optimal Control of Continuous-Time Markov Chains with Noise-Free Observation
- Author
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Alessandro Calvia and Calvia, A
- Subjects
0209 industrial biotechnology ,Partial observation control problem ,Control and Optimization ,93E20, 60J27, 60J25 ,Bellman equation ,Continuous-time Markov chains ,Piecewise deterministic Markov processes ,Viscosity solutions ,02 engineering and technology ,01 natural sciences ,020901 industrial engineering & automation ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,partial observation control problem, continuous-time Markov chains, piecewise deterministic Markov processes, Bellman equation, viscosity solutions ,Mathematics - Optimization and Control ,Finite set ,Mathematics ,Markov chain ,Applied Mathematics ,010102 general mathematics ,Process (computing) ,Optimal control ,Noise ,MAT/06 - PROBABILITA E STATISTICA MATEMATICA ,Optimization and Control (math.OC) ,Infinite horizon - Abstract
We consider an infinite horizon optimal control problem for a continuous-time Markov chain $X$ in a finite set $I$ with noise-free partial observation. The observation process is defined as $Y_t = h(X_t)$, $t \geq 0$, where $h$ is a given map defined on $I$. The observation is noise-free in the sense that the only source of randomness is the process $X$ itself. The aim is to minimize a discounted cost functional and study the associated value function $V$. After transforming the control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we provide a link between the value function $v$ associated with the latter control problem and the original value function $V$. Then, we present two different characterizations of $v$ (and indirectly of $V$): on one hand as the unique fixed point of a suitably defined contraction mapping and on the other hand as the unique constrained viscosity solution (in the sense of Soner) of a HJB integro-differential equation. Under suitable assumptions, we finally prove the existence of an optimal control
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- 2018
74. Linear-Quadratic-Gaussian Mixed Mean-Field Games with Heterogeneous Input Constraints
- Author
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Ying Hu, Jianhui Huang, Tianyang Nie, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), The Hong Kong Polytechnic University [Hong Kong] (POLYU), Shandong University, ANR-16-CE40-0015-01, Agence Nationale de la Recherche, 61573217, National Natural Science Foundation of China, ZR2016AQ13, Natural Science Foundation of Shandong Province, 15327516P, Research Grants Council, University Grants Committee, 2015HW023, Shandong University, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), The Hong Kong Polytechnic University [Hong Kong] ( POLYU ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and ANR-16-CE40-0015,MFG,Jeux Champs Moyen(2016)
- Subjects
[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC] ,Input constraint ,0209 industrial biotechnology ,Control and Optimization ,Minor (linear algebra) ,02 engineering and technology ,Space (mathematics) ,Linear-quadratic-Gaussian control ,01 natural sciences ,Projection (linear algebra) ,$ε$-Nash equilibrium ,Stochastic differential equation ,020901 industrial engineering & automation ,Consistency (statistics) ,FOS: Mathematics ,Applied mathematics ,Contraction mapping ,0101 mathematics ,Mathematics - Optimization and Control ,60H10, 60H30, 91A10, 91A23, 91A25, 93E20 ,Mathematics ,Applied Mathematics ,010102 general mathematics ,AMS Subject Classification: 60H10, 60H30, 91A10, 91A23, 91A25, 93E20 ,Regular polygon ,Projection operator ,Linear-quadratic mixed mean-field games ,Optimization and Control (math.OC) ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Forward-backward stochastic differential equation - Abstract
We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets $\Gamma_{k}$ of $\mathbb{R}^{m}.$ The decentralized strategies for individual agents and consistency condition system are represented in an unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on $\Gamma_{k}$. The well-posedness of consistency system is established in both the local and global cases by the contraction mapping and discounting method respectively. Related $\varepsilon-$Nash equilibrium property is also verified., Comment: 40 pages
- Published
- 2018
75. The Convex Feasible Set Algorithm for Real Time Optimization in Motion Planning
- Author
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Masayoshi Tomizuka, Chung-Yen Lin, and Changliu Liu
