Global Non-negative Approximate Controllability of Parabolic Equations with Singular Potentials

Abstract

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 \(L^\infty \). Besides the corresponding solution remains non-negative at all times.