Time optimal control of semilinear parabolic equations via bilinear controls

We establish the existence of the time optimal control for semilinear parabolic equations with gradient quadratic growth via bilinear controls. It is worth pointing out that there is no restriction on the growth of the nonlinearity f(s)f(s) with respect to the variable ss in the equation, which is a remarkable difference compared to the semilinear parabolic system with locally distributed controls. The technique used in this paper is the combination of the Hopf–Cole transformation, the a prior estimates on solutions of parabolic equations and the strategy of the stepwise control.