Existence of optimal controls for second-order nonlinear evolution equations.

Abstract

In this thesis we study the question of existence of optimal controls for systems governed by second order nonlinear evolution equations. Let I = ($0,\ T$), $(X, H, X\sp\*)$ be an evolution triple, with compact embedding $X\to H\to X\sp\*$ and Y a separable, reflexive Banach space, modeling the control space. Here $X\sp\*$ denote the dual of the Banach space X. Let $t\to U(t)$ be a measurable set-valued map with values $U(t)\in 2\sp{Y}.$ For admissible controls, we introduce the class ${\cal U}\sb{ad}$ given by $U\sb{ad}\equiv\{ u: I\mapsto Y$, strongly measurable, and $u(t)\in U(t) a.e.\}$. We consider the following Lagrange type optimal control problem: $$\left\{\eqalign{&J(x,u) = f\sbsp{0}{T} L(t,x (t), \dot x(t), u(t))dt\ \to\inf\cr&subject\ to\ the\ following\ state\ and\ control\ constraints:\cr&\ddot x(t) + A(t,\dot x(t) + Bx(t)) = f(t,x(t))u(t),\cr& x(0) =x\sb0\in X, \dot x(0) = x\sb1\in H, u(t)\in U(t)\ a.e.\cr}\right\}(P)$$ To establish the existence of an optimal pair $\{$x,u$\}$ for the problem (P), an appropriate hypotheses on the data have been introduced and some apriori bounds for the admissible trajectories of (P) have been derived.