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FOS: Computer and information sciences ,0209 industrial biotechnology ,Mathematical optimization ,Control and Optimization ,02 engineering and technology ,01 natural sciences ,Computer Science::Robotics ,Computer Science - Robotics ,020901 industrial engineering & automation ,Development (topology) ,FOS: Mathematics ,Motion planning ,Robot motion planning ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,business.industry ,Applied Mathematics ,010102 general mathematics ,Feasible region ,Regular polygon ,Robotics ,Optimization and Control (math.OC) ,Artificial intelligence ,business ,Robotics (cs.RO) - Abstract
With the development of robotics, there are growing needs for real time motion planning. However, due to obstacles in the environment, the planning problem is highly non-convex, which makes it difficult to achieve real time computation using existing non-convex optimization algorithms. This paper introduces the convex feasible set algorithm (CFS) which is a fast algorithm for non-convex optimization problems that have convex costs and non-convex constraints. The idea is to find a convex feasible set for the original problem and iteratively solve a sequence of subproblems using the convex constraints. The feasibility and the convergence of the proposed algorithm are proved in the paper. The application of this method on motion planning for mobile robots is discussed. The simulations demonstrate the effectiveness of the proposed algorithm., in SIAM Journal on Control and Optimization
- Published
- 2018
76. Risk Sensitive Portfolio Optimization in a Jump Diffusion Model with Regimes
- Author
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Milan Kumar Das, Nimit Rana, and Anindya Goswami
- Subjects
0209 industrial biotechnology ,Mathematical optimization ,Control and Optimization ,Jump diffusion ,Mathematics::Optimization and Control ,02 engineering and technology ,Risk sensitive ,01 natural sciences ,FOS: Economics and business ,010104 statistics & probability ,Mathematics - Analysis of PDEs ,020901 industrial engineering & automation ,Portfolio Management (q-fin.PM) ,Computer Science::Computational Engineering, Finance, and Science ,FOS: Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Quantitative Finance - Portfolio Management ,Mathematics ,Applied Mathematics ,Probability (math.PR) ,Process (computing) ,Finite horizon ,Optimization and Control (math.OC) ,Portfolio optimization problem ,Portfolio optimization ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
This article studies a portfolio optimization problem, where the market consisting of several stocks is modeled by a multi-dimensional jump-diffusion process with age-dependent semi-Markov modulated coefficients. We study risk sensitive portfolio optimization on the finite time horizon. We study the problem by using a probabilistic approach to establish the existence and uniqueness of the classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We also implement a numerical scheme to investigate the behavior of solutions for different values of the initial portfolio wealth, the maturity, and the risk of aversion parameter., 29 pages, 3 figures
- Published
- 2018
77. Equivalence between Minimal Time and Minimal Norm Control Problems for the Heat Equation
- Author
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Gengsheng Wang and Shulin Qin
- Subjects
Discrete mathematics ,0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Convex set ,02 engineering and technology ,01 natural sciences ,020901 industrial engineering & automation ,Time function ,Optimization and Control (math.OC) ,Norm (mathematics) ,Bounded function ,FOS: Mathematics ,Rising sun lemma ,Heat equation ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
This paper presents the equivalence between minimal time and minimal norm control problems for internally controlled heat equations. The target is an arbitrarily fixed bounded, closed and convex set with a nonempty interior in the state space. This study differs from [G. Wang and E. Zuazua, \textit{On the equivalence of minimal time and minimal norm controls for internally controlled heat equations}, SIAM J. Control Optim., 50 (2012), pp. 2938-2958] where the target set is the origin in the state space. When the target set is the origin or a ball, centered at the origin, the minimal norm and the minimal time functions are continuous and strictly decreasing, and they are inverses of each other. However, when the target is located in other place of the state space, the minimal norm function may be no longer monotonous and the range of the minimal time function may not be connected. These cause the main difficulty in our study. We overcome this difficulty by borrowing some idea from the classical raising sun lemma (see, for instance, Lemma 3.5 and Figure 5 on Pages 121-122 in [E. M. Stein and R. Shakarchi, \textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}, Princeton University Press, 2005]).
