An estimation of a kind of semilinear parabolic system for its time optimal control problem

Abstract

An observation estimation for a kind of semilinear parabolic system is considered in this paper, which is used to solve the time optimal control problem of the parabolic system indirectly for the functional <i>f</i> is globally Lipschitz continuous such that the controllability of the semilinear parabolic system (1) is equal to the null controllability of its linearized system (3). Therefore, the semilinear parabolic system firstly needs to be linearized according to the property of the functional <i>f</i>. Secondly, to get the main estimation of the linearized system (3), two lemmas about the linearized system (4) are listed, which are used to prove the main estimation which will be used to show the null controllability of linearize system (3). At last, the observation estimate for the linearized system (3) is shown by the two lemmas and several classical inequalities.