Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: a qualitative approach

We study the global "non-negative" approximate controllability property of a general semilinear heat equation with superlinear term, governed in a bounded domain by a multiplicative control in the reaction term. We show that any non-negative target state in L²(Ω) can approximately be reached from any non-negative, nonzero initial state by applying at most three static bilinear L^∞(Ω)-controls subsequently in time. Our approach is based on an asymptotic technique allowing us to distinguish and make use of the pure diffusion and/or pure reaction parts of the dynamics of the system at hand, while suppressing the effect of nonlinear term.