- Published
- 2018
78. On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces
- Author
-
Christian Kanzow and Daniel Steck
- Subjects
0209 industrial biotechnology ,021103 operations research ,Control and Optimization ,Augmented Lagrangian method ,Applied Mathematics ,0211 other engineering and technologies ,Banach space ,49K, 49M, 65K, 90C ,02 engineering and technology ,Characterization (mathematics) ,Local convergence ,Multiplier (Fourier analysis) ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Variational inequality ,FOS: Mathematics ,Applied mathematics ,Minification ,Mathematics - Optimization and Control ,Mathematics - Abstract
This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the (primal-dual) solution set and give a sufficient condition for such an error bound to hold. In the second part of the paper, we consider an algorithm of augmented Lagrangian type for the solution of such variational problems. We give some global convergence properties of the method and then use the error bound theory to provide estimates for the rate of convergence and to deduce boundedness of the sequence of penalty parameters. Finally, numerical results for optimal control, Nash equilibrium problems, and elliptic parameter estimation problems are presented., Comment: 27 pages
- Published
- 2018
79. Distributed Control for Spatial Self-Organization of Multi-agent Swarms
- Author
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Vishaal Krishnan and Sonia Martinez
- Subjects
Self-organization ,0209 industrial biotechnology ,34B45, 35B40, 35B35, 58J32, 58J35, 58E20 ,Control and Optimization ,Applied Mathematics ,Distributed computing ,Control (management) ,02 engineering and technology ,Computer Science::Multiagent Systems ,020901 industrial engineering & automation ,Density distribution ,Work (electrical) ,Optimization and Control (math.OC) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this work, we design distributed control laws for spatial self-organization of multi-agent swarms in 1D and 2D spatial domains. The objective is to achieve a desired density distribution over a simply-connected spatial domain. Since individual agents in a swarm are not themselves of interest and we are concerned only with the macroscopic objective, we view the network of agents in the swarm as a discrete approximation of a continuous medium and design control laws to shape the density distribution of the continuous medium. The key feature of this work is that the agents in the swarm do not have access to position information. Each individual agent is capable of measuring the current local density of agents and can communicate with its spatial neighbors. The network of agents implement a Laplacian-based distributed algorithm, which we call pseudo-localization, to localize themselves in a new coordinate frame, and a distributed control law to converge to the desired spatial density distribution. We start by studying self-organization in one-dimension, which is then followed by the two-dimensional case.
- Published
- 2018
80. Optimal Control of Partially Observable Piecewise Deterministic Markov Processes
- Author
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Nicole Bäuerle and Dirk Lange
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,Markov process ,Observable ,02 engineering and technology ,Optimal control ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Control theory ,FOS: Mathematics ,Jump ,Piecewise ,symbols ,Applied mathematics ,Piecewise-deterministic Markov process ,Markov decision process ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper we consider a control problem for a Partially Observable Piecewise Deterministic Markov Process of the following type: After the jump of the process the controller receives a noisy signal about the state and the aim is to control the process continuously in time in such a way that the expected discounted cost of the system is minimized. We solve this optimization problem by reducing it to a discrete-time Markov Decision Process. This includes the derivation of a filter for the unobservable state. Imposing sufficient continuity and compactness assumptions we are able to prove the existence of optimal policies and show that the value function satisfies a fixed point equation. A generic application is given to illustrate the results.
- Published
- 2018
81. Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations
- Author
-
Jinniao Qiu
- Subjects
Stochastic control ,Control and Optimization ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematics::Optimization and Control ,Hamilton–Jacobi–Bellman equation ,01 natural sciences ,Hamilton–Jacobi equation ,010104 statistics & probability ,Stochastic differential equation ,Viscosity ,Nonlinear system ,Mathematics - Analysis of PDEs ,Optimization and Control (math.OC) ,Computer Science::Systems and Control ,Bellman equation ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Viscosity solution ,Mathematics - Optimization and Control ,Mathematics - Probability ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well., Comment: 24 pages
- Published
- 2018
82. Hybrid Systems with Memory: Existence and Well-posedness of Generalized Solutions
- Author
-
Andrew R. Teel and Jun Liu
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Dynamical systems theory ,Applied Mathematics ,Uniform convergence ,Systems and Control (eess.SY) ,02 engineering and technology ,Topology ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Mathematics - Classical Analysis and ODEs ,Robustness (computer science) ,Hybrid system ,Phase space ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,0202 electrical engineering, electronic engineering, information engineering ,Computer Science - Systems and Control ,020201 artificial intelligence & image processing ,Mathematics - Optimization and Control ,Well posedness ,Hybrid data ,Mathematics - Abstract
Hybrid systems with memory refer to dynamical systems exhibiting both hybrid and delay phenomena. While systems of this type are frequently encountered in many physical and engineering systems, particularly in control applications, various issues centered around the robustness of hybrid delay systems have not been adequately dealt with. In this paper, we establish some basic results on a framework that allows to study hybrid systems with memory through generalized concepts of solutions. In particular, we develop the basic existence of generalized solutions using regularity conditions on the hybrid data, which are formulated in a phase space of hybrid trajectories equipped with the graphical convergence topology. In contrast with the uniform convergence topology that has been often used, adopting the graphical convergence topology allows us to establish well-posedness of hybrid systems with memory. We then show that, as a consequence of well-posedness, pre-asymptotic stability of well-posed hybrid systems with memory is robust.
