Variational Analysis in Nonsmooth Optimization and Discrete Optimal Control

The paper is devoted to applications of modern methods of variational· analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main attention is paid to the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and. operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the lack of compactness in infinite dimensions, which leads to imposing certain normal compactness properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions

Optimal Control of Delay-Differential Inclusions with Multivalued Initial Conditions in Infinite Dimensions

This paper is devoted to the study of a general class of optimal control problems described by delay-differential inclusions with infinite-dimensional state spaces, endpoints constraints, and multivalued initial conditions. To the best of our knowledge, problems of this type have not been considered in the literature, except some particular cases when either the state space is finite-dimensional or there is no delay in the dynamics. We develop the method of discrete approximations to derive necessary optimality conditions in the extended Euler-Lagrange form by using advanced tools of variational analysis and generalized differentiation in infinite dimensions. This method consists of the three major parts: (a) constructing a well-posed sequence of discrete-time problems that approximate in an appropriate sense the original continuous-time problem of dynamic optimization; (b) deriving necessary optimality conditions for the approximating discrete-time problems by reducing them to infinite-dimensional problems of mathematical programming and employing then generalized differential calculus; (c) passing finally to the limit in the obtained results for discrete approximations to establish necessary conditions for the given optimal solutions to the original problem. This method is fully realized in the delay-differential systems under consideration

Maximum Principle for Boundary Control Problems Arising in Optimal Investment with Vintage Capital

The paper concerns the study of the Pontryagin Maximum Principle for an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. The optimal control model has already been studied both in finite and infinite horizon with Dynamic Programming methods in a series of papers by the same author et al. [26, 27, 28, 29, 30]. Necessary and sufficient optimality conditions for open loop controls are established. Moreover the co-state variable is shown to coincide with the spatial gradient of the value function evaluated along the trajectory of the system, creating a parallel between Maximum Principle and Dynamic Programming. The abstract model applies, as recalled in one of the first sections, to optimal investment with vintage capital.Linear convex control, Boundary control, Hamilton–Jacobi–Bellman equations, Optimal investment problems, Vintage capital

Optimal Control of Semilinear Evolution Inclusions via Discrete Approximations

This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for constrained continuous-time evolution inclusions by passing to the limit from discrete approximations

On a Class of Infinite-Dimensional Singular Stochastic Control Problems

Federico S, Ferrari G, Riedel F, Röckner M. On a Class of Infinite-Dimensional Singular Stochastic Control Problems. Center for Mathematical Economics Working Papers. Vol 614. Bielefeld: Center for Mathematical Economics; 2019.We study a class of infinite-dimensional singular stochastic control problems with applications in economic theory and finance. The control process linearly affects an abstract evolution equation on a suitable partially-ordered infinite-dimensional space X, it takes values in the positive cone of X, and it has right-continuous and nondecreasing paths. We first provide a rigorous formulation of the problem by properly defining the controlled dynamics and integrals with respect to the control process. We then exploit the concave structure of our problem and derive necessary and sufficient first-order conditions for optimality. The latter are finally exploited in a specification of the model where we find an explicit expression of the optimal control. The techniques used are those of semigroup theory, vector-valued integration, convex analysis, and general theory of stochastic processes

On Infinite Dimensional Linear-Quadratic Problem with Fixed Endpoints. Continuity Question

In a Hilbert space setting, necessary and sufficient conditions for the minimum norm solution u to the equation Su = Rz to be continuously dependent on z are given. These conditions are used to study continuity of the minimum energy and linear-quadratic control problems for infinite dimensional linear systems with fixed endpoints

Nonlinear differential equations of Riccati type on ordered Banach spaces

In this paper we consider a general class of time-varying nonlinear differential equations on infinite dimensional ordered Banach spaces, which includes as special cases many known differential Riccati equations of optimal control. Using a linear matrix inequalities (LMIs) approach we provide necessary and sufficient conditions for the existence of some global solutions such as maximal, stabilizing and minimal solutions for this class of generalized Riccati equations. The obtained results extend to infinite dimensions and unify corresponding results in the literature. They provide useful tools for solving infinite-time linear quadratic (LQ) control problems for linear differential systems affected by countably-infinite-state Markovian jumps and/or multiplicative noise

Variational Analysis of Evolution Inclusions

The paper is devoted to optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities. with generally nonsmooth functions. We develop a variational analysis of such roblems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of Asplund (and hence any reflexive Banach) spaces. In this way we establish extended results on stability of discrete approximations (with the strong W^1,2-convergence of optimal solutions under consistent perturbations of endpoint constraints) and derive necessary optimality conditions for nonconvex discrete-time and continuous-time systems in the refined Euler-Lagrange and Weierstrass-Pontryagin forms accompanied by the appropriate transversality inclusions. In contrast to the case of geometric endpoint constraints in infinite dimensions, the necessary optimality conditions obtained in this paper do not impose any nonempty interiority /finite codimension/normal compactness assumptions. The approach and results developed in the paper make a bridge between optimal control/dynamic optimization and constrained mathematical programming problems in infinite-dimensional spaces