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In this paper we consider the distributed optimal control of the Navier—Stokes equations in the presence of pointwise mixed control-state constraints. After deriving a first order necessary condition, the regularity of the mixed constraint multiplier is investigated. Second order sufficient optimality conditions are studied as well. In the last part of the paper, a semismooth Newton method is applied for the numerical solution of the control problem. The convergence of the method is proved and numerical experiments are carried out. [ABSTRACT FROM AUTHOR]

In this paper we establish a comparison result for solutions to the problem minimize ∫Ω f(∇u(x)) dx on {u : u - u0 ϵ W01,1 (Ω)}. [ABSTRACT FROM AUTHOR]

This paper is concerned with the state-constrained optimal control of the threedimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness, and continuity for the state system as well as existence of optimal solutions, admitting global-in-time solutions, to the optimization problem were shown in the the companion paper of this work. In this part, we address further properties of the set of controls whose associated solutions exist globally, such as openness, which includes analysis of the linearized state system via maximal parabolic regularity. The adjoint system involving measures is investigated using a duality argument. These results allow us to derive first-order necessary conditions for the optimal control problem in the form of a qualified optimality system in which we do not need to refer to the set of controls admitting global solutions. The theoretical findings are illustrated by numerical results. This work is the second of two papers on the three-dimensional thermistor problem. [ABSTRACT FROM AUTHOR]

This paper investigates the concurrent path planning optimization and the built part structural optimization for powder bed fusion additive manufacturing processes. The state of the art studies rely on existing patterns for trajectories for a fixed built shape. The shape is often optimized for its mechanical performance but rarely in a combined way with its path planning building process. In this work, a two-dimensional model (in the layer plane) of the process is proposed under a steady state assumption. Then a systematic path optimization approach, free from a priori restrictions and previously developed in [M. Boissier, G. Allaire, and C. Tournier [Struct. Multidiscip. Optim., 61 (2020), pp. 2437--2466], is coupled to a structural optimization tool, both of them based on shape optimization theory. This multiphysics optimization leads to innovative and promising results. First, they confirm that it is essential to take into account the part shape in the scanning path optimization. Second, they also give hints to some design recipes: the material and the source parameters must be related to the thickness of the bars that compose the structure. Indeed, the thickness of a bar is a key ingredient which determines the type of path pattern to scan it: straight line, Omega-pattern, and Wave-pattern. [ABSTRACT FROM AUTHOR]

It is well known that ergodicity and the strong law of large numbers (SLLN) play important roles in stochastic control and stochastic approximations. For a class of regime-switching functional diffusion processes with infinite delay, this paper establishes exponential ergodicity in a Wasserstein distance under certain "averaging conditions." It follows from such an ergodicity property that an SLLN for additive functionals of the regime-switching diffusions is obtained based on the property of uniform mixing. Finally, an example is presented to illustrate the application of ergodicity and the SLLN to a numerical algorithm for stochastic optimization. [ABSTRACT FROM AUTHOR]

The originality of this paper is to deal with the particular case of null infimum jump costs in the infinite horizon impulse control problem. The value function of such problems is a viscosity solution of the classic quasi-variational inequality (QVI) associated, but not the unique one. This is a drawback to characterize it. In this paper, a new QVI for which the value function is the unique viscosity solution is given. This allows us to approximate the value function. So, we give some discrete approximations of the new QVI and prove that the approximate value function converges locally uniformly, toward the value function of the impulse control problem with zero lower bound of impulse cost. We choose the classic example of continuous time inventory control in Rn to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]

In this paper we consider a distributed stochastic optimization problem without gradient/subgradient information for local objective functions and subject to local convex constraints. Objective functions may be nonsmooth and observed with stochastic noises, and the network for the distributed design is time-varying. By adding stochastic dithers to local objective functions and constructing randomized differences motivated by the Kiefer--Wolfowitz algorithm, we propose a distributed subgradient-free algorithm for finding the global minimizer with local observations. Moreover, we prove that the consensus of estimates and global minimization can be achieved with probability one over the time-varying network, and we obtain the convergence rate of the mean average of estimates as well. Finally, we give numerical examples to illustrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper formulates the problem of extremum seeking for optimization of cost functions defined on Riemannian manifolds. We extend the conventional extremum seeking algorithms for optimization problems in Euclidean spaces to optimization of cost functions defined on smooth Riemannian manifolds. This problem falls within the category of online optimization methods. We introduce the notion of geodesic dithers, which is a perturbation of the optimizing trajectory in the tangent bundle of the ambient state manifolds, and obtain the extremum seeking closed loop as a perturbation of the averaged gradient system. The main results are obtained by applying closeness of solutions and averaging theory on Riemannian manifolds. The main results are further extended for optimization on Lie groups. Numerical examples on the Stiefel manifold V3,2 and the Lie group SEp3q are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

