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As far as numerical methods for optimization problems are concerned, in practice it is a matter of generating sequences of solutions and associated multipliers which are expected to obey, asymptotically, (at least) the first-order necessary optimality conditions. In mathematical programming, this procedure is theoretically validated by means of the so-called asymptotic KKT conditions. In optimal control theory, there is a lack of necessary optimality conditions of this kind. In this paper, we propose a new set of necessary optimality conditions for mixed-constrained optimal control problems in the form of an asymptotic (weak) maximum principle. We also propose, inspired by the augmented Lagrangian method for nonlinear programming, a method of multipliers for numerically solving mixed-constrained optimal control problems in which the generated sequences of solutions and multipliers satisfy the conditions of the asymptotic weak maximum principle. We discuss its convergence properties in terms of optimality and feasibility. To demonstrate the practical viability of the proposed theory, we provide some numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of optimal pairs for optimal control problems with their states constrained pointwise to Riemannian manifolds. For this purpose, by means of the Riemannian geometric tool, we introduce a crucial Cesari-type property, which is an extension of the classical Cesari property (see Definition 3.3, p. 51 in [L. D. Berkovitz, Optimal Control Theory, Appl. Math. Sci. 12, Springer-Verlag, New York, Heidelberg, 1974]) from the setting of Euclidean spaces to that of Riemannian manifolds. Moreover, we show the efficiency of our result by a concrete example. [ABSTRACT FROM AUTHOR]

We explore the relationship between the dual of a weighted minimum-energy control problem, a special case of linear-quadratic optimal control problems, and the Douglas--Rachford (DR) algorithm. We obtain an expression for the fixed point of the DR operator as applied to solving the optimal control problem, which in turn devises a certificate of optimality that can be employed for numerical verification. The fixed point and the optimality check are illustrated in two example optimal control problems. [ABSTRACT FROM AUTHOR]

We consider optimal control problems with integer-valued controls and a total variation regularization penalty in the objective on domains of dimension two or higher. The penalty yields that the feasible set is sequentially closed in the weak-\ast topology and closed in the strict topology in the space of functions of bounded variation. In turn, we derive first-order optimality conditions of the optimal control problem as well as trust-region subproblems with partially linearized model functions using local variations of the level sets of the feasible control functions. We also prove that a recently proposed function space trust-region algorithm---sequential linear integer programming---produces sequences of iterates whose limits are first-order optimal points. [ABSTRACT FROM AUTHOR]

This paper presents an interior point method for pure state and mixed-constraint optimal control problems for dynamics, mixed constraints, and cost function all affine in the control variable. This method relies on resolving a sequence of two-point boundary value problems of differential and algebraic equations. This paper establishes a convergence result for primal and dual variables of the optimal control problem. A primal and a primal-dual solving algorithm are presented, and a challenging numerical example is treated for illustration. [ABSTRACT FROM AUTHOR]

A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a type-II backward SVIE (BSVIE). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore, a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback. [ABSTRACT FROM AUTHOR]

A class of optimal control problems governed by fractional differential equations with control constraints and free right end point is considered. We first prove a result on the existence of optimal solutions for the case where the state equation may be nonlinear in control variable. Then we establish first- and second-order optimality conditions for locally optimal solutions to the general problem. When 1 2 < \alpha < 1, a theory of no-gap second-order conditions is obtained. A result on regularity of optimal solution for the case where the state equation is linear in control variable is also given. [ABSTRACT FROM AUTHOR]

In optimal control theory one sometimes extends the minimization domain of a given problem, with the aim of achieving the existence of an optimal control. However, this issue is naturally confronted with the possibility of a gap between the original infimum value and the extended one. Avoiding this phenomenon is not a trivial issue, especially when the trajectories are subject to endpoint constraints. However, since the seminal works by Warga, some authors have recognized "normality"" of an extended minimizer as a condition guaranteeing the absence of an infimum gap. Yet, normality is far from being necessary for this goal, a fact that makes the search for weaker assumptions a reasonable aim. In relation to a control-affine system with unbounded controls, in this paper we prove a sufficient no-gap condition based on a notion of higher order normality, which is less demanding than the standard normality and involves iterated Lie brackets of the vector fields defining the dynamics. [ABSTRACT FROM AUTHOR]

