Approximate null-controllability for parabolic variational inequality

O. NAKOULIMA,Jean VELIN

Université des Antilles et de La Guyane, France

We are interested in the study of the approximate null-controllability of a parabolic variational inequality with an obstacle denoted $\psi$:

$$\left\{ \begin{array}{rcll} y&\in &{\cal K}_\psi,\\ \int_Q\... ...l z\in {\cal V}, z\leq \psi \\ y(0)&=&y_0. \end{array}\right.$$ (1)

where $\Omega \subset\Bbb R^N$ is a bounded open whose boundary $\Gamma=\partial \Omega$ has $C^2$ regularity, $\omega$ is a nonempty open subset of $\Omega$, $Q=]0,T[\times\Omega ,$ $Q_\omega =]0,T[\times\omega ,$ ${\cal V}=L^2(0,T;H_0^1(\Omega )),$ ${\cal K}_\psi=\{y\in {\cal V};\,\, \displaystyle\frac {\partial y}{\partial t}\in {\cal V}^{\prime},\,\, y\leq \psi\}$ is a closed nonempty convex and $a$ is a continuous bilinear application on $H_0^1(\Omega )\times H_0^1(\Omega )$ defined by $a(y,z)=\int_\Omega \sum_{i,j=1}^N a_{i,j}\frac{\partial y} {\partial x_i}\frac{\partial z}{\partial x_i}dx+ \int_\Omega a_{0}yzdx,$ with $a_{i,j}\in C^2({\overline \Omega }),$ $a_0\in L^{\infty}(Q)$ satisfying elliptic conditons. After introducing a penalized term associated to the obstacle (cf [3]), we show that (1) can be formulated like a weak limit of a sequence of heat semi-linear problems. The approximate controllability theory is applied to this sequence, and by passing to the limit we prove that (1) satisfies approximate null-controllability condition in the sense that there exists an admissible control ${\hat v}\in L^2(Q_\omega )$ such that the corresponding solution ${\hat y}=y({\hat v})$ of (1) satisfies $\Vert{\hat y}(T)\Vert _{L^2(\Omega )}\leq \alpha$. To obtain our main theorem, we have combined some results on approximate controllability obtained by various authors (for instance [4], [1], [5]) for Laplace operators with results based on Carleman inequalities and obtained by Imanuvilov [2]for a class of problems involving second order differential operators with coefficients depending on spatial variables.