A Class of Multi-Objective Control Problems for Quasi-linear Parabolic Equations

This paper is devoted to studying a multi-objective control problem for a class of multi-dimensional quasi-linear parabolic equations. The considered system is driven by a leader control and two follower controls. For each leader control, a pair of follower controls is searched for as a Nash quasi-equilibrium (or Nash equilibrium) of cost functionals, while the aim for a leader control is to solve a controllability problem. This hierarchic control problem may be transformed into controllability of a strongly coupled system of quasi-linear parabolic equations through one control. Regarding controllability for quasi-linear parabolic equations of second order, the existing results usually require coefficients in principal parts to be independent of gradient of solutions, or spacial dimension to be limited. In this paper, the coefficients in principal parts for the controlled quasi-linear system contain not only the state itself but also its gradient with general spacial dimension.