The paper considers the problem of absolute stability of systems with time-periodic nonlinear blocks. The approach presented in this paper is based on the so-called quadratic criterion and relies on integral-quadratic inequalities (constraints), which also involve time delays. The results are given in the frequency domain. The paper also includes geometric interpretation and numerical treatment of the new stability criteria. [ABSTRACT FROM AUTHOR]

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 nonconvex, which makes it difficult to achieve real time computation using existing nonconvex optimization algorithms. This paper introduces the convex feasible set algorithm, which is a fast algorithm for nonconvex optimization problems that have convex costs and nonconvex 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. [ABSTRACT FROM AUTHOR]

Switched systems, or systems whose control parameters include a continuous-valued input and a discrete-valued input which corresponds to the mode of the system that is active at a particular instance in time, have shown to be highly effective in modeling a variety of physical phenomena. Unfortunately, the construction of an optimal control algorithm for such systems has proved difficult since it demands some form of optimal mode scheduling. In a pair of papers, we construct a first order optimization algorithm to address this problem. Our approach, which we prove in this paper converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical chattering lemma that we formalize. In the second part of this pair of papers, we describe how this conceptual algorithm can be recast in order to devise an implementable algorithm that constructs a sequence of points by recursive application that converge to local minimizers of the optimal control problem for switched systems. [ABSTRACT FROM AUTHOR]

Despite significant progress on stability analysis of conventional multiagent networked systems with weakly coupled state-network dynamics, most of the existing results have shortcomings in addressing multiagent systems with highly coupled state-network dynamics. Motivated by numerous applications of such dynamics, in our previous work [SIAM J. Control Optim., 57 (2019), pp. 1757-1782], we initiated a new direction for stability analysis of such systems that uses a sequential optimization framework. Building upon that, in this paper, we extend our results by providing another angle on multiagent network dynamics from a duality perspective, which allows us to view the network structure as dual variables of a constrained nonlinear program. Leveraging that idea, we show that the evolution of the coupled state-network multiagent dynamics can be viewed as iterates of a primal-dual algorithm for a static constrained optimization/saddle-point problem. This view bridges the Lyapunov stability of state-dependent network dynamics and frequently used optimization techniques such as block coordinated descent, mirror descent, the Newton method, and the subgradient method. As a result, we develop a systematic framework for analyzing the Lyapunov stability of state-dependent network dynamics using techniques from nonlinear optimization. Finally, we support our theoretical results through numerical simulations from social science. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable but not Gateaux-differentiable. By employing the limited differentiability properties of the control-to-state map, first-order necessary optimality conditions in qualified form are established, which are equivalent to the purely primal condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. The paper ends with the application of the general results to a semilinear heat equation. [ABSTRACT FROM AUTHOR]

Random convex programs are convex optimization problems that are robust with respect to a finite number of randomly sampled instances of an uncertain variable δ. This paper studies random convex programs in which there is uncertainty in the objective function. Specifically, let L(x, δ) be a loss function that is convex in x, the optimization variable, while it has an arbitrary dependence on the random variable d representing uncertainty in the optimization problem. After sampling N instances δ(1), δ(N) of the random variable δ, the random convex program can be written as follows: minx maxi L(x, δ(i)). The fundamental feature of this program is that its value LN* = maxi L(xN*,δ(i)), where xN* is the solution, remains guaranteed when xN* is applied to the vast majority of the other unseen instances of δ; that is, L(xN*, δ) ≤ LN* holds with high probability with respect to the uncertain variable δ. This generalization property has justified a systematic and rigorous use of randomization in robust optimization. In this paper, we introduce L1-regularization in random convex programs and show that L1-regularization boosts the above generalization property so that generalization is achieved with significantly fewer samples than in the standard convex program given above. Explicit bounds are derived that allow a rigorous and easy implementation of the method. [ABSTRACT FROM AUTHOR]

