Control problems governed by a pseudo-parabolic partial differential equation

Abstract

Let G be a bounded domain in R n {R^n} and Q = G × ( 0 , T ) Q\, = \,G\, \times \,\left ( {0,\,T} \right ) . We consider the solution y ( u ) y\left ( u \right ) of the pseudo-parabolic initial-value problem ( 1 + M ( x ) ) y t ( u ) + L ( x ) y ( u ) = u in L 2 ( Q ) , y ( ⋅ , 0 ; u ) = 0 in L 2 ( G ) , \begin{multline} \left ( {1\, + \,M\left ( x \right )} \right )\,{y_t}\,\left ( u \right )\, + \,L\left ( x \right )\,y\left ( u \right )\, = \,u\,{\text {in}}\,{L^2}\,\left ( Q \right ), \hfill \\ y\left ( { \cdot ,\,0;\,u} \right )\, = \,0\,{\text {in}}\,{L^2}\,\left ( G \right ), \hfill \\ \end{multline} , to be the state corresponding to the control u. Here M ( x ) M\left ( x \right ) and L ( x ) L\left ( x \right ) are symmetric uniformly strongly elliptic second-order partial differential operators. The control problem is to find a control u 0 {u_0} in a fixed ball in L 2 ( Q ) {L^2}\left ( Q \right ) such that (i) the endpoint of the corresponding state y ( ⋅ , T ; u 0 ) y\left ( { \cdot ,\,T;\,{u_0}} \right ) lies in a given neighborhood of a target Z in L 2 ( G ) {L^2}\left ( G \right ) and (ii) u 0 {u_0} minimizes a certain energy functional. In this paper we establish results concerning the controllability of the states and the compatibility of the constraints, existence and uniqueness of the optimal control, existence and properties of Lagrange multipliers associated with the constraints, and regularity properties of the optimal control.