We discuss state-constrained optimal control of a quasilinear parabolic partial differential equation. Existence of optimal controls and first-order necessary optimality conditions are derived for a rather general setting including pointwise in time and space constraints on the state. Second-order sufficient optimality conditions are obtained for averaged-in-time and pointwise in space state-constraints under general regularity assumptions for the equation. Moreover, we provide second-order sufficient optimality conditions also in the case of pointwise in space and time state-constraints, but in return for a more regular setting for the equation. [ABSTRACT FROM AUTHOR]

In optimal control theory, infimum gap means that there is a gap between the infimum values of a given minimum problem and an extended problem, obtained by enlarging the set of original solutions and controls. The gap phenomenon is somewhat "dual" to the problem of the controllability of the original control system to an extended solution. In this paper we present sufficient conditions for the absence of an infimum gap and for controllability for a wide class of optimal control problems subject to endpoint and state constraints. These conditions are based on a nondegenerate version of the nonsmooth constrained maximum principle, expressed in terms of subdifferentials. In particular, under some new constraint qualification conditions, we prove that (i) if an extended minimizer is a nondegenerate normal extremal, then no gap shows up; (ii) given an extended solution verifying the constraints, either it is a nondegenerate abnormal extremal or the original system is controllable to it. An application to the impulsive extension of a free end-time, nonconvex optimization problem with control-polynomial dynamics illustrates the results. [ABSTRACT FROM AUTHOR]

Nonconvex optimal control problems governed by evolution problems in infinite-dimensional spaces (as, e.g., parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a new class of optimal control problems constrained by elliptic equations with uncertain fractional exponents. We utilize risk measures to formulate the resulting optimization problem. We develop a functional analytic framework, study the existence of solution, and rigorously derive the first-order optimality conditions. Additionally, we employ a sample-based approximation for the uncertain exponent and the finite element method to discretize in space. We prove the rate of convergence for the optimal risk neutral controls when using quadrature approximation for the uncertain exponent and conclude with illustrative examples. [ABSTRACT FROM AUTHOR]

Controlling the shape and position of moving and pinned droplets on a solid surface is an important feature often found in microfluidic applications. In this work, we consider a well investigated phase field model including contact line dynamics as the state system for an (open-loop) optimal control problem. Here the spatially and temporally changeable contact angles between droplet and solid are considered as the control variables. We consider a suitable, energy stable, time discrete version of the state equation in our optimal control problem. We discuss regularity of the solution to the time discrete state equation and its continuity and differentiability properties. Furthermore, we show existence of solutions and state first order optimality conditions to the optimal control problem. We illustrate our results by actively pushing a droplet uphill against gravity in an optimal way. [ABSTRACT FROM AUTHOR]

An optimal ergodic control (EC) problem is investigated for a linear stochastic differential equation with quadratic cost functional. Constant nonhomogeneous terms, not all zero, appear in the state equation, which lead to the asymptotic limit of the state nonzero. Under the stabilizability condition, for any (admissible) closed-loop strategy, an invariant measure is proved to exist, which makes the ergodic cost functional well-defined and the EC problem well-formulated. Sufficient conditions, including those allowing the weighting matrices of cost functional to be indefinite, are introduced for finiteness and solvability for the EC problem. Some comparisons are made between the solvability of EC problem and the closed-loop solvability of stochastic linear-quadratic optimal control problem in the infinite horizon. The regularized EC problem is introduced to be used to obtain the optimal value of the EC problem. [ABSTRACT FROM AUTHOR]

We study a class of discounted autonomous infinite horizon optimal control problems where the state dynamics may undergo discontinuous changes when the state crosses from one region of the state space to another. Assuming the state and control space are one-dimensional, we provide structural results of the candidate optimal solutions. In particular, we show that candidate optimal state trajectories are monotonic. Further, we prove a theorem on absence of limit cycles, generalizing the Bendixson criterion to a hybrid context. Using these results, we derive necessary and sufficient conditions for the existence of tipping points for this class of problems. A tipping point is an initial condition starting from which there exist multiple candidate optimal solutions with different qualitative behaviors. Next, we specialize these results for piecewise linear quadratic models to obtain an analytic characterization of tipping points in terms of the model parameters. [ABSTRACT FROM AUTHOR]