In this paper we investigate a passivity approach to collective coordination and synchronization problems in the presence of quantized measurements and show that coordination tasks can be achieved in a practical sense for a large class of passive systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a theoretic and numerical study of utilizing squeezing and phase shift in coherent feedback control of linear quantum optical systems. A quadrature representation with built-in phase shifters is proposed for such systems. Fundamental structural characterizations of linear quantum optical systems are derived in terms of the new quadrature representation. These results reveal considerable insights into the issue of the physical realizability of such quantum systems. The problem of coherent quantum linear quadratic Gaussian (LQG) feedback control studied in H. I. Nurdin, M. R. James, and I. R. Petersen, Automatica, IFAC, 45 (2009), pp. 1837-1846; G. Zhang and M. R. James, IEEE Trans. Automat. Control, 56 (2011), pp. 1535-1550 is reinvestigated in depth. First, the optimization methods in these papers are extended to a multistep optimization algorithm which utilizes ideal squeezers. Second, a two-stage optimization approach is proposed on the basis of controller parametrization. Numerical studies show that closed-loop systems designed via the second approach may offer LQG control performance even better than that when the closed-loop systems are in the vacuum state. When ideal squeezers in a closed-loop system are replaced by (more realistic) degenerate parametric amplifiers, a sufficient condition is derived for the asymptotic stability of the resultant new closed-loop system; the issue of performance convergence is also discussed in the LQG control setting. [ABSTRACT FROM AUTHOR]

Dissipative systems have played an important role in the analysis and synthesis of dynamical systems. The commonly used definition of dissipativity often requires an assumption on the controllability of the system. In this paper we use a definition of dissipativity that is slightly different (and less often used in the literature) to study a linear, time-invariant, possibly uncontrollable dynamical system. We provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not "mixed"; i.e., no pair of uncontrollable poles is symmetric about the imaginary axis. This condition is known to be related to the solvability of a Lyapunov equation; we link Lyapunov functions for autonomous systems to storage functions of an uncontrollable system. The set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature. We show that for an uncontrollable system the set of storage functions is unbounded, and that the unboundedness arises precisely due to the set of Lyapunov functions for an autonomous linear system being unbounded. Further, we show that stabilizability of a system results in this unbounded set becoming bounded from below. Positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive. In this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions. In most of the results in this paper, we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis; we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity. [ABSTRACT FROM AUTHOR]

This paper considers an extension of the Merton optimal investment problem to the case where the risky asset is subject to transaction costs and capital gains taxes. We derive the dynamic programming equation in the sense of constrained viscosity solutions. We next introduce a family of functions (Vϵ)ϵ>0, which converges to our value function uniformly on compact subsets, and which is characterized as the unique constrained viscosity solution of an approximation of our dynamic programming equation. In particular, this result justifies the numerical results reported in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi (2005), Modeling Continuous-Time Financial Markets with Capital Gains Taxes, preprint, http://www.cmap.polytechnique.fr/~touzi/bst06.pdf]. [ABSTRACT FROM AUTHOR]

We revise our previous paper [SIAM J. Control Optim., 43 (2004), pp. 1063–1077]. We supply an assumption missing in the first part of the assertion of Corollary 4.6, and we prove that the second part of this corollary remains true also under conditions given in the previous paper. Finally, we comment on new contributions and pose the open question in the topic. [ABSTRACT FROM AUTHOR]