- Published
- 2018
83. Zero-Sum Stochastic Differential Games Without the Isaacs Condition: Random Rules of Priority and Intermediate Hamiltonians
- Author
-
Daniel Hernández-Hernández and Mihai Sîrbu
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Discretization ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Stochastic game ,Mathematics::Optimization and Control ,Zero (complex analysis) ,Representation (systemics) ,02 engineering and technology ,01 natural sciences ,Physics::History of Physics ,Physics::Popular Physics ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Bellman equation ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Probability ,Hamiltonian (control theory) ,Differential (mathematics) ,Mathematics - Abstract
For a zero-sum stochastic game which does not satisfy the Isaacs condition, we provide a value function representation for an Isaacs-type equation whose Hamiltonian lies in between the lower and upper Hamiltonians, as a convex combination of the two. For the general case (i.e. the convex combination is time and state dependent) our representation amounts to a random change of the rules of the game, to allow each player at any moment to see the other player's action or not, according to a coin toss with probabilities of heads and tails given by the convex combination appearing in the PDE. If the combination is state independent, then the rules can be set all in advance, in a deterministic way. This means that tossing the coin along the game, or tossing it repeatedly right at the beginning leads to the same value. The representations are asymptotic, over time discretizations. Space discretization is possible as well, leading to similar results., Comment: Preliminary version
- Published
- 2018
84. Measure Control of a Semilinear Parabolic Equation with a Nonlocal Time Delay
- Author
-
Eduardo Casas, Fredi Tröltzsch, Mariano Mateos, and Universidad de Cantabria
- Subjects
Parabolic equation ,Control and Optimization ,Applied Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Nonlocal time delay ,Measure control ,Optimal control ,Measure (mathematics) ,Term (time) ,Optimization and Control (math.OC) ,49K20, 35K58, 49M25 ,Kernel (statistics) ,FOS: Mathematics ,Control (linguistics) ,Mathematics - Optimization and Control ,Mathematics - Abstract
We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a non-differentiable term with the measure-norm of the control. Existence, uniqueness and regularity of the solution of the state equation, as well as differentiability properties of the control-to-state operator are obtained. Next, we provide first order optimality conditions for local solutions. Finally, the control space is suitably discretized and we prove convergence of the solutions of the discrete problems to the solutions of the original problem. Several numerical examples are included to illustrate the theoretical results., Comment: 27 pages, 3 figures
- Published
- 2018
85. The Regular Indefinite Linear Quadratic Optimal Control Problem: Stabilizable Case
- Author
-
Angela P. Schoellig, Mireille E. Broucke, and Marijan Vukosavljev
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,Feedback control ,Open problem ,Optimal cost ,02 engineering and technology ,Linear quadratic ,Linear quadratic optimal control ,Algebraic Riccati equation ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,020201 artificial intelligence & image processing ,Mathematics - Optimization and Control ,Mathematics - Abstract
This paper addresses an open problem in the area of linear quadratic optimal control. We consider the regular, infinite-horizon, stability-modulo-a-subspace, indefinite linear quadratic problem under the assumption that the dynamics are stabilizable. Our result generalizes previous works dealing with the same problem in the case of controllable dynamics. We explicitly characterize the unique solution of the algebraic Riccati equation that gives the optimal cost and optimal feedback control, as well as necessary and sufficient conditions for the existence of optimal controls., Comment: 16 pages
- Published
- 2018
86. Rank Deficiency of Kalman Error Covariance Matrices in Linear Time-Varying System With Deterministic Evolution
- Author
-
Karthik S. Gurumoorthy, Alberto Carrassi, Amit Apte, Christopher K. R. T. Jones, Colin Grudzien, Gurumoorthy K.S., Grudzien C., Apte A., Carrassi A., and Jones C.K.R.T.
- Subjects
Lyapunov function ,0209 industrial biotechnology ,Control and Optimization ,010504 meteorology & atmospheric sciences ,Rank (linear algebra) ,Covariance matrix ,Dynamical Systems (math.DS) ,02 engineering and technology ,Lyapunov exponent ,01 natural sciences ,symbols.namesake ,020901 industrial engineering & automation ,Data assimilation ,Control theory ,93E11, 93C05, 93B05, 60G35, 15A03 ,FOS: Mathematics ,Applied mathematics ,Deterministic system (philosophy) ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control ,0105 earth and related environmental sciences ,Mathematics ,Linear dynamic ,Applied Mathematics ,Kalman filter ,Covariance ,Rank ,Optimization and Control (math.OC) ,symbols - Abstract
We prove that for linear, discrete, time-varying, deterministic system (perfect model) with noisy outputs, the Riccati transformation in the Kalman filter asymptotically bounds the rank of the forecast and the analysis error covariance matrices to be less than or equal to the number of non-negative Lyapunov exponents of the system. Further, the support of these error covariance matrices is shown to be confined to the space spanned by the unstable-neutral backward Lyapunov vectors, providing the theoretical justification for the methodology of the algorithms that perform assimilation only in the unstable-neutral subspace. The equivalent property of the autonomous system is investigated as a special case.