We here consider optimal control problems governed by nonlinear stochastic equations on a Hilbert space H with nonconvex payoff, which is rewritten as a deterministic optimal control problem governed by a Kolmogorov equation in H. We prove the existence and first-order necessary condition of closed-loop optimal controls for the above control problem. The strategy is based on solving a deterministic bilinear optimal control problem for the corresponding Kolmogorov equation on the space L2(H, \nu), where \nu is the related infinitesimally invariant measure for the Kolmogorov operator. [ABSTRACT FROM AUTHOR]

An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection technique. First-order necessary optimality conditions are derived and two types of second-order sufficient optimality conditions are established. A first theorem invokes a well-known assumption on the set of zeros of the switching function. A second relies on coercivity of the second derivative of the reduced objective functional. The results are applied to the convergence of optimal state functions for a finite element discretizion of the control problem. [ABSTRACT FROM AUTHOR]

We consider a problem of optimal distribution of conductivities in a system governed by a nonlocal diffusion law. The problem stems from applications in optimal design and more specifically topology optimization. We propose a novel parametrization of nonlocal material properties. With this parametrization the nonlocal diffusion law in the limit of vanishing nonlocal interaction horizons converges to the famous and ubiquitously used generalized Laplacian with SIMP (solid isotropic material with penalization) material model. The optimal control problem for the limiting local model is typically ill-posed and does not attain its infimum without additional regularization. Surprisingly, its nonlocal counterpart attains its global minima in many practical situations, as we demonstrate in this work. In spite of this qualitatively different behavior, we are able to partially characterize the relationship between the nonlocal and the local optimal control problems. We also complement our theoretical findings with numerical examples, which illustrate the viability of our approach to optimal design practitioners. [ABSTRACT FROM AUTHOR]

A periodical cycle of body deformation is a common strategy for locomotion (see, for instance, birds, fishes, humans). The aim of this paper is to establish that the autopropulsion of deformable objects is optimally achieved using periodic strategies of body deformations. This property is proved for a simple model using an optimal control theory framework. [ABSTRACT FROM AUTHOR]

In stochastic control applications, typically only an ideal model (controlled transition kernel) is assumed and the control design is based on the given model, raising the problem of performance loss due to the mismatch between the assumed model and the actual model. Toward this end, we study continuity properties of discrete-time stochastic control problems with respect to system models (i.e., controlled transition kernels) and robustness of optimal control policies designed for incorrect models applied to the true system. We study both fully observed and partially observed setups under an infinite horizon discounted expected cost criterion. We show that continuity can be established under total variation convergence of the transition kernels under mild assumptions and with further restrictions on the dynamics and observation model under weak and setwise convergence of the transition kernels. Using these continuity properties, we establish convergence results and error bounds due to mismatch that occurs by the application of a control policy which is designed for an incorrectly estimated system model to a true model, thus establishing positive and negative results on robustness. Compared to the existing literature, we obtain strictly refined robustness results that are applicable even when the incorrect models can be investigated under weak convergence and setwise convergence criteria (with respect to a true model), in addition to the total variation criteria. These entail positive implications on empirical learning in (data-driven) stochastic control since often system models are learned through empirical training data where typically a weak convergence criterion applies but stronger convergence criteria do not. [ABSTRACT FROM AUTHOR]

The optimal control problem governed by the stochastic reflection problem associated with a closed convex set K in Rd is reduced via the corresponding Kolmogorov equation to a deterministic bilinear parabolic optimal control problem on (0, T)Ã--K. In this way, one gets directly a stochastic optimal feedback controller by avoiding the standard dynamic programming equation associated with the stochastic optimal control problem. [ABSTRACT FROM AUTHOR]

We study linear-quadratic optimal control problems with time-inconsistent cost functionals for stochastic evolution equations in a Hilbert space. A closed-loop equilibrium strategy is introduced for these control problems. By means of multiperson differential games, we prove the existence of such a strategy. [ABSTRACT FROM AUTHOR]

This paper establishes a convergence result for implicit Euler discretizations of optimal control problems with differential-algebraic equations of higher index and mixed controlstate constraints. The main difficulty of the analysis is caused by a structural discrepancy between the necessary conditions of the continuous optimal control problem and the necessary conditions of the discretized problems. This discrepancy does not allow one to compare the respective necessary conditions directly. We use an equivalent reformulation of the discretized problems to overcome this discrepancy and to prove first order convergence of the discretized states, algebraic states, controls, and multipliers of the reformulation. [ABSTRACT FROM AUTHOR]

We study an optimal distributed control problem associated to a stochastic Cahn--Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of a relaxed optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analyzed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved. [ABSTRACT FROM AUTHOR]

