84 results on '"Mathematics - Optimization and Control"'
Search Results
2. Local minimizers for variational obstacle avoidance on Riemannian manifolds
- Author
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Goodman, Jacob R.
- Subjects
Mathematics - Differential Geometry ,Control and Optimization ,Differential Geometry (math.DG) ,Optimization and Control (math.OC) ,Mechanics of Materials ,Applied Mathematics ,FOS: Mathematics ,Geometry and Topology ,Mathematics - Optimization and Control - Abstract
This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories and subsequently classified, with local uniqueness results obtained in both cases., 11 pages
- Published
- 2023
3. Null controllability of a nonlinear age, space and two-sex structured population dynamics model
- Author
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Yacouba Simporé and Oumar Traore
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education.field_of_study ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Null (mathematics) ,Population ,Zero (complex analysis) ,Fixed-point theorem ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Controllability ,Mathematics - Analysis of PDEs ,Schauder fixed point theorem ,Optimization and Control (math.OC) ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Observability ,0101 mathematics ,education ,Mathematics - Optimization and Control ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we study the null controllability of a nonlinear age, space and two-sex structured population dynamics model. This model is such that the nonlinearity and the couplage are at birth level. We consider a population with males and females and we are dealing with two cases of null controllability problems. The first problem is related to the total extinction, which means that, we estimate a time \begin{document}$ T $\end{document} to bring the male and female subpopulation density to zero. The second case concerns null controllability of male or female subpopulation. Since the absence of males or females in the population stops births; so, if we have the total extinction of the females at time \begin{document}$ T, $\end{document} and if \begin{document}$ A $\end{document} is the life span of the individuals, at time \begin{document}$ T+A $\end{document} one will get certainly the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after the Schauder's fixed point theorem.
- Published
- 2023
4. Causal state feedback representation for linear quadratic optimal control problems of singular Volterra integral equations
- Author
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Han, Shuo, Lin, Ping, and Yong, Jiongmin
- Subjects
Control and Optimization ,Optimization and Control (math.OC) ,Applied Mathematics ,FOS: Mathematics ,Mathematics - Optimization and Control - Abstract
This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Under proper convexity conditions, optimal control uniquely exists, and it could be characterized via Frechet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations have a shortcoming that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. The main purpose of this paper is to obtain a causal state feedback representation of the optimal control., Comment: 25 pages
- Published
- 2022
5. Pointwise control of the linearized Gear-Grimshaw system
- Author
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Ademir F. Pazoto, Vilmos Komornik, and Roberto de A. Capistrano-Filho
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Pointwise ,Control and Optimization ,Dirac measure ,Applied Mathematics ,Mathematical analysis ,93B07, 35Q53 (Primary), 93B52, 93B05 (Secondary) ,Control function ,Controllability ,symbols.namesake ,Mathematics - Analysis of PDEs ,Unit circle ,Optimization and Control (math.OC) ,Modeling and Simulation ,FOS: Mathematics ,symbols ,Uniqueness ,Observability ,Mathematics - Optimization and Control ,Fourier series ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability., 27 pages
- Published
- 2020
6. Projection methods for solving split equilibrium problems
- Author
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Dang Van Hieu
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Approximations of π ,Strategy and Management ,Proximal point method ,0211 other engineering and technologies ,02 engineering and technology ,Projection (linear algebra) ,symbols.namesake ,020901 industrial engineering & automation ,Component (UML) ,Convergence (routing) ,FOS: Mathematics ,Projection method ,Applied mathematics ,Business and International Management ,Electrical and Electronic Engineering ,Mathematics - Optimization and Control ,Mathematics ,021103 operations research ,Applied Mathematics ,Hilbert space ,Inverse problem ,Atomic and Molecular Physics, and Optics ,65K10, 65K15, 90C33 ,Optimization and Control (math.OC) ,symbols - Abstract
The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others., Comment: 19 pages, 8 figures (Accepted for publication on January 24, 2019)
- Published
- 2020
7. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix
- Author
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Mohamed Fadili, El Mustapha Ait Ben Hassi, and Lahcen Maniar
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Control and Optimization ,Applied Mathematics ,Degenerate energy levels ,Null (mathematics) ,Mathematical analysis ,Diagonalizable matrix ,Parabolic partial differential equation ,Controllability ,Optimization and Control (math.OC) ,Rank condition ,FOS: Mathematics ,Algebraic number ,Constant (mathematics) ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called \textit{Kalman's rank} condition.
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- 2020
8. Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems
- Author
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Sérgio S. Rodrigues
- Subjects
Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Oblique projection ,Orthogonal complement ,01 natural sciences ,Parabolic partial differential equation ,93D15, 93C10, 93B52, 93C20 ,010101 applied mathematics ,Arbitrarily large ,Nonlinear system ,Optimization and Control (math.OC) ,Modeling and Simulation ,FOS: Mathematics ,Initial value problem ,Degree of a polynomial ,0101 mathematics ,Mathematics - Optimization and Control ,Operator norm ,Mathematics - Abstract
It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depends on the operator norm of the oblique projection, on the polynomial degree of the nonlinearity, on the norm of the initial condition, and on the total volume covered by the actuators. The range of the feedback controller coincides with the range of the oblique projection, which is the linear span of the actuators. The oblique projection is performed along the orthogonal complement of a subspace spanned by a suitable finite number of eigenfunctions of the diffusion operator. For rectangular domains, it is possible to explicitly construct/place the actuators so that the stability of the closed-loop system is guaranteed. Simulations are presented, which show the semiglobal stabilizing performance of the nonlinear feedback.
