Control Problems In Hilbert Spaces Research Articles

Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

Stochastic Control Problems with Unbounded Control Operators: Solutions Through Generalized Derivatives

This paper deals with a family of stochastic control problems in Hilbert spaces which arises in typical applications (such as boundary control and control of delay equations with delay in the control) and for which is difficult to apply the dynamic programming approach due to the unboudedness of the control operator and to the lack of regularity of the underlying transition semigroup. We introduce a specific concept of partial derivative, designed for this situation, and we develop a method to prove that the associated HJB equation has a solution with enough regularity to find optimal controls in feedback form.

Stabilization of partial differential equations by sequential action control

Abstract We present a framework of sequential action control (SAC) for stabilization of systems of partial differential equations which can be posed as abstract semilinear control problems in Hilbert spaces. We follow a late-lumping approach and show that the control action can be explicitly obtained from variational principles using adjoint information. Moreover, we analyse the closed-loop system obtained from the SAC feedback for the linear problem with quadratic stage costs. We apply this theory to a prototypical example of an unstable heat equation and provide numerical results as the verification and demonstration of the framework.

Approximate controllability results for Sobolev‐type delay differential system of fractional order without uniqueness

AbstractThe main aim of this article is to focus on approximate controllability for Sobolev‐type fractional control problems in Hilbert spaces without uniqueness. By using the fixed point theorem for multivalued maps with nonconvex values, the main results of our article are proved. Moreover, we obtain some results on the continuity of the solution map and the topological structure of the solution set of the considered Sobolev‐type fractional differential system. Finally, we provide a theoretical application to assist in the effectiveness of our discussion.

Verification theorems for stochastic optimal control problems in Hilbert spaces by means of a generalized Dynkin formula

Verification theorems are key results to successfully employ the dynamic programming approach to optimal control problems. In this paper, we introduce a new method to prove verification theorems for infinite dimensional stochastic optimal control problems. The method applies in the case of additively controlled Ornstein–Uhlenbeck processes, when the associated Hamilton–Jacobi–Bellman (HJB) equation admits a mild solution (in the sense of [J. Differential Equations 262 (2017) 3343–3389]). The main methodological novelty of our result relies on the fact that it is not needed to prove, as in previous literature (see, e.g., [Comm. Partial Differential Equations 20 (1995) 775–826]), that the mild solution is a strong solution, that is, a suitable limit of classical solutions of approximating HJB equations. To achieve the goal, we prove a new type of Dynkin formula, which is the key tool for the proof of our main result.

Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces

In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reflects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking terms.

The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

We consider the stochastic linear quadratic optimal control problem for state equations of the Ito-Skorokhod type, where the dynamics are driven by strongly continuous semigroup. We provide a numerical framework for solving the control problem using a polynomial chaos expansion approach in white noise setting. After applying polynomial chaos expansion to the state equation, we obtain a system of infinitely many deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of deterministic linear quadratic regulator problems. Solving these control problems, we find optimal coefficients for the state and the control. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to a direct approach. Moreover, we apply our result to a fully stochastic problem, in which the state, control and observation operators can be random, and we also consider an extension to state equations with memory noise.

Regular Solutions of First-Order Hamilton?Jacobi Equations for Boundary Control Problems and Applications to Economics

This is the first of two papers regarding a family of linear convex control problems in Hilbert spaces and the related Hamilton–Jacobi–Bellman equations. The framework is motivated by an application to boundary control of a PDE modeling investments with vintage capital. Existence and uniqueness of a strong solution (namely, the limit of classic solutions of approximating equations, introduced by Barbu and Da Prato) is investigated. Moreover, such a solution is proved to be C1 in the space variable.

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

ON THE DYNAMIC PROGRAMMING APPROACH FOR OPTIMAL CONTROL PROBLEMS OF PDE'S WITH AGE STRUCTURE

A survey and some new results are presented concerning the dynamic programming for a class of optimal control problems of partial differential equations with age-structure and of delay systems that include some applied examples from economic theory and from population dynamics. A general optimal control problem in Hilbert spaces applying to all examples is investigated, with particular stress on one family of applications: optimal investment models with vintage capital. Some new results are given for the case of constrained investments, including a study of the properties of the optimal trajectories.

A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes

We consider a nonlinear controlled stochastic evolution equation in a Hilbert space, with a Wiener process affecting the control, assuming Lipschitz conditions on the coefficients. We take a cost functional quadratic in the control term, but otherwise with general coefficients that may even take infinite values. Under a mild finiteness condition, and after appropriate formulation, we prove existence and uniqueness of the optimal control. We construct the optimal feedback law by means of an associated backward stochastic differential equation. In this Hilbert space setting we are able to treat some state constraints and in some cases to recover conditioned processes as optimal trajectories of appropriate optimal control problems. Applications to optimal control of stochastic partial differential equations are also given.

