Null-exact Controllability of a Semilinear Cascade System of Parabolic-hyperbolic Equations

This paper is concerned with the null-exact controllability of a cascade system formed by a semilinear heat and a semilinear wave equation in a cylinder Ω×(0, T ). More precisely, we intend to drive the solution of the heat equation (resp. the wave equation) exactly to zero (resp. exactly to a prescribed but arbitrary final state). The control acts only on the heat equation and is supported by a set of the form ω× (0, T ), where ω ⊂ Ω. In the wave equation, the restriction of the solution to the heat equation to another set O × (0, T ) appears. The nonlinear terms are assumed to be globally Lipschitz-continuous. In the main result in this paper, we show that, under appropriate assumptions on T , ω and O, the equations are simultaneously controllable.