Optimal Control of Nonsmooth Distributed Parameter Systems, Issue 1459The book is devoted to the study of distributed control problems governed by various nonsmooth state systems. The main questions investigated include: existence of optimal pairs, first order optimality conditions, state-constrained systems, approximation and discretization, bang-bang and regularity properties for optimal control. In order to give the reader a better overview of the domain, several sections deal with topics that do not enter directly into the announced subject: boundary control, delay differential equations. In a subject still actively developing, the methods can be more important than the results and these include: adapted penalization techniques, the singular control systems approach, the variational inequality method, the Ekeland variational principle. Some prerequisites relating to convex analysis, nonlinear operators and partial differential equations are collected in the first chapter or are supplied appropriately in the text. The monograph is intended for graduate students and for researchers interested in this area of mathematics. |
Contents
INTRODUCTION | 1 |
SEMILINEAR EQUATION | 29 |
VARIATIONAL INEQUALITIES | 77 |
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adjoint admissible pair approximate problem assume Banach space bounded in L²(Q Ch.I coercive compact control constraints control problems governed convergence convex function cost functional deduce define denote dom(A equation equivalent existence Gateaux differentiable given grad Hilbert space hypothesis implies indicator function integrate over 0,t J.L.Lions L²(E least one optimal Lemma limit limsup linear locally Lipschitzian lower semicontinuous mapping maximal monotone graph maximal monotone operator Moreover multiply nonlinear obtain Obviously optimal control optimal pair problem 3.1 Proposition regularity results Remark satisfies sequence Sobolev embedding theorem solution of 5.2 ß² ß³ Stefan problem strongly in C(0,T;H strongly in L²(Q subdifferential subsequence subset Theorem two-phase Stefan problems unique solution V.Barbu variational inequalities weak solution weak topology weakly in L²(0,T;H Xi Xi y₁ yields yn+1 Yosida approximation દ દ