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- 2020
9. Convergence of simultaneous distributed-boundary parabolic optimal control problems
- Author
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Carolina M. Bollo, Claudia M. Gariboldi, and Domingo A. Tarzia
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PARABOLIC VARIATIONAL EQUALITIES ,Control and Optimization ,MIXED BOUNDARY CON-DITIONS ,media_common.quotation_subject ,Boundary (topology) ,01 natural sciences ,Omega ,purl.org/becyt/ford/1 [https] ,Combinatorics ,OPTIMAL CONTROL ,EXISTENCE AND UNIQUENESS ,FOS: Mathematics ,Boundary value problem ,Uniqueness ,0101 mathematics ,Mathematics - Optimization and Control ,media_common ,Physics ,Computer Science::Information Retrieval ,Applied Mathematics ,010102 general mathematics ,purl.org/becyt/ford/1.1 [https] ,State (functional analysis) ,Optimal control ,Infinity ,010101 applied mathematics ,Optimization and Control (math.OC) ,OPTIMALITY CONDITIONS ,Modeling and Simulation ,Domain (ring theory) ,49J20, 35K05, 49K20 - Abstract
We consider a heat conduction problem $S$ with mixed boundary conditions in a n-dimensional domain $\Omega$ with regular boundary $\Gamma$ and a family of problems $S_{\alpha}$, where the parameter $\alpha>0$ is the heat transfer coefficient on the portion of the boundary $\Gamma_{1}$. In relation to these state systems, we formulate simultaneous \emph{distributed-boundary} optimal control problems on the internal energy $g$ and the heat flux $q$ on the complementary portion of the boundary $\Gamma_{2}$. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient $\alpha$ goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author., Comment: This paper has been accepted for publication in Evolution Equations and Control Theory
- Published
- 2020
10. Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions
- Author
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Do Duc Thuan, Stefanie Sonner, Bao Quoc Tang, and Klemens Fellner
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Physics ,Steady state ,Applied Mathematics ,010102 general mathematics ,Multiplicative function ,Mathematical analysis ,Boundary (topology) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Range (mathematics) ,Mathematics - Analysis of PDEs ,Exponential stability ,Optimization and Control (math.OC) ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Boundary value problem ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Noise (radio) ,Analysis of PDEs (math.AP) - Abstract
The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative It\^o noise is studied. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the trivial steady state. This differs from previous works on the stabilisation by noise of parabolic PDEs, where the noise acts inside the domain and stabilisation typically occurs for an infinite range of noise intensities. To the best of our knowledge, this is the first result on the stabilisation of PDEs by boundary noise., Comment: to appear in Discrete and Continuous Dynamical Systems - Series B
- Published
- 2019
11. Strong stabilization of (almost) impedance passive systems by static output feedback
- Author
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George Weiss and Ruth F. Curtain
- Subjects
Output feedback ,0209 industrial biotechnology ,Passive systems ,Control and Optimization ,colocated ,CONTROLLABILITY ,Boundary (topology) ,BEAM ,02 engineering and technology ,positive transfer function ,01 natural sciences ,Bounded operator ,Combinatorics ,STABILIZABILITY ,symbols.namesake ,020901 industrial engineering & automation ,contraction semigroup ,scattering passive system ,FOS: Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,PART II ,EXPONENTIAL STABILIZATION ,Physics ,weak stability ,DIMENSIONAL LINEAR-SYSTEMS ,output feedback ,STABILITY ,well-posed linear system ,Semigroup ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,impedance passive system ,OPERATOR ,THIN AIR ,System node ,strong stability ,UNBOUNDED CONTROL ,Optimization and Control (math.OC) ,symbols ,Contraction semigroup - Abstract
The plant to be stabilized is a system node $\Sigma$ with generating triple $(A,B,C)$ and transfer function $\bf G$, where $A$ generates a contraction semigroup on the Hilbert space $X$. The control and observation operators $B$ and $C$ may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator $E$ such that, if we replace ${\bf G}(s)$ by ${\bf G}(s)+E$, the new system $\Sigma_E$ becomes impedance passive. An easier case is when $\bf G$ is already impedance passive and a special case is when \mm $\Sigma$ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback $u=-\kappa y+v$, where $u$ is the input of the plant and $\kappa>0$, stabilizes $\Sigma$, strongly or even exponentially. Here, $y$ is the output of \m $\Sigma$ and $v$ is the new input. Our main result is that if for some $E\in{\mathcal L}(U)$, $\Sigma_E$ is impedance passive, and \m $\Sigma$ is approximately observable or approximately controllable in infinite time, then for sufficiently small $\kappa$ the closed-loop system is weakly stable. If, moreover, $\sigma(A)\cap i{\mathbb R}$ is countable, then the closed-loop semigroup and its dual are both strongly stable., Comment: 29 pages
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- 2019
12. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I
- Author
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Tianxiao Wang
- Subjects
0209 industrial biotechnology ,Class (set theory) ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Markov process ,Contrast (statistics) ,02 engineering and technology ,Optimal control ,01 natural sciences ,Stochastic differential equation ,symbols.namesake ,020901 industrial engineering & automation ,Mean field theory ,Optimization and Control (math.OC) ,FOS: Mathematics ,symbols ,Applied mathematics ,Dynamic inconsistency ,Uniqueness ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper, a class of time inconsistent linear quadratic optimal control problems for mean-field stochastic differential equations (SDEs) are considered under Markovian framework. Open-loop equilibrium controls and their particular closed-loop representations are introduced and characterized via variational ideas. Several interesting features are revealed and a system of coupled Riccati equations is derived. In contrast with the analogue optimal control problems of SDEs, the mean-field terms in state equation, which is another reason of time inconsistency, prompts us to define the above two notions in new manners. An interesting result, which is almost trivial in the counterpart problems of SDEs, is given and plays significant role in the previous characterizations. As application, the uniqueness of open-loop equilibrium controls is discussed.
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- 2019
13. On construction of upper and lower bounds for the HOMO-LUMO spectral gap
- Author
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Daniel Sevcovic and Sona Pavlikova
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Quantitative Biology::Biomolecules ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,Maximization ,Upper and lower bounds ,law.invention ,Combinatorics ,Invertible matrix ,Optimization and Control (math.OC) ,law ,05C50, 15A09, 15B36, 90C11, 90C22 ,FOS: Mathematics ,Physics::Atomic and Molecular Clusters ,Bipartite graph ,Schur complement ,Spectral gap ,Adjacency matrix ,Mathematics - Optimization and Control ,Eigenvalues and eigenvectors ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.
