Approximation Of Optimal Control Problems Research Articles

Approximation of optimal control problems governed by non-linear evolution equations

Abstract In this paper an optimal control problem involving a non-linear evolution equation and an approximating problem are formulated. Then the conditions of convergence are obtained An optimal control problem for systems governed by retarded fuctional differential equations, and its approximation by finite difference methods, are considered as an example.

Approximations for functionals and optimal control problems on jump diffusion processes

The paper treats approximations to stochastic differential equations with both a diffusion and a jump component, and to associated functionals and partial-differential-integral equations of the (degenerate or not) elliptic or parabolic type. Approximations for the optimal control problem on such a model, or for the associated nonlinear partial-differential-integral equation are discussed. The techniques are purely probabilistic and are extensions of those in [3], which dealt with the diffusion case.

The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solution

In the first part of this paper, we summarize and complete earlier results of Delfour-Mitter [3] on the optimal control problem for linear hereditary differential systems (HDS) with a linear-quadratic cost function. The properties of the operator Π(t) which characterizes the feedback gains and the reference functionr(t) are fully detailed. In a second part, we construct an approximation to the linear HDS in state form and to the linear adjoint state equation and prove convergence. In a third part we construct and solve the approximate optimal control problem following the method of J. C. Nedelec. In the last part we construct the approximation to Π(t) andr(t) and prove convergence. Finally we give a number of typical examples to illustrate the main features of the kernel of Π(t).

On the approximation of extremal problems. I

NECESSARY and sufficient conditions are obtained for the approximation of an extremal problem with respect to a functional, and their application to the investigation of the convergence of some methods for solving constrained extremal problems is described. Aspects of the regularization of incorrectly posed extremal problems are discussed. In the present paper, to be published in two parts, we discuss the construction of a sequence of extremal problems, approximating an initial extremal problem both in the sense of the optimal value of a functional, and in the sense of the set of elements realizing this value, together with the related question of the stability of extremal problems. In Part I, necessary and sufficient conditions are found for the approximation of the initial problem “with respect to a functional” (Section 1). The results obtained are then applied to the investigation of some constrained minimization methods. In particular, Section 2 deals with the Ritz method, and Section 3 with finite-difference approximations for optimal control problems. Taking A. N. Tikhonov's regularization method [1, 2] as starting-point, Section 4 discusses the construction of elements close to the set on which the optimal value of the functional is realized in the initial problem.

Approximations for optimal control problems described by partial differential equations

Approximations for optimal control problems described by partial differential equations

The convergence of difference approximations for optimal control problems

NUMERICAL methods mostly have to be used to solve optimal control problems, where we have to minimize a functional specified on the trajectories of a set of differential equations. The initial “differential” problem is usually replaced by a difference problem, and we have to consider whether the solution of the difference optimization problem is convergent to the solution of the differential problem. We discuss this convergence in the present paper for a wide class of optimal control problems. Apart from proving “convergence with respect to the functional”, we also consider regularization of the difference approximations in order to obtain a minimizing sequence of controls, strongly convergent in L 2. We shall assume that the computations involve errors. The results obtained are independent of the method of solving the difference optimization problem.

The equivalence and approximation of optimal control problems

Discontinuities occurring in closed loop optimal control problems when using maximum principle