We analyze the sensitivity of the extremal equations that arise from the first order optimality conditions for time dependent optimization problems. More specifically, we consider parabolic PDEs with distributed or boundary control and a linear quadratic performance criterion. We prove the solution's boundedness with respect to the right-hand side of the first order optimality condition which includes initial data. If the system fulfills a particular stabilizability and detectability assumption, the bound is independent of the time horizon. As a consequence, the influence of a perturbation of the right-hand side decreases exponentially backward in time. We use this property for the construction of efficient numerical discretizations in a model predictive control scheme. Moreover, a quantitative turnpike theorem in the W([0,T])-norm is derived. [ABSTRACT FROM AUTHOR]

We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first-order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach, where the control is not discretized---the so-called variational discretization approach; and a fully discrete approach, where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with two-dimensional numerical tests. [ABSTRACT FROM AUTHOR]

We study some optimal control problems on networks with junctions, approximate the junctions by a switching rule of delay-relay type, and study the passage to the limit when ", the parameter of the approximation, goes to zero. First, for a twofold junction problem we characterize the limit value function as a viscosity solution and maximal subsolution of a suitable Hamilton{ Jacobi problem. Then, for a threefold junction problem we consider two different approximations, recovering in both cases some uniqueness results in the sense of a maximal subsolution. [ABSTRACT FROM AUTHOR]

We establish a variety of results extending the well-known Pontryagin maximum principle of optimal control to discrete-time optimal control problems posed on smooth manifolds. These results are organized around a new theorem on critical and approximate critical points for discrete-time geometric control systems. We show that this theorem can be used to derive Lie group variational integrators in Hamiltonian form; to establish a maximum principle for control problems in the absence of state constraints; and to provide sufficient conditions for exact penalization techniques in the presence of state or mixed constraints. Exact penalization techniques are used to study sensitivity of the optimal value function to constraint perturbations and to prove necessary conditions for optimality, including in the form of a maximum principle, for discrete-time geometric control problems with state or mixed constraints. [ABSTRACT FROM AUTHOR]

In this work we revisit the linear-quadratic optimal control problem for differential-algebraic systems on the infinite time horizon with zero terminal state. Based on the recently developed Lur'e equation for differential-algebraic equations we obtain new equivalent conditions for feasibility. These are related to the existence of a stabilizing solution of the Lur'e equation. This approach also allows us to determine optimal controls if they exist. In particular, we can characterize regularity of the optimal control problem. The latter refers to existence and uniqueness of optimal controls for any consistent initial condition. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem of tracking type governed by a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process is formulated by the corresponding Euler–Lagrange equations that contain a regularization term that penalizes the violation of the irreversibility condition in the evolution of the fracture. We prove convergence of solutions of the regularized problem when taking the limit with respect to the penalty term and obtain an estimate for the constraint violation in terms of the penalty parameter. To this end, we make use of convexity of the energy functional due to a viscous regularization which corresponds to a time-step restriction in the temporal discretization of the problem. Numerical experiments underline our theoretical findings. [ABSTRACT FROM AUTHOR]

We investigate the hard-thresholding method applied to optimal control problems with L0(Ω) control cost, which penalizes the measure of the support of the control. As the underlying measure space is nonatomic, arguments of convergence proofs in l² or Rn cannot be applied. Nevertheless, we prove the surprising property that the values of the objective functional are lower semicontinuous along the iterates. That is, the function value in a weak limit point is less than or equal to the lim-inf of the function values along the iterates. Under a compactness assumption, we can prove that weak limit points are strong limit points, which enables us to prove certain stationarity conditions for the limit points. Numerical experiments are carried out, which show the performance of the method. These indicate that the method is robust with respect to discretization. In addition, we show that solutions obtained by the thresholding algorithm are superior to solutions of L¹(Ω)-regularized problems. [ABSTRACT FROM AUTHOR]

Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin maximum principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions which are essentially sharp, Pontryagin extremals depend continuously on the parameters delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy; however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint vector with respect to the parameter delays opens a new perspective for the numerical implementation of indirect methods, such as the shooting method. [ABSTRACT FROM AUTHOR]

