Abstract:

This paper studies the controllability and the existence of time optimal control for a class of nonlinear parabolic equations, which has drift-diffusion term with nonlinear terms $h(u,v)$ and $g(u,v)$. It applies the $L^{p}$-$L^{q}$ estimate of semigroups and the recently developed maximal regularity theory to study the regularity and the cost estimate of control functions. Moreover, this paper establishes the local exact controllability of the nonlinear control system by utilizing the Kakutani's fixed point theorem. Therefore, it is applied to the existence of time optimal controls.

Key words: Parabolic equations, Observability estimate, Exact controllability, Kakutani's fixed point