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- 2019
14. A resilient convex combination for consensus-based distributed algorithms
- Author
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Shaoshuai Mou, Shreyas Sundaram, and Xuan Wang
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Discrete mathematics ,0209 industrial biotechnology ,021103 operations research ,Control and Optimization ,Algebra and Number Theory ,Computational complexity theory ,Intersection (set theory) ,Computer science ,Applied Mathematics ,Multi-agent system ,0211 other engineering and technologies ,Regular polygon ,Systems and Control (eess.SY) ,02 engineering and technology ,Set (abstract data type) ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Distributed algorithm ,Bounded function ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,Computer Science - Systems and Control ,Convex combination ,Mathematics - Optimization and Control - Abstract
Consider a set of vectors in $\mathbb{R}^n$, partitioned into two classes: normal vectors and malicious vectors. The number of malicious vectors is bounded but their identities are unknown. The paper provides a way for achieving a resilient convex combination, which is a convex combination of only normal vectors. Compared with existing approaches based on Tverberg points, the proposed method based on the intersection of convex hulls has lower computational complexity. Simulations suggest that the proposed method can be applied to resilience for consensus-based distributed algorithms against Byzantine attacks., Comment: Version 2
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- 2019
15. A discrete districting plan
- Author
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Alberto Saracco and Giorgio Maria Saracco
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Statistics and Probability ,Physics - Physics and Society ,Computer science ,Applied Mathematics ,Gerrymandering ,91D20, 49Q10, 52C99 ,General Engineering ,FOS: Physical sciences ,Discrete geometry ,Physics and Society (physics.soc-ph) ,Plan (drawing) ,01 natural sciences ,Outcome (game theory) ,Computer Science Applications ,Continuous analysis ,010101 applied mathematics ,Optimization and Control (math.OC) ,Order (exchange) ,FOS: Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematical economics - Abstract
The outcome of elections is strongly dependent on the districting choices, making thus possible (and frequent) the gerrymandering phenomenon, i.e.\ politicians suitably changing the shape of electoral districts in order to win the forthcoming elections. While so far the problem has been treated using continuous analysis tools, it has been recently pointed out that a more reality-adherent model would use the discrete geometry of graphs or networks. Here we propose a parameter-dependent discrete model for choosing an "optimal" districting plan. We analyze several properties of the model and lay foundations for further analysis on the subject.
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- 2019
16. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance
- Author
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Shuaiqi Zhang, Yi Zhuang, and Jie Xiong
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TheoryofComputation_MISCELLANEOUS ,Computer Science::Computer Science and Game Theory ,0209 industrial biotechnology ,Class (set theory) ,Control and Optimization ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Stochastic differential equation ,020901 industrial engineering & automation ,Maximum principle ,Differential game ,FOS: Mathematics ,Filtration (mathematics) ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Equilibrium point ,Applied Mathematics ,010102 general mathematics ,TheoryofComputation_GENERAL ,Zero-sum game ,Optimization and Control (math.OC) ,Nash equilibrium ,symbols - Abstract
In this article, we concern a kind of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding open-loop Nash equilibrium control is required to adapted to the filtration that the observation process generated. To find this open-loop Nash equilibrium point, we prove the maximum principle as a necessary condition of the existence of this point, and give a verification theorem as a sufficient condition to verify it is the real open-loop Nash equilibrium point. Combined this with reality, a financial investment problem is raised. We can obtain the explicit observable investment strategy by using stochastic filtering theory and the results above., Comment: 21 pages
- Published
- 2019
17. From mean field games to the best reply strategy in a stochastic framework
- Author
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Matt Barker
- Subjects
Statistics and Probability ,Mean field limit ,Applied Mathematics ,Linear quadratic ,Best reply ,Mean field game ,symbols.namesake ,Model predictive control ,Mean field theory ,Optimization and Control (math.OC) ,Nash equilibrium ,Modeling and Simulation ,FOS: Mathematics ,symbols ,Applied mathematics ,Differential (infinitesimal) ,Mathematics - Optimization and Control ,Mathematics - Abstract
This paper builds on the work of Degond, Herty and Liu in [ 16 ] by considering \begin{document}$ N $\end{document} -player stochastic differential games. The control corresponding to a Nash equilibrium of such a game is approximated through model predictive control (MPC) techniques. In the case of a linear quadratic running-cost, considered here, the MPC method is shown to approximate the solution to the control problem by the best reply strategy (BRS) for the running cost. We then compare the MPC approach when taking the mean field limit with the popular mean field game (MFG) strategy. We find that our MPC approach reduces the two coupled PDEs to a single PDE, greatly increasing the simplicity and tractability of the original problem. We give two examples of applications of this approach to previous literature and conclude with future perspectives for this research.
- Published
- 2019
18. Construction of the minimum time function for linear systems via higher-order set-valued methods
- Author
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Robert Baier and Thuy T. T. Le
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Convex analysis ,Class (set theory) ,Control and Optimization ,Discretization ,Applied Mathematics ,Linear system ,49N60, 93B03(49N05, 49M25, 52A27) ,Set (abstract data type) ,Hausdorff distance ,Compact space ,Rate of convergence ,Optimization and Control (math.OC) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., Holder continuity) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Holder continuous or even discontinuous., Comment: arXiv admin note: substantial text overlap with arXiv:1512.08617, arXiv:1512.08630
- Published
- 2019
19. Necessary optimality conditions for average cost minimization problems
- Author
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Piernicola Bettiol and Nathalie T. Khalil
- Subjects
Mathematical optimization ,Computer science ,Applied Mathematics ,Robust optimization ,Function (mathematics) ,Optimal control ,Minimax ,Separable space ,Constraint (information theory) ,Optimization and Control (math.OC) ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Minification ,Mathematics - Optimization and Control ,Average cost ,49J15, 49K35 - Abstract
Control systems involving unknown parameters appear a natural framework for applications in which the model design has to take into account various uncertainties. In these circumstances the performance criterion can be given in terms of an average cost, providing a paradigm which differs from the more traditional minimax or robust optimization criteria. In this paper, we provide necessary optimality conditions for a nonrestrictive class of optimal control problems in which unknown parameters intervene in the dynamics, the cost function and the right end-point constraint. An important feature of our results is that we allow the unknown parameters belonging to a mere complete separable metric space (not necessarily compact)., Comment: 30 pages
- Published
- 2019
20. An SBV relaxation of the Cross-Newell energy for modeling stripe patterns and their defects
- Author
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Venkataramani, Shankar C.
