Approximate controllability for linear degenerate parabolic problems with bilinear control

Abstract

In this work we study the global approximate multiplicative controllability for the linear degenerate parabolic Cauchy-Neumann problem $$\{arrayl{v}_t-(a(x)v_x{)}_x=\alpha (t,x)vin{Q}_T=(0,T)\times (-1,1)[2.5ex]a(x)v_x(t,x){|}_{x=\pm 1}=0t\in (0,T)[2.5ex]v(0,x)=v_0(x)x\in (-1,1),array.$$ with the bilinear control $\alpha (t,x)\in L^{\infty }(Q_T).$ The problem is strongly degenerate in the sense that $a\in C^1([-1,1]),$ positive on $(-1,1),$ is allowed to vanish at $\pm 1$ provided that a certain integrability condition is fulfilled. We will show that the above system can be steered in $L^2(\Omega )$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear static controls. Moreover, we extend the above result relaxing the sign constraint on $v_0$.