*preprint*

**Inserted:** 12 dec 2017

**Year:** 2011

**Abstract:**

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