Multiplicative Controllability of the Semilinear Parabolic Equation: A Qualitative Approach

Abstract

In this chapter we establish the global non-negative approximate controllability property for a rather general semilinear heat equation with superlinear term, governed in a bounded domain Ω ⊂ Rⁿ by a multiplicative control in the reaction term like vu(x, t), where v is the control. 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. This result is further applied to discuss the controllability properties of the nonhomogeneous version of this problem with bilinear term like v(u(x, t).θ (x)), where θ is given. 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 a nonlinear term.