Methods for determining ARMA(n,m) filters from covariance and cepstral estimates are proposed. In [C. I. Byrnes, P. Enqvist, and A. Lindquist, SIAM J. Control Optim., 41 (2002), pp. 23-59], we have shown that an ARMA(n,n) model determines and is uniquely determined by a window r0, r1, . . . , rn of covariance lags and c1, c2, . . . , cn of cepstral lags. This unique model can be determined from a convex optimization problem which was shown to be the dual of a maximum entropy Problem. in this paper, generalizations of this problem are analyzed. Problems with covariance lags r0, r1, . . . , rn and cepstral lags c1, c2, . . . , cm of different lengths are considered, and by considering different combinations of covariances, cepstral parameters, poles, and zeros, it is shown that only zeros and covariances give a parameterization that is consistent with generic data. However, the main contribution of this paper is a regularization of the optimization Problems that is Proposed in order to handle generic data. For the covariance and cepstral problem, if the data does not correspond to a system of desired order, solutions with zeros on the boundary occur and the cepstral coefficients are not interpolated exactly. In order to achieve strictly minimum phase filters for estimated covariance and cepstral data, a barrier-like term is introduced to the optimization problem. This term is chosen so that convexity is maintained and so that the unique solution will still interpolate the covariances but only approximate the cepstral lags. Furthermore, the solution will depend analytically on the covariance and cepstral data, which provides robustness, and the barrier term increases the entropy of the solution. [ABSTRACT FROM AUTHOR]

In this paper we consider an optimization problem with equality constraints given in operator form as F(x) = 0, where F : X → Y is an operator between Banach spaces. The paper addresses the case when the equality constraints are not regular in the sense that the Fréchet derivative F[sup1](x*) is not onto. In the first part of the paper, we pursue an approach based on the construction of p-regularity. For p-regular constrained optimization problems, we formulate necessary conditions for optimality and derive sufficient conditions for optimality. In the second part of the paper, we consider a generalization of the concept of p-regularity and derive generalized necessary conditions for optimality for an optimization problem that is neither regular nor p-regular. For this problem, we show that the tangent cone to a level surface of F can consist of rays (rather than lines). This is in contrast to the regular and the p-regular cases, for which the tangent cone is always "two-sided." We state that if the gradient of the generalized p-regular problem is nonzero, it can belong to an open set, despite the fact that all constructions are usually closed. Both p-regular and generalized conditions for optimality reduce to classical conditions for regular cases, but they give new and nontrivial conditions for nonregular cases. The presented results can be considered as a part of the p-regularity theory. [ABSTRACT FROM AUTHOR]

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal controls, for systems governed by the Ito stochastic differential equations (SDEs), where both the drift and diffusion terms are allowed to depend on controls and the systems are allowed to be degenerate. Necessary and sufficient conditions for a control to be near-optimal are studied. It is shown that any near-optimal control nearly maximizes the ``H-function'' (which is a generalization of the usual Hamiltonian and is quadratic with respect to the diffusion coefficients) in some integral sense, and vice versa if certain additional concavity conditions are imposed. Error estimates for both the near-optimality of the controls and the near-maximum of the H-function are obtained, based on some delicate estimates of the adjoint processes. Examples are presented to demonstrate the results. [ABSTRACT FROM AUTHOR]

This paper studies distributed algorithms for the nonsmooth extended monotropic optimization problem, which is a general convex optimization problem with a certain separable structure. The considered nonsmooth objective function is the sum of local objective functions as-signed to agents in a multiagent network, with local set constraints and affine equality constraints. Each agent only knows its local objective function, local set constraint, and the information ex-changed between neighbors. To solve the constrained convex optimization problem, we propose two novel distributed continuous-time subgradient-based algorithms, with projected output feedback and derivative feedback, respectively. Moreover, we prove the convergence of proposed algorithms to the optimal solutions under some mild conditions and analyze convergence rates, with the help of the techniques of variational inequalities, decomposition methods, and differential inclusions. Finally, we give an example to illustrate the efficacy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper considers a dynamic optimization problem for a class of switched systems characterized by two key attributes: (i) the switching mechanism is invoked automatically when the state variables satisfy certain switching conditions; and (ii) the subsystem dynamics involve time-delays in the state variables. The decision variables in the problem, which must be selected optimally to minimize system cost, consist of a set of time-invariant system parameters in the initial state functions. To solve the dynamic optimization problem, we first show that the partial derivatives of the system state with respect to the system parameters can be expressed in terms of the solution of a set of variational switched systems. Then, on the basis of this result, we develop a gradient-based optimization algorithm to determine the optimal parameter values. Finally, we validate the proposed algorithm by solving an example problem arising in the production of 1,3-propanediol. [ABSTRACT FROM AUTHOR]