A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes

We consider a nonlinear controlled stochastic evolution equation in a Hilbert space, with a Wiener process affecting the control, assuming Lipschitz conditions on the coefficients. We take a cost functional quadratic in the control term, but otherwise with general coefficients that may even take infinite values. Under a mild finiteness condition, and after appropriate formulation, we prove existence and uniqueness of the optimal control. We construct the optimal feedback law by means of an associated backward stochastic differential equation. In this Hilbert space setting we are able to treat some state constraints and in some cases to recover conditioned processes as optimal trajectories of appropriate optimal control problems. Applications to optimal control of stochastic partial differential equations are also given.

Updating Procedures For Iterative Learning Control In Hilbert Space

Abstract The research results presented here are extensions of previous work on Iterative Learning Control (ILC) using projection-based update techniques. These updates were mainly developed for Quasi-Newton optimization and also for solving systems of simultaneous equations. Recently, such updates have been used by the authors to estimate the dynamics of a control system which is asked to perform a periodic task using a discrete formulation. In addition, recently, preliminary extensions of these updating techniques to continuous systems through formulation in Hilbert Space have been proposed. This paper provides a formulation of the Iterative Learning Control problem in Hilbert Space with a convergence proof of the proposed solution based on Broyden’s update.

Linear quadratic problem with unbounded control in Hilbert spaces

A finite horizon linear quadratic optimal control problem in Hilbert space is considered, in which the controls are unbounded and the state weight operators are allowed to be indefinite and not pecessarily smoothing. It is proved that any optimal control and optimal trajectory are continuous. The closed-loop synthesis for optimal control is given via the solution to Fredholm integral equation and also the solution to Riccati integral equation, which exists under some mild conditions.

Sequential and Parallel Splitting Methods for Bilinear Control Problems in Hilbert Spaces

An augmented Lagrangian method with second-order update is developed, and its relationship to the sequential quadratic programming method is described. The rate of convergence proof depends on a second-order sufficient optimality condition, which is shown to be satisfied for a class of nonlinear optimal control problems of tracking type. Numerical examples are included which demonstrate the globalizing effect of the augmented Lagrangian method.

Approximation of H∞-control problem in hilbert spaces

We consider the approximation of a H∞-control problem, with boimded input and output. The approximating operators involved converge pointwise, except the disturbance operators Dh which converge in the uniform norm. We show that the existence of solution for the continuous H∞-control problem implies existence of solution for the approximating H∞ problem. We also prove pointwise convergence of the solutions to the Riccati equations associated to the approximating H∞ problem to the solution to the Riccati equations associated to the continuous problem. We show that the solution to the approximating H∞ problem solves in a weak sense the continuous problem. As an example we considered a control system governed by a first order hyperbolic equation.

An Approximation Theorem for the Algebraic Riccati Equation

For an infinite-dimensional linear quadratic control problem in Hilbert space, approximation of the solution of the algebraic Riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems. As an application, strong onvergence of the approximating Riccati operators in case of a previously developed spline approximation scheme for delay systems is established. Finally, convergence of the transfer-functions of the approximating systems is investigated.

Constrained least-squares solutions of linear inclusions and singular control problems in Hilbert spaces

LetH1 andH2 be Hilbert spaces and letN be an algebraic subspace ofH1. The least-squares problem for a linear relationLźH1źH2 restricted to an algebraic cosetS:=g+N, g ź H1, is considered. Various characterizations of a minimizer are derived in the form of inclusion relations that incorporate the constraints, and conditions under which a minimizer exists are developed. In particular, it is shown that generalized forms of the "normal equations" for constrained least-squares problems become "normal inclusions" that involve a multivalued adjoint. The setting includes large classes of constrained least-squares minimization problems and optimal control problems subject to generalized boundary conditions. An application of the abstract theory is given to a singular control problem involving ordinary differential equations with generalized boundary conditions, where the control may generate multiresponses. Characterizations of an optimal solution are developed in the form of inclusions that involve either an integrodifferential equation or a differential equation, and adjoint subspaces and/or solutions of certain linear equations.

Continuous dependence on data in abstract control problems

The coefficients of a plant of an abstract control problem are perturbed. It is shown that the characterizations given by Zolezzi for the abstract quadratic control problem in Hilbert spaces remain true for more general functions and spaces.

Optimality principle and synthesis for a stochastic control problem in hilbert spaces

We consider a linear system with additive noise in Hilbert space and minimize a convex functional associated with this process. A necessary and sufficient condition for a control to be optimal is derived by evaluating the subdifferential of the cost function. Then the subdifferential of the value function is characterized. Finally using these results and a conditional value function, optimal controls are characterized as a feedback law in terms of the value function.