Variational Stability Analysis of Optimal Control Problems for Systems Governed by Nonlinea.pdf

c Journal of Mathematical Systems Estimation and Control BirkhauserBoston Vol No pp Variational Stability Analysis of Optimal Control Problems for Systems Governed by Nonlinear Second Order Evolution Equations Stanislaw Migorski Abstract The variational stability of optimal control problems governed by second order nonlinear evolution abstract equations is studied First we prove an existence theorem for optimal solutions Then admitting p erturbations in all the data of the control problem we show the results on the asymptotic b ehavior of optimal solutions to control problems as well as on the convergence of minimal values and reachable sets The notions of  convergence of functionals and the KuratowskiMosco convergence of sets are employed Finally an example of nonlinear hyp erb olic control problem demonstrates the applicability of the results Key words control problem  convergence KuratowskiMosco convergence monotone op erator compact emb edding hyp erb olic system AMS Sub ject Classications G J K J Intro duction and Notation In this pap er we investigate the optimal control problems governed by abstract second order evolution equations We consider a sequence of such problems indexed by the parameter n IN which app ears in all the data including the nonlinear op erator of the state equation the integrand of cost functional and the control constraint set The goal is to identify a limit problem which is obtained from the p erturb ed control problems as n tends to innity First we deal with a nonlinear second order evolution equation for which we show th