- Subjects
Applied Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Pattern Formation and Solitons (nlin.PS) ,Nonlinear Sciences - Pattern Formation and Solitons ,Mathematics - Analysis of PDEs ,Optimization and Control (math.OC) ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Optimization and Control ,35B36, 49J45 (Primary), 26A45, 90C46 (Secondary) ,Mathematical Physics ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We investigate stripe patterns formation far from threshold using a combination of topological, analytic, and numerical methods. We first give a definition of the mathematical structure of `multi-valued' phase functions that are needed for describing layered structures or stripe patterns containing defects. This definition yields insight into the appropriate `gauge symmetries' of patterns, and leads to the formulation of variational problems, in the class of special functions with bounded variation, to model patterns with defects. We then discuss approaches to discretize and numerically solve these variational problems. These energy minimizing solutions support defects having the same character as seen in experiments., 28 pages, 9 figures
- Published
- 2022
21. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints
- Author
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Zhang, Ling and Sun, Xiaoqi
- Subjects
Quantitative Biology::Neurons and Cognition ,Optimization and Control (math.OC) ,Applied Mathematics ,Computer Science::Neural and Evolutionary Computation ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Discrete Mathematics and Combinatorics ,Dynamical Systems (math.DS) ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control - Abstract
In this paper, a kind of neural network with time-varying delays is proposed to solve the problems of quadratic programming. The delay term of the neural network changes with time t. The number of neurons in the neural network is n + h, so the structure is more concise. The equilibrium point of the neural network is consistent with the optimal solution of the original optimization problem. The existence and uniqueness of the equilibrium point of the neural network are proved. Application inequality technique proved global exponential stability of the network. Some numerical examples are given to show that the proposed neural network model has good performance for solving optimization problems., This is a paper in submission. I want to contribute to the Journal of Nonlinear Analysis-Theory Methods & Applications
- Published
- 2022
22. Well-posedness and stability of non-autonomous semilinear input-output systems
- Author
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Schmid, Jochen
- Subjects
Mathematics - Analysis of PDEs ,Control and Optimization ,Optimization and Control (math.OC) ,Applied Mathematics ,Modeling and Simulation ,Mathematics::Analysis of PDEs ,FOS: Mathematics ,Mathematics - Optimization and Control ,Analysis of PDEs (math.AP) - Abstract
We establish well-posedness results for non-autonomous semilinear input-output systems, the central assumption being the scattering-passivity of the considered semilinear system. We consider both systems with distributed control and observation and systems with boundary control and observation. Applications are given to nonlinearly controlled collocated systems and to nonlinearly controlled port-Hamiltonian systems., Comment: 31 pages, 1 figure. Added an application example (Section 5.1). Also, added proofs (Section 3.1 and 4.1) and stability statements to the well-posedness theorems (Section 3.2 and 4.2). And finally, removed some errors, namely a slight imprecision in the definition of generalized solutions and outputs (see discussion at the very end of Section 2.1) and an error in the assumptions of Corollary 5.1
- Published
- 2022
23. Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives
- Author
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Ammi, Moulay Rchid Sidi, Tahiri, Mostafa, and Torres, Delfim F. M.
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Optimality conditions ,Reaction-diffusion equations ,Optimization and Control (math.OC) ,Epidemic model ,Atangana–Baleanu–Caputo fractional derivatives ,Applied Mathematics ,FOS: Mathematics ,Numerical simulations ,34A08, 49K20, 35K57, 47H10 ,Discrete Mathematics and Combinatorics ,Mathematics - Optimization and Control ,Analysis - Abstract
The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators., This is a preprint of a paper whose final and definite form is with 'Discrete and Continuous Dynamical Systems -- Series S' (DCDS-S), ISSN 1937-1632 (print), ISSN 1937-1179 (online), available at [http://aimsciences.org/journal/1937-1632]
- Published
- 2022
24. Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games
- Author
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Christopher Griffin and James Fan
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Optimization and Control (math.OC) ,Computer Science - Computer Science and Game Theory ,Applied Mathematics ,Modeling and Simulation ,FOS: Mathematics ,91A22, 92A15, 34D45 ,Mathematics - Optimization and Control ,Computer Science and Game Theory (cs.GT) - Abstract
We study an optimal control problem arising from a generalization of rock-paper-scissors in which the number of strategies may be selected from any positive odd number greater than 1 and in which the payoff to the winner is controlled by a control variable $\gamma$. Using the replicator dynamics as the equations of motion, we show that a quasi-linearization of the problem admits a special optimal control form in which explicit dynamics for the controller can be identified. We show that all optimal controls must satisfy a specific second order differential equation parameterized by the number of strategies in the game. We show that as the number of strategies increases, a limiting case admits a closed form for the open-loop optimal control. In performing our analysis we show necessary conditions on an optimal control problem that allow this analytic approach to function., Comment: 23 pages, 6 figures
- Published
- 2022
25. Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations
- Author
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Manil T. Mohan
- Subjects
Physics ,Pure mathematics ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,Isotropy ,Motion (geometry) ,Order (ring theory) ,Optimal control ,Enstrophy ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Optimization and Control (math.OC) ,Modeling and Simulation ,FOS: Mathematics ,Exponent ,Absorption (logic) ,Nabla symbol ,0101 mathematics ,Mathematics - Optimization and Control - Abstract
The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. In this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent $r=1,2$ and $3$. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case $r=3$, we show the second order necessary and sufficient conditions of optimality. We also investigate an another control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is 2D CBF equations, using optimal control techniques., Comment: arXiv admin note: text overlap with arXiv:1909.09308
- Published
- 2022
26. Robust policy selection and harvest risk quantification for natural resources management under model uncertainty
- Author
-
Papayiannis, Georgios I.