- Published
- 2017
87. Model-Independent Bounds for Asian Options: A Dynamic Programming Approach
- Author
-
Alexander M. G. Cox and Sigrid Källblad
- Subjects
Computer Science::Computer Science and Game Theory ,050208 finance ,Control and Optimization ,Applied Mathematics ,Probability (math.PR) ,05 social sciences ,91G20 (Primary), 93E20, 49L20, 60G48 (Secondary) ,01 natural sciences ,FOS: Economics and business ,Dynamic programming ,010104 statistics & probability ,Optimization and Control (math.OC) ,0502 economics and business ,FOS: Mathematics ,Econometrics ,Asian option ,Pricing of Securities (q-fin.PR) ,0101 mathematics ,Quantitative Finance - Pricing of Securities ,Mathematics - Optimization and Control ,Mathematics - Probability ,Mathematics - Abstract
We consider the problem of finding model-independent bounds on the price of an Asian option, when the call prices at the maturity date of the option are known. Our methods differ from most approaches to model-independent pricing in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the asset at the maturity date. By formulating the problem in this manner, we are able to determine the model-independent price through a PDE formulation. Notably, this approach does not require specific constraints on the payoff function (e.g. convexity), and would appear to generalise to many related problems., Comment: Updated version with some technical changes, and a new appendix containing a proof of the DPP
- Published
- 2017
88. Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics
- Author
-
Huyên Pham, Xiaoli Wei, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Centre de Recherche en Économie et Statistique (CREST), Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] (ENSAI)-École polytechnique (X)-École Nationale de la Statistique et de l'Administration Économique (ENSAE Paris)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE05-0024,CAESARS,Contrôle et simulation des systèmes électriques, interaction et robustesse(2015)
- Subjects
0209 industrial biotechnology ,viscosity solutions ,Control and Optimization ,02 engineering and technology ,01 natural sciences ,Bellman equation ,020901 industrial engineering & automation ,FOS: Mathematics ,Stochastic McKean-Vlasov SDEs ,Applied mathematics ,Uniqueness ,Differentiable function ,0101 mathematics ,Mathematics - Optimization and Control ,Probability measure ,Mathematics ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,State (functional analysis) ,16. Peace & justice ,Optimal control ,dynamic programming principle ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Dynamic programming ,Wasserstein space ,Flow (mathematics) ,Optimization and Control (math.OC) ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Mathematics - Probability - Abstract
We study the optimal control of general stochastic McKean-Vlasov equation. Such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to P.L. Lions [32], and It{\^o}'s formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise., Comment: 33 pages, to appear in SIAM Journal on Control and Optimization
- Published
- 2017
89. A Probabilistic Representation for the Value of Zero-Sum Differential Games with Incomplete Information on Both Sides
- Author
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Catherine Rainer, Fabien Gensbittel, Groupe de recherche en économie mathématique et quantitative (GREMAQ), Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Université Toulouse 1 Capitole (UT1), Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National de la Recherche Agronomique (INRA)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées
- Subjects
Computer Science::Computer Science and Game Theory ,0209 industrial biotechnology ,Control and Optimization ,zerosum games ,MathematicsofComputing_NUMERICALANALYSIS ,Combinatorial game theory ,02 engineering and technology ,01 natural sciences ,Convexity ,010104 statistics & probability ,020901 industrial engineering & automation ,Example of a game without a value ,Complete information ,Differential game ,FOS: Mathematics ,incomplete information ,0101 mathematics ,Mathematics - Optimization and Control ,B- ECONOMIE ET FINANCE ,Mathematics ,Discrete mathematics ,Applied Mathematics ,Probabilistic logic ,Optimization and Control (math.OC) ,stochastic differential game ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Hamilton-Jacobi equations ,Martingale (probability theory) - Abstract
International audience; We prove that for a class of zero-sum differential games with incomplete information on both sides, the value admits a probabilistic representation as the value of a zero-sum stochastic differential game with complete information, where both players control a continuous martingale. A similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side (see Cardaliaguet-Rainer [8]), and our result is a continuous-time analog of the so called splitting-game introduced in Laraki [20] and Sorin [27] in order to analyze discrete-time models. It was proved by Cardaliaguet [4, 5] that the value of the games we consider is the unique solution of some Hamilton-Jacobi equation with convexity constraints. Our result provides therefore a new probabilistic representation for solutions of Hamilton-Jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility.
