Infinite dimensional parametric optimal control problems

Abstract

In this paper we study parametric optimal control problems monitored by nonlinear evolution equations. The parameter appears in all the data, including the nonlinear operator. First we show that for every value of the parameter, the optimal control problem has a solution. Then we study how these solutions as well as the value of the problem respond to changes in the parazneter. Finally, we work out in detail two examples of nonlinear parabolic optimal control systems.The purpose of this paper is to examine parametric infinite dimensional optimal control problems. The parameter appears in all the data of the problem, including the generally nonlinear operator modeling the partial differential operator of the problem. First we show that for each value of the parameter, the optimal control problem admits an optimal "state-control" pair. Then we investigate how the set of such optimal pairs, as well as the value of the problem, respond to variations of the parameter. Such a sensitivity analysis (also known in the literature as "variational analysis"), is important because it generates useful information about the tolerances that are permitted in the specification of mathematical models, it produces continuous dependence results and also can lead to robust computational schemes for the numerical treatment of the problem.Previous works on the subject primarily concentrated on finite dimensional systems, while the few in¡ dimensional works dealt only with linear or semilinear systems. For details we refer to Pieri [24], Stassinopoulos-Vinter [29] and Zolezzi [33] who considered finite dimensional control systems and to Benatti [4], Carja [7], Papageorgiou [19], [20], [21], [22] and Przyluski [25], who examined inŸ dimensional control systems. In particular, Benatti [4], Carja [7], Papageorgiou [20], [21] and Przyluski [25] investigated linear systems, using rather strong hypotheses on the data and their modes of convergence, while Papageorgiou [19], [22], studied

Optimal control for non‐linear integrodifferential functional equations

“…To prove the existence of admissible state-control pairs satisfying the constraints of (P) we apply the method of Theorem 1 of Reference [2] (or Theorem 3.1 in [9]) to our following result. …”

“…Let the operator A be given as a single valued operator, which is hemicontinuous and coercive from V to V n : Here V n stands for the dual space of V : The controller B is a bounded linear operator from some Banach space Y to H and the non-linear mapping g is Lipschitz continuous from R  V into H : When f ðÁ; xðÁÞÞ þ Bu 2 L 2 ð0; T ; V n Þ; it is well known as the quasiautonomous differential equation (see Theorem 2.6 in Barbu [1] In References [2][3][4] Aizicovici and Papageorgiou investigated parametric optimal control problems of non-linear evolution systems with slightly different conditions. The existence and uniqueness of solution of the system model consisting of a non-linear monotone operator with controls appearing linearly based on Galerkin approximation method has been studied by Ahmed and Xiang [5][6][7].…”

Approximate controllability and regularity for nonlinear differential equations

“…We refer to [2,4,5] to see the existence of solutions for a class of nonlinear evolution equations with monotone perturbations First, we begin with the existence, and a variational constant formula for solutions of the equation (SE) on L (0, T; V* ), which is also applicable to optimal control problem. We prove the existence and uniqueness for solution of the equation by converting the problem into a fixed point problem.…”

Nonlinear variational evolution inequalities in Hilbert spaces

“…Here V * stands for the dual space of V . 2) where the nonlinear mapping f is a Lipschitz continuous from R × V into H. Equation (1.2) is caused by the following nonlinear variational inequality problem:…”