- Subjects
FOS: Economics and business ,Statistics and Probability ,Optimization and Control (math.OC) ,Risk Management (q-fin.RM) ,Applied Mathematics ,Modeling and Simulation ,FOS: Mathematics ,Mathematics - Optimization and Control ,Quantitative Finance - Risk Management - Abstract
In this work the problem of optimal harvesting policy selection for natural resources management under model uncertainty is investigated. Under the framework of the neoclassical growth model dynamics, the associated optimal control problem is investigated by introducing the concept of model uncertainty on the initial conditions of the operational procedure. At this stage, the notion of convex risk measures, and in particular the class of Fr\'echet risk measures, is employed in order to quantify the total operational and marginal risk, whereas simultaneously obtaining robust to model uncertainty harvesting strategies., Comment: 15 pages
- Published
- 2022
27. Computation of open-loop inputs for uniformly ensemble controllable systems
- Author
-
Michael Schönlein
- Subjects
Control and Optimization ,Computer science ,Applied Mathematics ,Computation ,Linear system ,Open-loop controller ,Construct (python library) ,Topology ,Integral equation ,Task (project management) ,Controllability ,Optimization and Control (math.OC) ,Control theory ,FOS: Mathematics ,Mathematics - Optimization and Control - Abstract
This paper presents computational methods for families of linear systems depending on a parameter. Such a family is called ensemble controllable if for any family of parameter-dependent target states and any neighborhood of it there is a parameter-independent input steering the origin into the neighborhood. Assuming that a family of systems is ensemble controllable we present methods to construct suitable open-loop input functions. Our approach to solve this infinite-dimensional task is based on a combination of methods from the theory of linear integral equations and finite-dimensional control theory.
- Published
- 2022
28. A backward SDE method for uncertainty quantification in deep learning
- Author
-
Archibald, Richard, Bao, Feng, Cao, Yanzhao, and Zhang, He
- Subjects
FOS: Computer and information sciences ,Computer Science - Machine Learning ,Statistics - Machine Learning ,ComputingMethodologies_SIMULATIONANDMODELING ,Optimization and Control (math.OC) ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Machine Learning (stat.ML) ,Mathematics - Optimization and Control ,Analysis ,Machine Learning (cs.LG) - Abstract
We develop a backward stochastic differential equation based probabilistic machine learning method, which formulates a class of stochastic neural networks as a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced with the gradient computed through a backward stochastic differential equation. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.
- Published
- 2022
29. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control
- Author
-
Mohamed Ouzahra
- Subjects
Semigroup ,Applied Mathematics ,Multiplicative function ,State (functional analysis) ,Interval (mathematics) ,93B05 ,Minimax approximation algorithm ,Term (time) ,Controllability ,Optimization and Control (math.OC) ,Reaction–diffusion system ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Moreover, in the case of a globally supported control, we prove the approximate controllability within any time-interval given in advance which does not depend on the initial and target states. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions., Comment: 20 pages
- Published
- 2022
30. Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications
- Author
-
Leonid Faybusovich and Cunlu Zhou
- Subjects
0209 industrial biotechnology ,Class (set theory) ,Mathematical optimization ,Work (thermodynamics) ,021103 operations research ,Control and Optimization ,Algebra and Number Theory ,Optimization problem ,Computer science ,Path following algorithm ,Applied Mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Type (model theory) ,Quantum key distribution ,90C22, 90C25, 90C30, 90C51, 90C90, 81-08 ,Quantum relative entropy ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,FOS: Mathematics ,Quantum information ,Mathematics - Optimization and Control - Abstract
We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult and important optimization problem in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options., Comment: 26 pages; added a new appendix A; added more details in eq.(58), eq.(60), eq.(75) - eq.(90); other small improvements
- Published
- 2022
31. Existence of complete Lyapunov functions with prescribed orbital derivative
- Author
-
Giesl, Peter, Hafstein, Sigurdur, and Suhr, Stefan
- Subjects
Optimization and Control (math.OC) ,Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Dynamical Systems (math.DS) ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control ,34D05, 93D30, 37C10 - Abstract
Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we can prescribe the (negative) values of the derivative along orbits in any compact set, which is contained in the complement of the chain-recurrent set. Further, the complete Lyapunov function is as smooth as the vector field defining the dynamics. This delivers a theoretical foundation for numerical methods to construct complete Lyapunov functions and renders them accessible for further theoretical analysis and development., 14 pages, comments are welcome
- Published
- 2022
32. Characterization of toric systems via transport costs
- Author
-
Sonja Hohloch
- Subjects
Physics ,Control and Optimization ,Integrable system ,Computer Science::Information Retrieval ,Applied Mathematics ,Torus ,Dynamical Systems (math.DS) ,Mathematics::Geometric Topology ,Hamiltonian system ,Combinatorics ,symbols.namesake ,Optimization and Control (math.OC) ,Mathematics - Symplectic Geometry ,28A75, 37J35, 49K21, 49Q10, 53D99, 70H05, 70H06 ,Mechanics of Materials ,FOS: Mathematics ,symbols ,Symplectic Geometry (math.SG) ,Geometry and Topology ,Mathematics - Dynamical Systems ,Hamiltonian (quantum mechanics) ,Mathematics - Optimization and Control ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$T$ map where $T \in {\mathbb R}^n$ is the period of the acting $n$-torus., 6 pages, 0 figures
- Published
- 2020
33. Optimal actuator location of the minimum norm controls for stochastic heat equations
- Author
-
Donghui Yang and Jie Zhong
- Subjects
Control and Optimization ,Optimization problem ,35K05, 49J20, 93B05, 93B07, 93E20 ,Applied Mathematics ,Null (mathematics) ,Controllability ,symbols.namesake ,Minimum norm ,Zero-sum game ,Optimization and Control (math.OC) ,Nash equilibrium ,FOS: Mathematics ,symbols ,Applied mathematics ,Heat equation ,Actuator ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper, we study the approximate null controllability for the stochastic heat equation with the control acting on a measurable subset, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.
- Published
- 2018
34. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations
- Author
-
Daniel Wachsmuth and Frank Pörner
- Subjects
021103 operations research ,Control and Optimization ,Partial differential equation ,Applied Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Linear partial differential equations ,Optimal control ,01 natural sciences ,Regularization (mathematics) ,Tikhonov regularization ,Optimization and Control (math.OC) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,49M20 (Primary), 49K20, 49N45 (Secondary) ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under suitable conditions. These conditions comprise second-order sufficient optimality conditions as well as regularity conditions on the control, which consists of a source condition and a condition on the active sets. In addition, we show that these conditions are necessary for convergence rates under certain conditions. We also consider sparse optimal control problems and derive regularization error estimates for them. Numerical experiments underline the theoretical findings., Comment: 25 pages, 3 pictures
- Published
- 2018
35. Optimal control of a rate-independent evolution equation via viscous regularization
- Author
-
Daniel Wachsmuth, Gerd Wachsmuth, and Ulisse Stefanelli
- Subjects
01 natural sciences ,Regularization (mathematics) ,Viscosity ,Mathematics - Analysis of PDEs ,Quadratic equation ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Applied mathematics ,49K20, 35K87 ,Necessary optimality conditions ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,Dissipation ,Optimal control ,010101 applied mathematics ,Optimization and Control (math.OC) ,Evolution equation ,Rate-independent system ,Analysis ,Smoothing ,Analysis of PDEs (math.AP) - Abstract
We study the optimal control of a rate-independent system that is driven by a convex, quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.
