Global Nonnegative Controllability of the 1-D Semilinear Parabolic Equation

Abstract

In this chapter we study the global approximate controllability of the one dimensional semilinear convection-diffusion-reaction equation governed in a bounded domain via a coefficient in the reaction term. Even in the linear case, due to the maximum principle, such system is not globally or locally controllable in any reasonable linear space. It is also well known that for the superlinear terms admitting a power growth at infinity the global approximate controllability by traditional additive controls of localized support is out of question. However, we will show that a system like that can be steered in L²(0,1) from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static (x-dependent only) multiplicative controls, applied subsequently in time, while only one such control is needed in the linear case.