This paper studies near-controllability of a class of discrete-time bilinear systems via a root locus approach. A necessary and sufficient criterion for the systems to be nearly controllable is given. In particular, by using the root locus approach, the control inputs which achieve the state transition for the nearly controllable systems can be computed. Furthermore, for the nonnearly-controllable systems, nearly controllable subspaces are derived and near-controllability index is defined. Accordingly, the controllability properties of such a class of discrete-time bilinear systems are fully characterized. Finally, examples are provided to demonstrate the results of the paper. [ABSTRACT FROM AUTHOR]

In the first part of this two-paper series, we presented a conceptual algorithm for the optimal control of constrained switched systems and proved that this algorithm generates a sequence of points that converge to a necessary condition for optimality. However, since our algorithm requires the exact solution of a differential equation, the numerical implementation of this algorithm is impractical. In this paper, we address this shortcoming by constructing an implementable algorithm that discretizes the differential equation, producing a finite-dimensional nonlinear program. We prove that this implementable algorithm constructs a sequence of points that asymptotically satisfy a necessary condition for optimality for the constrained switched system optimal control problem. Four simulation experiments are included to validate the theoretical developments. [ABSTRACT FROM AUTHOR]

This paper deals with shape optimization for elastic materials under stochastic loads. It transfers the paradigm of stochastic dominance, which allows for flexible risk aversion via comparison with benchmark random variables, from finite-dimensional stochastic programming to shape optimization. Rather than handling risk aversion in the objective, this enables risk aversion by including dominance constraints that single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. This new class of stochastic shape optimization problems arises by optimizing over such feasible sets. The analytical description is built on risk-averse cost measures. The actual optimized cost functional measures the volume and perimeter of the structure. In the implementation, shapes are represented by a phase field which permits an easy estimate of a regularized perimeter. The analytical description and the numerical implementation of dominance constraints are built on risk-averse measures for the cost functional. A suitable numerical discretization is obtained using finite elements both for the displacement and the phase field function. Different numerical experiments demonstrate the potential of the proposed stochastic shape optimization model and in particular the impact of high variability of forces or probabilities in the different realizations. [ABSTRACT FROM AUTHOR]

We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. This paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem, and finalizes with a range of numerical tests. [ABSTRACT FROM AUTHOR]

This paper discusses asymptotic analysis and adaptive design of convex estimators over the Hölder class under the sup-norm risk and the pointwise risk using constrained optimization and asymptotic statistical techniques. Specifically, convex B-spline estimators are proposed to achieve uniform optimal convergence rates and adaptive procedures. The presence of the convex shape constraint complicates asymptotic performance analysis, particularly uniform convergence analysis. This in turn requires deep understanding of a family of size varying constrained optimization problems on spline coeficients. To address these issues, we establish the uniform Lipschitz property of optimal spline coeficients in the l∞-norm by exploiting the structure of underlying constrained optimization 8 problems. By using this property, polyhedral theory, and statistical techniques, we show that the convex B-spline estimator attains uniform consistency and optimal rates of convergence on the entire interval of interest over the Hölder class under the sup-norm risk and the pointwise risk. In addition, adaptive estimates are constructed under both risks when the Hölder exponent is between one and two. These estimates achieve a maximal risk within a constant factor of the minimax risk over the Hölder class. [ABSTRACT FROM AUTHOR]

Optimum experimental design (OED) problems are optimization problems in which an experimental setting and decisions on when to measure--the so-called sampling design--are to be determined such that a follow-up parameter estimation yields accurate results for model parameters. In this paper we use the interpretation of OED as optimal control problems with a very particular structure for the analysis of optimal sampling decisions. We introduce the information gain function, motivated by an analysis of necessary conditions of optimality. We highlight differences between problem formulations and propose to use a linear penalization of sampling decisions to overcome the intrinsic ill-conditioning of OED. The results of this paper are independent from the actual numerical method to compute the solution to the OED problem and of the question of local and global optima. [ABSTRACT FROM AUTHOR]