- Published
- 2017
36. A continuous-time approach to online optimization
- Author
-
Joon Kwon, Panayotis Mertikopoulos, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Performance analysis and optimization of LARge Infrastructures and Systems (POLARIS ), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'Informatique de Grenoble (LIG ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Centre de mathématiques appliquées, Ecole Polytechnique, Centre National de la Recherche Scientifique (CNRS), French National Research Agency (ANR) ANR-13-JS01-0004-01 ANR-13-INFR-004 ANR-16-CE33-0004-01, and ANR-16-CE33-0004,ORACLESS,Stratégies adaptatives d'allocation des ressources dans les réseaux sans fil dynamiques(2016)
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,0209 industrial biotechnology ,Class (set theory) ,convex optimization ,[SDV]Life Sciences [q-bio] ,0211 other engineering and technologies ,Machine Learning (stat.ML) ,02 engineering and technology ,Machine Learning (cs.LG) ,Fictitious play ,020901 industrial engineering & automation ,Online optimization ,Statistics - Machine Learning ,mirror descent ,FOS: Mathematics ,Mathematics - Optimization and Control ,gradient descent ,regret minimization ,Mathematics ,continuous time ,Discrete mathematics ,021103 operations research ,Applied Mathematics ,Regret ,Exponential function ,Term (time) ,Computer Science - Learning ,Optimization and Control (math.OC) ,Modeling and Simulation ,Convex optimization ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Gradient descent - Abstract
Mathematics Subject Classification: Primary: 68Q32, 68T05, 91A26; Secondary: 90C25.; International audience; We consider a family of mirror descent strategies for online optimization in continuous-time and we show that they lead to no regret. From a more traditional, discrete-time viewpoint, this continuous-time approach allows us to derive the no-regret properties of a large class of discrete-time algorithms including as special cases the exponential weights algorithm, online mirror descent, smooth fictitious play and vanishingly smooth fictitious play. In so doing, we obtain a unified view of many classical regret bounds, and we show that they can be decomposed into a term stemming from continuous-time considerations and a term which measures the disparity between discrete and continuous time. This generalizes the continuous-time based analysis of the exponential weights algorithm from [29]. As a result, we obtain a general class of infinite horizon learning strategies that guarantee an regret bound without having to resort to a doubling trick.
- Published
- 2017
37. Performance bounds for the mean-field limit of constrained dynamics
- Author
-
Mattia Zanella and Michael Herty
- Subjects
Mean-field limits ,0209 industrial biotechnology ,Work (thermodynamics) ,Relation (database) ,FOS: Physical sciences ,02 engineering and technology ,01 natural sciences ,NO ,020901 industrial engineering & automation ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Model predictive control ,Limit (mathematics) ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Applied Mathematics ,Horizon ,Consensus models ,Analysis ,Optimal control ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,System dynamics ,010101 applied mathematics ,Optimization and Control (math.OC) ,Ordinary differential equation ,Adaptation and Self-Organizing Systems (nlin.AO) - Abstract
In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.
- Published
- 2017
38. Decompositions and bang-bang properties
- Author
-
Yubiao Zhang and Gengsheng Wang
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,02 engineering and technology ,01 natural sciences ,Combinatorics ,020901 industrial engineering & automation ,Optimization and Control (math.OC) ,Norm (mathematics) ,FOS: Mathematics ,Uniqueness ,0101 mathematics ,Bang bang ,Mathematics - Optimization and Control ,Mathematics - Abstract
We study the bang-bang properties of minimal time and minimal norm control problems (where the target set is the origin of the state space and the controlled system is linear and time-invariant) from a new perspective. More precisely, we study how the bang-bang property of each minimal time (or minimal norm) problem depends on a pair of parameters \begin{document} $(M, y_0)$ \end{document} (or \begin{document} $(T,y_0)$ \end{document} ), where \begin{document} $M>0$ \end{document} is a bound of controls and \begin{document} $y_0$ \end{document} is the initial state (or \begin{document} $T>0$ \end{document} is an ending time and \begin{document} $y_0$ \end{document} is the initial state). The controlled system may have neither the \begin{document} $L^∞$ \end{document} -null controllability nor the backward uniqueness property.