We consider a control constrained nonlinear optimal control problem under perturbations represented by a parameter which is a function of time. Our main assumptions involve smoothness of the functions appearing in the integral functional and the state equations, an integral coercivity condition, and a condition that the reference optimal control is an isolated solution of the variational inequality for the control appearing in the maximum principle. We also consider a corresponding discrete-time optimal control problem obtained from the continuous-time one by applying the Euler finite-difference scheme. Based on an enhanced version of Robinson's implicit function theorem, we establish that there exists a natural number \= N such that if the number N of the grid points is greater than \= N, then the solution mapping of the discrete-time problem has a Lipschitz continuous single-valued localization with respect to the parameter, whose Lipschitz constant and the sizes of the neighborhoods involved do not depend on N. As an application, we show that the Newton/SQP method converges uniformly with respect to the step-size of the discretization and small changes of the parameter. Numerical experiments with a satellite optimal control problem illustrate the convergence result. [ABSTRACT FROM AUTHOR]

We study sparse solutions of optimal control problems governed by PDEs with uncertain coefficients. We propose two formulations, one where the solution is a deterministic control optimizing the mean objective, and a formulation aiming at stochastic controls that share the same sparsity structure. In both formulations, regions where the controls do not vanish can be interpreted as optimal locations for placing control devices. In this paper, we focus on linear PDEs with linearly entering uncertain parameters. Under these assumptions, the deterministic formulation reduces to a problem with known structure, and thus we mainly focus on the stochastic control formulation. Here, shared sparsity is achieved by incorporating the L1-norm of the mean of the pointwise squared controls in the objective. We reformulate the problem using a norm reweighting function that is defined over physical space only and thus helps to avoid approximation of the random space using samples or quadrature. We show that a fixed point algorithm applied to the norm reweighting formulation leads to a variant of the well-studied iterative reweighted least squares (IRLS) algorithm, and we propose a novel preconditioned Newton-conjugate gradient method to speed up the IRLS algorithm. We combine our algorithms with low-rank operator approximations, for which we provide estimates of the truncation error. We carefully examine the computational complexity of the resulting algorithms. The sparsity structure of the optimal controls and the performance of the solution algorithms are studied numerically using control problems governed by the Laplace and Helmholtz equations. In these experiments the Newton variant clearly outperforms the IRLS method. [ABSTRACT FROM AUTHOR]

We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval (0, T) that depends on the boundary traces of the solution. If the time horizon T is sufficiently large, the solution of the dynamic optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for T → ∞ the approximation error converges to zero in the sense of the norm in L2(0, 1) with the rate 1/T, if the time interval (0, T) is transformed to the fixed interval (0, 1). Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If T is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of sufficiently regular representations of Hamilton{Jacobi equations in optimal control theory with the compact control set. We introduce a new method to construct representations for a wide class of Hamiltonians, wider than was achieved before. Our result is proved by means of these conditions on the Hamiltonian that are necessary for the existence of a representation. In particular, we solve an open problem of Rampazzo [SIAM J. Control Optim., 44 (2005), pp. 867{884]. We apply the obtained results to reduce a variational problem to an optimal control problem. [ABSTRACT FROM AUTHOR]

We consider bilinear optimal control problems whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient secondorder conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in L1. In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization and prove error estimates of bang-bang controls in L1-norms. [ABSTRACT FROM AUTHOR]

In this paper, we study second-order necessary and sufficient optimality conditions of Karush{Kuhn{Tucker type for local optimal solutions in the sense of Pareto to a class of multiobjec-tive optimal control problems with mixed pointwise constraints. To deal with the problems, we first derive second-order optimality conditions for abstract multiobjective optimal control problems which satisfy the Robinson constraint qualification. We then apply the obtained results to our concrete problems. The proofs of obtained results are direct and self-contained without using scalarization techniques. [ABSTRACT FROM AUTHOR]

The aim of this paper is to consider a sensitivity analysis of optimal control problems for a class of systems governed by differential hemivariational inequalities (DHVIs) on Banach spaces. The first one is to investigate the nonemptiness as well as the compactness of the mild solutions set S(ζ) (ζ being the initial condition) for DHVIs and also to present an extension of Filippov's theorem, whose proof differs from previous work only in some technical details. The second aim is to study the optimal control problems described by DHVIs depending on the initial condition ζ and some further parameter η and establish the sensitivity properties of the optimal control problems for DHVIs. Finally, a mathematical model is provided to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order Hamilton--Jacobi--Bellman equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity, and stability of the scheme. These properties are verified for a specific semi-Lagrangian scheme. [ABSTRACT FROM AUTHOR]

