Existence and Uniqueness of Optimal Controls for a Quasilinear Parabolic Equation
Thomas I. Seidman, Zhou Hong-xing · SIAM Journal on Control and Optimization · 1982
We consider the quasilinear parabolic equation $\dot y + {\bf A}y + {\bf F}(y) = \varphi $ on $Q: = (0,T) \times \Omega $. Viewing $\varphi $ as a control, we seek to minimize $J(\varphi ): = \| {\varphi - \hat \varphi } \|^2 + \lambda \left\| {y - \hat y} \|^2 + \mu \| {y(T) - \hat \eta } \|^2 $. Under suitable hypotheses it is shown that one has existence of an optimal control $\varphi _ * $ , and that this satisfies the appropriate optimality system. Further, for small data $J_ * : = \min J$ is small enough) one has global uniqueness and continuous dependence on the data.