- Published
- 2017
39. Optimal control of a Tuberculosis model with state and control delays
- Author
-
Cristiana J. Silva, Delfim F. M. Torres, and Helmut Maurer
- Subjects
Time delays ,Pediatrics ,medicine.medical_specialty ,Time Factors ,Tuberculosis ,Control (management) ,Control variable ,Tuberculin ,010103 numerical & computational mathematics ,Models, Biological ,01 natural sciences ,010305 fluids & plasmas ,Active tb ,0103 physical sciences ,FOS: Mathematics ,medicine ,Humans ,0101 mathematics ,Quantitative Biology - Populations and Evolution ,Mathematics - Optimization and Control ,34D30, 92D30, 49M05, 93A30 ,business.industry ,Applied Mathematics ,Populations and Evolution (q-bio.PE) ,General Medicine ,medicine.disease ,Optimal control ,3. Good health ,Computational Mathematics ,Optimization and Control (math.OC) ,FOS: Biological sciences ,Modeling and Simulation ,State (computer science) ,General Agricultural and Biological Sciences ,business ,Stability - Abstract
We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis., This is a preprint of a paper whose final and definite form will be published in the journal 'Mathematical Biosciences and Engineering', ISSN 1547-1063 (print), ISSN 1551-0018 (online). Submitted 30/Oct/2015; revised 24/May/2016; accepted for publication 26/Jun/2016
- Published
- 2017
40. Feedback stabilization for a coupled PDE-ODE production system
- Author
-
Stephan Knapp, Vanessa Baumgärtner, and Simone Göttlich
- Subjects
Lyapunov function ,Conservation law ,Control and Optimization ,Applied Mathematics ,Ode ,Bottleneck ,65Mxx, 93D05, 90B30 ,symbols.namesake ,Exponential stability ,Optimization and Control (math.OC) ,Ordinary differential equation ,FOS: Mathematics ,symbols ,Applied mathematics ,Production (computer science) ,Mathematics - Optimization and Control ,Numerical stability ,Mathematics - Abstract
We consider an interlinked production model consisting of conservation laws (PDE) coupled to ordinary differential equations (ODE). Our focus is the analysis of control laws for the coupled system and corresponding stabilization questions of equilibrium dynamics in the presence of disturbances. These investigations are carried out using an appropriate Lyapunov function on the theoretical and numerical level. The discrete $L^2-$stabilization technique allows to derive a mixed feedback law that is able to ensure exponential stability also in bottleneck situations. All results are accompanied by computational examples., Comment: 18 pages
- Published
- 2019
41. Numerical simulations of a rolling ball robot actuated by internal point masses
- Author
-
Vakhtang Putkaradze and Stuart Rogers
- Subjects
Control and Optimization ,Algebra and Number Theory ,Computer science ,Applied Mathematics ,Direct method ,Internal point ,Equations of motion ,Dynamical Systems (math.DS) ,Physics::Classical Physics ,Optimal control ,Continuation method ,Computer Science::Robotics ,37J60, 70E18, 70E60, 49K15, 34A34, 49J15, 65L10 ,Optimization and Control (math.OC) ,Control theory ,Obstacle avoidance ,FOS: Mathematics ,Robot ,Ball (mathematics) ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control - Abstract
The controlled motion of a rolling ball actuated by internal point masses that move along arbitrarily-shaped rails fixed within the ball is considered. The controlled equations of motion are solved numerically using a predictor-corrector continuation method, starting from an initial solution obtained via a direct method, to realize trajectory tracking and obstacle avoidance maneuvers., 51 pages, 59 figures, 13 tables, 2 algorithms. arXiv admin note: substantial text overlap with arXiv:1708.03829
- Published
- 2021
42. Funnel control for boundary control systems
- Author
-
Timo Reis, Felix L. Schwenninger, Marc Puche, Mathematical Systems Theory, and Digital Society Institute
- Subjects
Control and Optimization ,business.product_category ,Hyperbolic PDEs ,MathematicsofComputing_NUMERICALANALYSIS ,Boundary (topology) ,Dissipative operator ,01 natural sciences ,Signal ,Mathematics - Analysis of PDEs ,Control theory ,FOS: Mathematics ,Uniqueness ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Dissipative operators ,Applied Mathematics ,010102 general mathematics ,Boundary control systems ,Parabolic PDEs ,Parabolic partial differential equation ,Primary: 93C20, 93C40, Secondary: 47H06 ,010101 applied mathematics ,Nonlinear feedback ,Nonlinear system ,Optimization and Control (math.OC) ,Funnel control ,Modeling and Simulation ,Control system ,Funnel ,business ,Analysis of PDEs (math.AP) - Abstract
We study a nonlinear, non-autonomous feedback controller applied to boundary control systems. Our aim is to track a given reference signal with prescribed performance. Existence and uniqueness of solutions to the resulting closed-loop system is proved by using nonlinear operator theory. We apply our results to both hyperbolic and parabolic equations., Comment: 26 pages, thoroughly revised version. The system class has been generalized considerably. Added general example class of parabolic problems
- Published
- 2021
43. The entry and exit game in the electricity markets: A mean-field game approach
- Author
-
Peter Tankov, René Aïd, Roxana Dumitrescu, Laboratoire d'Economie de Dauphine (LEDa), Institut de Recherche pour le Développement (IRD)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [London], King‘s College London, Ecole Nationale de la Statistique et de l'Analyse Economique (ENSAE), and Ecole Nationale de la Statistique et de l'Analyse Economique
- Subjects
Statistics and Probability ,020209 energy ,02 engineering and technology ,Mean-field games ,7. Clean energy ,Microeconomics ,symbols.namesake ,Demand curve ,Order (exchange) ,electricity markets ,0502 economics and business ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Market price ,Economics ,Electricity market ,Optimal stopping ,Mathematics - Optimization and Control ,JEL: C - Mathematical and Quantitative Methods/C.C7 - Game Theory and Bargaining Theory/C.C7.C73 - Stochastic and Dynamic Games • Evolutionary Games • Repeated Games ,050208 finance ,[QFIN]Quantitative Finance [q-fin] ,business.industry ,Applied Mathematics ,05 social sciences ,renewable energy ,Renewable energy ,JEL: Q - Agricultural and Natural Resource Economics • Environmental and Ecological Economics/Q.Q4 - Energy/Q.Q4.Q43 - Energy and the Macroeconomy ,optimal stopping ,Optimization and Control (math.OC) ,Nash equilibrium ,Modeling and Simulation ,symbols ,business ,Aggregate supply - Abstract
We develop a model for the industry dynamics in the electricity market, based on mean-field games of optimal stopping. In our model, there are two types of agents: the renewable producers and the conventional producers. The renewable producers choose the optimal moment to build new renewable plants, and the conventional producers choose the optimal moment to exit the market. The agents interact through the market price, determined by matching the aggregate supply of the two types of producers with an exogenous demand function. Using a relaxed formulation of optimal stopping mean-field games, we prove the existence of a Nash equilibrium and the uniqueness of the equilibrium price process. An empirical example, inspired by the UK electricity market is presented. The example shows that while renewable subsidies clearly lead to higher renewable penetration, this may entail a cost to the consumer in terms of higher peakload prices. In order to avoid rising prices, the renewable subsidies must be combined with mechanisms ensuring that sufficient conventional capacity remains in place to meet the energy demand during peak periods.