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- 2017
90. Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems
- Author
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Lassi Paunonen, Tampere University, Mathematics, and Research group: Computer Science and Applied Logics
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Controller design ,0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Linear system ,Internal model ,Error feedback ,02 engineering and technology ,113 Computer and information sciences ,93C05, 93B52 (47D06) ,01 natural sciences ,Transfer function ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Robustness (computer science) ,Control theory ,111 Mathematics ,FOS: Mathematics ,Heat equation ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
We construct two error feedback controllers for robust output tracking and disturbance rejection of a regular linear system with nonsmooth reference and disturbance signals. We show that for sufficiently smooth signals the output converges to the reference at a rate that depends on the behaviour of the transfer function of the plant on the imaginary axis. In addition, we construct a controller that can be designed to achieve robustness with respect to a given class of uncertainties in the system, and present a novel controller structure for output tracking and disturbance rejection without the robustness requirement. We also generalize the internal model principle for regular linear systems with boundary disturbance and for controllers with unbounded input and output operators. The construction of controllers is illustrated with an example where we consider output tracking of a nonsmooth periodic reference signal for a two-dimensional heat equation with boundary control and observation, and with periodic disturbances on the boundary., Comment: 30 pages, 3 figures, to appear in SIAM Journal on Control & Optimization
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- 2017
91. Time-Inconsistent Recursive Stochastic Optimal Control Problems
- Author
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Qingmeng Wei, Zhiyong Yu, and Jiongmin Yong
- Subjects
Stochastic control ,0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Mathematics::Optimization and Control ,Zero (complex analysis) ,Hamilton–Jacobi–Bellman equation ,02 engineering and technology ,Interval (mathematics) ,01 natural sciences ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Bellman equation ,FOS: Mathematics ,Partition (number theory) ,Applied mathematics ,Dynamic inconsistency ,0101 mathematics ,Differential (infinitesimal) ,Mathematics - Optimization and Control ,Mathematics - Abstract
A time-inconsistent stochastic optimal control problem with a recursive cost functional is studied. Equilibrium strategy is introduced, which is time-consistent and locally approximately optimal. By means of multiperson hierarchical differential games associated with partitions of the time interval, a family of approximate equilibrium strategy is constructed, and by sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB) equation is derived through which the equilibrium value function can be identified and the equilibrium strategy can be obtained. Moreover, a well-posedness result of the equilibrium HJB equation is established under certain conditions, and a verification theorem is proved. Finally, an illustrative example is presented, and some comparisons of different definitions of equilibrium strategy are put in order.
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- 2017
92. Pointwise Second-Order Necessary Conditions for Stochastic Optimal Controls, Part II: The General Case
- Author
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Haisen Zhang and Xu Zhang
- Subjects
Stochastic control ,Pointwise ,0209 industrial biotechnology ,Control and Optimization ,Series (mathematics) ,Applied Mathematics ,010102 general mathematics ,Control variable ,02 engineering and technology ,01 natural sciences ,Constraint (information theory) ,020901 industrial engineering & automation ,Maximum principle ,Optimization and Control (math.OC) ,Adjoint equation ,Control system ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
This paper is the second part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, we consider the general cases, i.e., the control region is allowed to be nonconvex, and the control variable enters into both the drift and the diffusion terms of the control systems. By introducing four variational equations and four adjoint equations, we obtain the desired necessary conditions for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle., Comment: 49 pages
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- 2017
93. Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience
- Author
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Ulrich Horst and Paulwin Graewe
- Subjects
0209 industrial biotechnology ,Mathematical optimization ,Control and Optimization ,Mathematics::Optimization and Control ,02 engineering and technology ,FOS: Economics and business ,Terminal value ,Stochastic differential equation ,020901 industrial engineering & automation ,Bellman equation ,0502 economics and business ,FOS: Mathematics ,Trading strategy ,Uniqueness ,Mathematics - Optimization and Control ,Mathematics ,Quantitative Finance - Trading and Market Microstructure ,050208 finance ,Applied Mathematics ,Probability (math.PR) ,05 social sciences ,Absolute continuity ,Trading and Market Microstructure (q-fin.TR) ,Terminal (electronics) ,Optimization and Control (math.OC) ,Asymptotic expansion ,Mathematics - Probability - Abstract
We study an optimal execution problem in illiquid markets with both instantaneous and persistent price impact and stochastic resilience when only absolutely continuous trading strategies are admissible. In our model the value function can be described by a three-dimensional system of backward stochastic differential equations (BSDE) with a singular terminal condition in one component. We prove existence and uniqueness of a solution to the BSDE system and characterize both the value function and the optimal strategy in terms of the unique solution to the BSDE system. Our existence proof is based on an asymptotic expansion of the BSDE system at the terminal time that allows us to express the system in terms of a equivalent system with finite terminal value but singular driver.
