Optimal control in non well posed problem for Stokes equations

V. A. Annenkov

Dalʹnevostočnyj matematičeskij žurnal, Tome 4 (2003) no. 1, pp. 18-26

Résumé

This paper deals with the optimal control problem for Stokes system in bounded domain $\Omega$. The bound of $\Omega$ is a union of two smooth disjoint parts: $\partial{\Omega}=\Gamma_{0}{\cup}\Gamma_{1}$, $\Gamma_{0}\cap\Gamma_{1}=\emptyset$. Fluid velocity and stress vector on the part $\Gamma_{0}$ on the boundary simultaneously play a role of control parameters. So, we deal with a system governed by non well posed problem. Extremal condition on the part $\Gamma_{1}$ leads to the necessary a priory estimation for velocity. We use penalization tecnique and study convergence of penalized problem to original problem when penalization parameter tends to zero. The tecnique we use was developed by J.-L. Lions. One of possible applications of obtained result is obtaining the conditions for optimal state (singular optimality system).