Control of a Pseudo-Parabolic Initial-Value Problem to a Target Function
Luther W. White · SIAM Journal on Control and Optimization · 1979
Let G be a bounded domain in $R^n $ with a smooth boundary and let $Q = G \times (0,T]$ We consider the solution $y(u)$ of the pseudo-parabolic initial-value problem \[\begin{gathered}M(x)y_t (u) + L(x)y(u) = u\quad {\text{in }}L^2 (Q),\hfill \\ y( \cdot ,0;u) = 0\quad {\text{in }}L^2 (G) \hfill \\ \] to be the state corresponding to the control u. Here $M(x)$ and $L(x)$ are second order symmetric uniformly strongly elliptic operators on G. The control problem is to find a control $u_0 $ in a given ball in $L^2 (Q)$ such that, for a given Z, the trace $y( \cdot ,T;u) = Z( \cdot )$ is in $L^2 (G)$ and such that $u_0 $ minimizes a certain noncoercive energy functional arising naturally from the differential equation. In this paper we give controllability results for the pseudo-parabolic initial-value problem and regularity results for $u_0 $. Furthermore, we establish results that $u_0$ lies on the surface of the constraint ball in $L^2 (Q)$ and that the optimal controls of similar problems that steer to balls centered at Z converge to $u_0 $ in $L^2 (Q)$ as the target radii shrink to zero. The regularity results indicate that convergence in $L^2 (Q)$ is as strong as we may expect. Finally, we include a simple example to illustrate some of our results.