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- 2017
94. On Near Optimal Control of Systems with Slow Observables
- Author
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Vladimir Gaitsgory and Sergey Rossomakhine
- Subjects
0209 industrial biotechnology ,State variable ,Control and Optimization ,Basis (linear algebra) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Value (computer science) ,Observable ,02 engineering and technology ,Optimal control ,01 natural sciences ,020901 industrial engineering & automation ,Asymptotically optimal algorithm ,Optimization and Control (math.OC) ,34E15, 34C29, 34A60, 93C70 ,FOS: Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
The paper deals with a problem of control of a system characterized by the fact that the influence of controls on the dynamics of certain functions of state variables (called observables) is relatively weak and the rates of change of these observables are much slower than the rates of change of the state variables themselves. The contributions of the paper are twofold. Firstly, the averaged system whose solutions approximate the trajectories of the slow observables is introduced, and it is shown that the optimal value of the problem of optimal control with time discounting criterion considered on the solutions of the system with slow observables (this problem is referred to as perturbed) converges to the optimal value of the corresponding problem of optimal control of the averaged system. Secondly, a way how an asymptotically optimal control of the perturbed problem can be constructed on the basis of an optimal solution of the averaged problem is indicated, sufficient and necessary optimality conditions for the averaged problem are stated, and a way how a near optimal solution of the latter can be constructed numerically is outlined (the construction being illustrated with an example)., 40 pages, 4 figures
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- 2017
95. On Viscosity Solution of HJB Equations with State Constraints and Reflection Control
- Author
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Lin Wang, Subhamay Saha, Hitoshi Ishii, and Anup Biswas
- Subjects
93E20, 60H30, 35J25, 49L25 ,Control and Optimization ,Euclidean space ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Mathematics::Optimization and Control ,Hamilton–Jacobi–Bellman equation ,Type (model theory) ,01 natural sciences ,Orthant ,010104 statistics & probability ,Nonlinear system ,Mathematics - Analysis of PDEs ,Optimization and Control (math.OC) ,Bellman equation ,FOS: Mathematics ,Boundary value problem ,0101 mathematics ,Viscosity solution ,Mathematics - Optimization and Control ,Mathematics - Probability ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Motivated by a control problem of a certain queueing network we consider a control problem where the dynamics is constrained in the nonnegative orthant $\mathbb{R}_+$ of the $d$-dimensional Euclidean space and controlled by the reflections at the faces/boundaries. We define a discounted value function associated to this problem and show that the value function is a viscosity solution to a certain HJB equation in $\mathbb{R}_+$ with nonlinear Neumann type boundary condition. Under certain conditions, we also characterize this value function as the unique solution to this HJB equation., Comment: 32 pages, To appear in SICON
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- 2017
96. Solving Two-Point Boundary Value Problems for a Wave Equation via the Principle of Stationary Action and Optimal Control
- Author
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Peter M. Dower and William M. McEneaney
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Stochastic game ,02 engineering and technology ,Wave equation ,Optimal control ,01 natural sciences ,Principle of least action ,020901 industrial engineering & automation ,Quadratic equation ,Optimization and Control (math.OC) ,FOS: Mathematics ,Fundamental solution ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Optimization and Control ,Hamiltonian (control theory) ,Mathematics - Abstract
A new approach to solving two-point boundary value problems for a wave equation is developed. This new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control. In particular, an infinite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated Hamiltonian and choice of terminal payoff respectively. In order to solve this optimal control problem for any such terminal payoff, and hence solve any two-point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff, a fundamental solution to the optimal control problem is constructed. Specifically, the optimal control problem corresponding to any given terminal payoff can be solved via a max-plus convolution of this fundamental solution with the specified terminal payoff. Crucially, the fundamental solution is shown to be a quadratic functional that is defined with respect to the unique solution of a set of operator differential equations, and computable using spectral methods. An example is presented in which this fundamental solution is computed and applied to solve a two-point boundary value problem for the wave equation of interest., 30 pages, 5 figures, journal version submitted January 2015
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- 2017
97. ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances
- Author
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Iasson Karafyllis and Miroslav Krstic
