Multiplicative controllability for semilinear reaction–diffusion equations with finitely many changes of sign

We study the global approximate controllability properties of a one-dimensional semilinear reaction–diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state, with as many changes of sign in the same order as the given initial data, can be approximately reached in the L2(0, 1)-norm at some time T>0. Our method employs shifting the points of sign change by making use of a finite sequence of initial-value pure diffusion problems

GLOBAL NON-NEGATIVE APPROXIMATE CONTROLLABILITY OF PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

In this work, we consider the linear 1 − d heat equation with some singular potential (typically the so-called inverse square potential). We investigate the global approximate controllability via a multiplicative (or bilinear) control. Provided that the singular potential is not super-critical, we prove that any non-zero and non-negative initial state in L 2 can be steered into any neighborhood of any non-negative target in L 2 using some static bilinear control in L ∞. Besides the corresponding solution remains non-negative at all times