The extremum seeking control (ESC) approach with a first-order sliding mode has been proposed for searching a setpoint by making a sliding mode occur, where a performance index tracks a reference signal to reach its extremum. The system dynamics is assumed to be much faster than the parameter updating rate determined by the ESC rule and thus is omitted in the stability analysis for the ESC system with a first-order sliding mode. As a result only a static optimization can be obtained. In this paper, an ESC approach with a second-order sliding mode is proposed with the consideration of the system dynamics such that the performance index can be optimized dynamically even if the system dynamics is not faster than the parameter updating rate for the extremum seeking. The asymptotic second-order sliding mode relay control algorithm is implemented to guarantee the asymptotic convergence to the second-order sliding mode without using the derivative of the performance index. Simulation results show that the second-order sliding mode can be reached asymptotically and the system converges to the setpoint on the second-order sliding mode in the speed determined by the reference signal. [ABSTRACT FROM AUTHOR]

This paper is concerned with optimization of uncertain stochastic systems, in which uncertainty is described by a total variation distance constraint between the measures induced by the uncertain systems and the measure induced by the nominal system, while the payoff is a linear functional of the uncertain measure. Robustness at the abstract setting is formulated as a minimax game, in which the control seeks to minimize the payoff over the admissible controls while the uncertainty aims at maximizing it over the total variation distance constraint. It is shown that the maximizing measure in the total variation distance constraint exists, while the resulting payoff is a linear combination of L1 and L∞ norms. Further, the maximizing measure is characterized by a linear combination of a tilted measure and the nominal measure, giving rise to a payoff which is a nonlinear functional on the space of measures to be minimized over the admissible controls. The abstract formulation and results are subsequently applied to continuous-time uncertain stochastic controlled systems, in which the control seeks to minimize the payoff while the uncertainty aims to maximize it over the total variation distance constraint. The minimization over the admissible controls of the nonlinear functional payoff is addressed by developing a generalized principle of optimality or dynamic programming equation satisfied by the value function. Subsequently, it is proved that the value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation. It is also shown that the value function is also a viscosity solution of the HJB equation. Finally, the linear quadratic case is studied, and it is shown that the infinity norm of a quadratic payoff is well defined and finite. Throughout the paper the formulation and conclusions are related to previous work found in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations. An existence result of time optimal controls for such a problem is established. The rectangular type of control constraints originates from the study of time optinial control problems for ordinary differential equations. In the finite dimensional case, there is no difference between such problems with control constraints of the rectangular type and those of the ball type, from the perspective of the study on the existence of optimal controls. Interestingly, in the infinite dimensional case, the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type. For infinite dimensional systems, the existence for time optimal controls with constraints of the ball type has already been discussed in the literature, while the study of the rectangular type has not been touched upon as far as we know. [ABSTRACT FROM AUTHOR]

The article presents a research paper that discusses the relationship between five theorems including Yukubovich's S-lemma, Yuan's alternative theorem and nonstrict Finsler's theorem. The use of S-procedure as a very useful tool in control theory and robust optimization analysis is mentioned. It also include some extensions of these results to an infinite-dimensional vector space.

The article presents a research paper that discusses the problem related to finding the minimal state space realization for a specific category of fractional order systems. Both for commensurate and incommensurate cases the fractional order transfer function is determined by the sense of inner dimension. There is no specific method to check the minimality of incommensurate realizations. Hadamard product-based transformation is introduced to transform different minimal realizations.

The article presents a research paper that discusses ways to obtain the optimal time to reach a target for a controlled nonlinear system, under state constraints. It provides a continuous level-set approach for describing the optimal times and the backwardreachability sets. It presents an approach through a Hamilton-Jacobi equation, without assuming any controllability assumption. The paper includes some numerical illustrations and approximations.

A methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane. The sensing-pursuit problem considered in this paper is analogous to the Marco Polo game, in which a pursuer Marco must capture multiple mobile targets that are sensed intermittently, and with very limited information. The competing objectives exhibited by this problem arise in a number of surveillance and monitoring applications. In this paper, the mobile sensor network consists of a set of robotic sensors that must track and capture mobile targets based on the information obtained through cooperative detections. When these detections form a satisfactory target track, a mobile sensor is switched to pursuit mode and deployed to capture the target in minimum time. Since the sensors are installed on robotic platforms and have limited range, the geometry of the platforms and of the sensors' fields-of-view play a key role in obstacle avoidance and target detection. A new cell-decomposition approach is presented to determine the probability of detection and the cost of operating the sensors from the geometric properties of the network and its workspace. The correctness and complexity of the algorithm are analyzed, proving that the termination time is a function of the network parameters and of the number of required detections. [ABSTRACT FROM AUTHOR]

