Analysis and approximations of terminal-state tracking optimal control problems and controllability problems constrained by linear and semilinear parabolic partial differential equations

Abstract

Terminal-state tracking optimal control problems for linear and semilinear parabolic equations are studied. The control objective is to track a desired terminal state and the control is of the distributed type. A distinctive feature of this work is that the controlled state and the target state are allowed to have nonmatching boundary conditions. In the linear case, analytic solution formulae for the optimal control problems are derived in the form of eigen series. Pointwise-in-time L2 norm estimates for the optimal solutions are obtained and approximate controllability results are established. Exact controllability is shown when the target state and the controlled state have matching boundary conditions. One-dimensional computational results are presented which il­ lustrate the terminal-state tracking properties for the solutions expressed by the series formulae. In the semilinear case, the existence of an optimal control solution is shown. The dynamics of the optimal control solution is analyzed. Error estimates are obtained for semidiscrete (spatially discrete) approximations of the optimal control problem in two and three space dimensions. A gradient algorithm is discussed and numerical results are presented.