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Boundary (topology) ,Systems and Control (eess.SY) ,02 engineering and technology ,01 natural sciences ,Stability (probability) ,Mathematics - Analysis of PDEs ,020901 industrial engineering & automation ,Maximum principle ,Exponential stability ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Partial differential equation ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,Eigenfunction ,Optimization and Control (math.OC) ,Computer Science - Systems and Control ,Heat equation ,Analysis of PDEs (math.AP) - Abstract
For 1-D parabolic PDEs with disturbances at both boundaries and distributed disturbances we provide ISS estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an eigenfunction expansion of the solution, and (ii) a finite-difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation. Finally, the obtained results are applied to quasi-static thermoelasticity models that involve nonlocal boundary conditions. Small-gain conditions that guarantee the global exponential stability of the zero solution for such models are derived., 32 pages, submitted to SIAM Journal on Control and Optimization for possible publication
- Published
- 2017
98. An Extension of the Projected Gradient Method to a Banach Space Setting with Application in Structural Topology Optimization
- Author
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Luise Blank and Christoph Rupprecht
- Subjects
Pointwise ,0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,49M05, 49M15, 65K, 74P05, 90C ,Banach space ,02 engineering and technology ,01 natural sciences ,Pattern search ,Random search ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Metric (mathematics) ,FOS: Mathematics ,Random optimization ,Differentiable function ,0101 mathematics ,Mathematics - Optimization and Control ,Gradient method ,Mathematics - Abstract
For the minimization of a nonlinear cost functional $j$ under convex constraints the relaxed projected gradient process $\varphi_{k+1} = \varphi_{k} + \alpha_k(P_H(\varphi_{k}-\lambda_k \nabla_H j(\varphi_{k}))-\varphi_{k})$ is a well known method. The analysis is classically performed in a Hilbert space $H$. We generalize this method to functionals $j$ which are differentiable in a Banach space. Thus it is possible to perform e.g. an $L^2$ gradient method if $j$ is only differentiable in $L^\infty$. We show global convergence using Armijo backtracking in $\alpha_k$ and allow the inner product and the scaling $\lambda_k$ to change in every iteration. As application we present a structural topology optimization problem based on a phase field model, where the reduced cost functional $j$ is differentiable in $H^1\cap L^\infty$. The presented numerical results using the $H^1$ inner product and a pointwise chosen metric including second order information show the expected mesh independency in the iteration numbers. The latter yields an additional, drastic decrease in iteration numbers as well as in computation time. Moreover we present numerical results using a BFGS update of the $H^1$ inner product for further optimization problems based on phase field models.
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- 2017
99. Mean-Field SDE Driven by a Fractional Brownian Motion and Related Stochastic Control Problem
- Author
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Shuai Jing and Rainer Buckdahn
- Subjects
Stochastic control ,Hurst exponent ,Control and Optimization ,Fractional Brownian motion ,Applied Mathematics ,Probability (math.PR) ,93E20, 60H05, 60H35 ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,010104 statistics & probability ,Stochastic differential equation ,Maximum principle ,Mean field theory ,Optimization and Control (math.OC) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Probability ,Brownian motion ,Mathematics - Abstract
We study a class of mean-field stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H\in(1/2,1)$ and a related stochastic control problem. We derive a Pontryagin type maximum principle and the associated adjoint mean-field backward stochastic differential equation driven by a classical Brownian motion, and we prove that under certain assumptions, which generalise the classical ones, the necessary condition for the optimality of an admissible control is also sufficient., Comment: 34 pages
- Published
- 2017
100. Kernel Methods for the Approximation of Nonlinear Systems
- Author
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Jake Bouvrie and Boumediene Hamzi
- Subjects
FOS: Computer and information sciences ,0209 industrial biotechnology ,Control and Optimization ,Machine Learning (stat.ML) ,Systems and Control (eess.SY) ,Dynamical Systems (math.DS) ,02 engineering and technology ,Nonlinear control ,Dynamical system ,01 natural sciences ,Reduction (complexity) ,020901 industrial engineering & automation ,Statistics - Machine Learning ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Mathematics - Dynamical Systems ,0101 mathematics ,Representation (mathematics) ,Mathematics - Optimization and Control ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Nonlinear system ,Kernel method ,Optimization and Control (math.OC) ,Computer Science - Systems and Control ,Embedding ,Reproducing kernel Hilbert space - Abstract
We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Empirical simulations illustrating the approach are also provided., Rewritten to improve readability. arXiv admin note: text overlap with arXiv:1011.2952
- Published
- 2017
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