The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Grass(p, n) are treated as original examples. A link is also drawn with the many existing results on the circle. [ABSTRACT FROM AUTHOR]

The object of this paper is to revisit the results of Bernhard [J. Optim. Theory Appl., 27 (1979), pp. 51-69] on two-person zero-sum linear quadratic differential games and generalize them to utility functions without positivity assumptions on the matrices acting on the state variable and to linear dynamics with bounded measurable data matrices. Our paper specializes to state feedback via Lebesgue measurable affine closed loop strategies with possible non-L2-integrable singularities. After sharpening the recent results of Delfour [SIAM J. Control Optim., 46 (2007), pp. 750-774] on the characterization of the open loop lower and upper values of the game, it first deals with L2-integrable closed loop strategies and then with the larger family of strategies that may have non-L2-integrable singularities. A new conceptually meaningful and mathematically precise definition of a closed loop saddle point is introduced to simultaneously handle state feedbacks of the L2 type and smooth locally bounded ones, except at most in the neighborhood of finitely many instants of time. A necessary and sufficient condition is that the free end problem be normalizable almost everywhere. This relaxation of the classical notion allows singularities in the feedback law at an infinite number of instants, including accumulation points that are not isolated. A complete classification of closed loop saddle points is given in terms of the convexity/concavity properties of the utility function, and connections are given with the open loop lower value, upper value, and value of the game. [ABSTRACT FROM AUTHOR]

This paper analyzes a class of singular control problems for which value functions are not necessarily smooth. Necessary and sufficient conditions for the well-known smooth fit principle, along with the regularity of the value functions, are given. Explicit solutions for the optimal policy and for the value functions are provided. In particular, when payoff functions satisfy the usual Inada conditions, the boundaries between action and continuation regions are smooth and strictly monotonic, as postulated and exploited in the existing literature (see [A. K. Dixit and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, NJ, 1994]; [M. H. A. Davis et al., Adv. in Appl. Probab., 19 (1987), pp. 156-176]; [T. Ø. Kobila, Stochastics Stochastics Rep., 43 (1993), pp. 29-63]; [A. B. Abel and J. C. Eberly, J. Econom. Dynam. Control, 21 (1997), pp. 831-852]; [A. Øksendal, Finance Stoch., 4 (2000), pp. 223-250]; [J. A. Scheinkman and T. Zariphopoulou, J. Econom. Theory, 96 (2001), pp. 180-207]; [A. Merhi and M. Zervos, SIAM J. Control Optim., 46 (2007), pp. 839-876]; and [L. H. Alvarez, A General Theory of Optimal Capacity Accumulation under Price Uncertainty and Costly Reversibility, Working Paper, Helsinki Center of Economic Research, Helsinski, Finland, 2006]). Illustrative examples for both smooth and nonsmooth cases are discussed to emphasize the pitfall of solving singular control problems with a priori smoothness assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraint for the cost functional described by the solution of a reflected backward stochastic differential equation. We give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton–Jacobi–Bellman equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of a second order method for recovering the a priori unknown shape of an inclusion ? inside a body Ω from boundary measurement. This inverse problem--known as electrical impedance tomography--has many important practical applications and hence has been the focus of much attention during the past few years. However, to the best of our knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: We investigate the existence of second order derivative of the state u with respect to perturbations of the shape of the interface ∂?. Then we choose a cost function in order ∂to recover the geometry of ∂? and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer. [ABSTRACT FROM AUTHOR]

In this paper the authors study a model for the optimal operation of a bank or insurance company which was recently introduced by Peura and Keppo. The model generalizes a previous one of Milne and Robertson by allowing the bank to raise capital as well as to pay out dividends. Optimal operation of the bank is determined by solving an optimal control problem. In this paper it is shown that the solution of the optimal control problem proposed by Peura and Keppo exists for all values of the parameters and is unique. [ABSTRACT FROM AUTHOR]