- Published
- 2021
44. Optimal control of ODEs with state suprema
- Author
-
Gerd Wachsmuth, Daniel Wachsmuth, and Tobias Geiger
- Subjects
0209 industrial biotechnology ,Control and Optimization ,Differential equation ,Applied Mathematics ,010102 general mathematics ,Ode ,02 engineering and technology ,State (functional analysis) ,Optimal control ,49K21, 34K35, 49J21 ,01 natural sciences ,Regularization (mathematics) ,020901 industrial engineering & automation ,Maximum principle ,Optimization and Control (math.OC) ,FOS: Mathematics ,Applied mathematics ,Standard theory ,Limit (mathematics) ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Abstract
We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.
- Published
- 2021
45. Stability of non-linear filter for deterministic dynamics
- Author
-
Amit Apte and Anugu Sumith Reddy
- Subjects
Context (language use) ,Filter (signal processing) ,Stability (probability) ,Nonlinear system ,Distribution (mathematics) ,Discrete time and continuous time ,Optimization and Control (math.OC) ,Nonlinear filter ,FOS: Mathematics ,60G35, 93B07, 93E11, 62M20, 93C10 ,Applied mathematics ,State space ,Mathematics - Optimization and Control ,Mathematics - Abstract
This papers shows that nonlinear filter in the case of deterministic dynamics is stable with respect to the initial conditions under the conditions that observations are sufficiently rich, both in the context of continuous and discrete time filters. Earlier works on the stability of the nonlinear filters are in the context of stochastic dynamics and assume conditions like compact state space or time independent observation model, whereas we prove filter stability for deterministic dynamics with more general assumptions on the state space and observation process. We give several examples of systems that satisfy these assumptions. We also show that the asymptotic structure of the filtering distribution is related to the dynamical properties of the signal., Comment: 24 pages, 1 figure. In V4, typos are corrected and few proofs are modified
- Published
- 2021
46. A regularization operator for source identification for elliptic PDEs
- Author
-
Bjørn Fredrik Nielsen and Ole Løseth Elvetun
- Subjects
Control and Optimization ,Helmholtz equation ,Elliptic pdes ,Dirichlet distribution ,Tikhonov regularization ,Parameter identification problem ,symbols.namesake ,Optimization and Control (math.OC) ,Modeling and Simulation ,Norm (mathematics) ,FOS: Mathematics ,symbols ,Regularization operator ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Pharmacology (medical) ,Quadratic programming ,Mathematics - Optimization and Control ,Analysis ,Mathematics - Abstract
We study a source identification problem for a prototypical elliptic PDE from Dirichlet boundary data. This problem is ill-posed, and the involved forward operator has a significant nullspace. Standard Tikhonov regularization yields solutions which approach the minimum $L^2$-norm least-squares solution as the regularization parameter tends to zero. We show that this approach 'always' suggests that the unknown local source is very close to the boundary of the domain of the PDE, regardless of the position of the true local source. We propose an alternative regularization procedure, realized in terms of a novel regularization operator, which is better suited for identifying local sources positioned anywhere in the domain of the PDE. Our approach is motivated by the classical theory for Tikhonov regularization and yields a standard quadratic optimization problem. Since the new methodology is derived for an abstract operator equation, it can be applied to many other source identification problems. This paper contains several numerical experiments and an analysis of the new methodology., Comment: 29 pages
- Published
- 2021
47. Optimal switching at Poisson random intervention times
- Author
-
Wei Wei and Gechun Liang
- Subjects
Sequence ,60H10, 60G40, 93E20 ,Computer science ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Structure (category theory) ,Markov process ,Poisson distribution ,01 natural sciences ,010104 statistics & probability ,Stochastic differential equation ,symbols.namesake ,Optimization and Control (math.OC) ,Bellman equation ,FOS: Mathematics ,symbols ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Optimal stopping ,Infinite horizon ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics - Probability - Abstract
This paper introduces a new class of optimal switching problems, where the player is allowed to switch at a sequence of exogenous Poisson arrival times, and the underlying switching system is governed by an infinite horizon backward stochastic differential equation system. The value function and the optimal switching strategy are characterized by the solution of the underlying switching system. In a Markovian setting, the paper gives a complete description of the structure of switching regions by means of the comparison principle., Comment: 26 pages, 1 figure
- Published
- 2016
48. Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type
- Author
-
Rosanna Manzo and Olha P. Kupenko
- Subjects
Dirichlet problem ,Solenoidal vector field ,Applied Mathematics ,Direct method ,Mathematical analysis ,Perturbation (astronomy) ,Nonlinear monotone Dirichlet problem ,Optimal control ,domain perturbation ,Nonlinear system ,Monotone polygon ,Optimization and Control (math.OC) ,FOS: Mathematics ,equation of Hammerstein type ,control in coefficients ,Discrete Mathematics and Combinatorics ,Calculus of variations ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with matrix-valued $L^\infty(\Omega;\mathbb{R}^{N\times N} )$-controls in coefficients and a nonlinear equation of Hammerstein type. Since problems of this type have no solutions in general, we make a special assumption on the coefficients of the state equation and introduce the class of so-called solenoidal admissible controls. Using the direct method in calculus of variations, we prove the existence of an optimal control. We also study the stability of the optimal control problem with respect to the domain perturbation. In particular, we derive the sufficient conditions of the Mosco-stability for the given class of OCPs.
- Published
- 2015
49. Classical converse theorems in Lyapunov's second method
- Author
-
Christopher M. Kellett
- Subjects
Lyapunov function ,Dynamical systems theory ,Applied Mathematics ,Lyapunov optimization ,Lyapunov exponent ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Optimization and Control (math.OC) ,Stability theory ,FOS: Mathematics ,symbols ,Calculus ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Lyapunov equation ,Lyapunov redesign ,Mathematics - Optimization and Control ,Mathematics ,Control-Lyapunov function - Abstract
Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.
- Published
- 2015
50. Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares
- Author
-
Reza Kamyar and Matthew M. Peet
- Subjects
Mathematical optimization ,Applied Mathematics ,Explained sum of squares ,Stability (learning theory) ,Gröbner basis ,Optimization and Control (math.OC) ,Quantifier elimination ,Convex optimization ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Kharitonov's theorem ,Focus (optics) ,Mathematics - Optimization and Control ,Parametric statistics ,Mathematics - Abstract
In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H_infinity control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature., Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series B
- Published
- 2015
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