Calculus of Variations and Geometric Measure Theory

P. Cannarsa - G. Floridia

Approximate controllability for linear degenerate parabolic problems with bilinear control

created by floridia on 12 Dec 2017



Inserted: 12 dec 2017

Year: 2011

ArXiv: 1106.4232 PDF


In this work we study the global approximate multiplicative controllability for the linear degenerate parabolic Cauchy-Neumann problem $$ \{{array}{l} \displaystyle{vt-(a(x) vx)x =\alpha (t,x)v\,\,\qquad {in} \qquad QT \,=\,(0,T)\times(-1,1)} 2.5ex \displaystyle{a(x)vx(t,x)
{x=\pm 1} = 0\,\,\qquad\qquad\qquad\,\, t\in (0,T)} 2.5ex \displaystyle{v(0,x)=v0 (x) \,\qquad\qquad\qquad\qquad\quad\,\, x\in (-1,1)}, {array}. $$ 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 $v_0$.