Relaxed controls have proved to be very useful in studying the existence of optimal controls in optimal control theory. Many positive results have been obtained in the literature. However, negative results have also made their rare appearances. The optimal control problem considered in this paper looks quite simple. Yet, by treating such a problem, we can get interesting results, substantiating our idea as to whether an optimal control exists or not. In our opinion, the method used in the paper can be applied to more generalized cases. [ABSTRACT FROM AUTHOR]

This papers analyzes a class of nonlinear additive-increase multiplicative-decrease (AIMD) protocols that are widely deployed in communication networks. It is demonstrated that the use of these protocols guarantees that the system has a unique stable outcome to which it converges geometrically under all starting points. The development is based on a contraction argument and the derivation of explicit bounds on the contraction coefficient of corresponding operators in terms of the network parameters. In particular, bounds on the corresponding rate of convergence are obtained, improving upon known bounds for standard (linear) AIMD networks. [ABSTRACT FROM AUTHOR]

In this paper, we study some optimal control problems governed by certain semilinear parabolic equations with K-approximate periodic solutions. We first prove the existence and uniqueness theorems for K-approximate periodic solutions of the equations. We then use these results to establish the qualified Pontryagin maximum principle. The existence for such optimal controls is also investigated in the paper. [ABSTRACT FROM AUTHOR]

An optimal control problem for a two-dimensional elliptic equation is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise linear functions. The state and the adjoint state are discretized by linear finite elements. Approximation of order $h$ in the $L^\infty$-norm is proved in the main result. [ABSTRACT FROM AUTHOR]

This paper considers a class of optimization problems arising in wireless communication systems. We analyze the optimal control and the associated Hamilton--Jacobi--Bellman (HJB) equations. It turns out that the value function is a unique viscosity solution of the HJB equation in a certain function class. To deal with the fast growth condition of the value function in establishing uniqueness, we construct particular semiconvex/semiconcave approximations for the viscosity sub/supersolutions, and obtain a maximum principle on unbounded domains. The localized envelope function technique introduced in this paper permits an analysis of the uniqueness of viscosity solutions defined on unbounded domains in cases with very general growth conditions when combined with appropriate system dynamics. The optimization problem with state constraints is also considered. [ABSTRACT FROM AUTHOR]

This paper concerns optimal control problems for dynamical systems described by a parametric family of discrete/finite-difference approximations of continuous-time control systems. Control theory for parametric systems governed by discrete approximations plays an important role in both qualitative and numerical aspects of optimal control and occupies an intermediate position in dynamic optimization: between optimal control of discrete-time (with fixed steps) and continuous-time control systems. The central result in optimal control of discrete approximation systems is the approximate maximum principle (AMP), which gives the necessary optimality condition in a perturbed maximum principle form with no a priori convexity assumptions and thus ensures the stability of the Pontryagin maximum principle (PMP) under discrete approximation procedures. The AMP has been justified for optimal control problems of smooth dynamical systems with endpoint constraints under some properness assumption imposed on the sequence of optimal controls. in this paper we show, by a series of counterexamples, that the properness assumption is essential for the validity of the AMP, and that the AMP does not hold, in its expected (lower) subdifferential form, for nonsmooth problems. Moreover, a new upper subdifferential form of the AMP is established for ordinary and time-delay control systems. The results obtained surprisingly solve (in both negative and positive directions) a long-standing and well-recognized question about the possibility of extending the AMP to nonsmooth control problems, for which the affirmative answer has been expected in the conventional lower subdifferential form. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal singular stochastic control problem. By using a time transformation, this problem is shown to be equivalent to an auxiliary control problem defined as a combination of an optimal stopping problem and a classical control problem. For this auxiliary control problem, the controller must choose a stopping time (optimal stopping), and the new control variables belong to a compact set. This equivalence is obtained by showing that the (discontinuous) state process governed by a singular control is given by a time transformation of an auxiliary state process governed by a classical bounded control. it is proved that the value functions for these two problems are equal. For a general form of the cost, the existence of an optimal singular control is established under certain technical hypotheses. Moreover, the problem of approximating singular optimal control by absolutely continuous controls is discussed in the same class of admissible controls. [ABSTRACT FROM AUTHOR]