An infinite-horizon optimal control problem subject to an infinite-dimensional state equation with state and control variables appearing in a bilinear form is investigated. A sensitivity analysis with respect to the initial condition is carried out. We show in particular that the value function is infinitely differentiable in the neighborhood of the steady state, under a stabilizability assumption. In a second part, we derive error estimates for controls generated by polynomial feedback laws, which are derived from Taylor expansions of the value function. [ABSTRACT FROM AUTHOR]

This paper studies the impulse control of a general Markov process under the average (or ergodic) cost when the impulse instants are restricted to be the arrival times of an exogenous process, and this restriction is referred to as a constraint. A detailed setting is described, a characterization of the optimal cost is obtained as a solution of an HJB equation, and an optimal impulse control is identified. [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]

We consider a class of (ill-posed) optimal control problems in which a distributed vector-valued control is enforced to take values pointwise in a finite set M ⊂ Rm. After convex relaxation, one obtains a well-posed optimization problem, which still promotes control values in M. We state the corresponding well-posedness and stability analysis and exemplify the results for two specific cases of quite general interest, optimal control of the Bloch equation and optimal control of an elastic deformation. We finally formulate a semismooth Newton method to numerically solve a regularized version of the optimal control problem and illustrate the behavior of the approach for our example cases. [ABSTRACT FROM AUTHOR]

This paper is devoted to second-order necessary optimality conditions for strong local minima for a Mayer type optimal control problem with a general control constraint U ⊂ Rm, and state and final-point constraints described by a finite number of inequalities. We use the second-order linearization of a relaxed differential inclusion associated to the control system to find a convex subset of second-order tangents to the set of its trajectories. This leads to second-order necessary optimality conditions via a straightforward way, based on separation theorems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents an additional difficulty for the analysis. The nonlinearities in the bulk and surface free energies are of logarithmic type, which entails that the thermodynamic forces driving the phase separation process may become singular. We show existence for the optimal control problem under investigation, prove the Fréchet differentiability of the associated control-to-state mapping in suitable Banach spaces, and derive the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system. Due to the strong nonlinear couplings between state variables and control, the corresponding proofs require a considerable analytical effort. [ABSTRACT FROM AUTHOR]

We consider an infinite horizon optimal control problem for a continuous-time Markov chain X in a finite set I with noise-free partial observation. The observation process is defined as Yt = h(Xt), t ≥ 0, where h is a given map defined on I. The observation is noise-free in the sense that the only source of randomness is the process X itself. The aim is to minimize a discounted cost functional and study the associated value function V . After transforming the control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we provide a link between the value function v associated with the latter control problem and the original value function V . Then, we present two different characterizations of v (and indirectly of V ): on one hand as the unique fixed point of a suitably defined contraction mapping and on the other hand as the unique constrained viscosity solution (in the sense of Soner) of a HJB integro-differential equation. Under suitable assumptions, we finally prove the existence of an optimal control. [ABSTRACT FROM AUTHOR]

This paper addresses an open problem in the area of linear quadratic optimal control. We consider the regular, infinite-horizon, stability-modulo-a-subspace, indefinite linear quadratic problem under the assumption that the dynamics are stabilizable. Our result generalizes previous works dealing with the same problem in the case of controllable dynamics. We explicitly characterize the unique solution of the algebraic Riccati equation that gives the optimal cost and optimal feedback control, as well as necessary and sufficient conditions for the existence of optimal controls. [ABSTRACT FROM AUTHOR]

For the Borel model of the continuous-time Markov decision process, we introduce a wide class of control strategies. In a particular case, such strategies transform to the standard relaxed strategies, intensively studied in the last decade. In another special case, if one restricts to another special subclass of the general strategies, the model transforms to the semi-Markov decision process. Further, we show that the relaxed strategies are not realizable. For the constrained optimal control problem with total expected costs, we describe the suffic ient class of realizable strategies, the so-called Poisson-related strategies. Finally, we show that, for so lving the formulated optimal control problems, one can use all the tools developed earlier for the classical discrete-time Markov decision processes. [ABSTRACT FROM AUTHOR]

We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingaie approach developed in [A. M. G. Cox and S. Källblad, SIAM J. Control Optim., 55 (2017), pp. 3409-3436] allows us to obtain an equivalent infinite dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions. [ABSTRACT FROM AUTHOR]