Homogenization of an Optimal Control Problem
S. Kesavan, Jeannine Saint Jean Paulin · SIAM Journal on Control and Optimization · 1997
We consider an optimal control problem in which both the state equation and the cost functional have rapidly oscillating coefficients (characterized respectively by matrices $A_\varepsilon$ and $B_\varepsilon$, where $\varepsilon$ is a small parameter). We make no periodicity assumption. We study the limit of the problem when $\varepsilon \rightarrow 0$ and work in the framework of H-convergence. We prove that the limit satisfies a problem similar to the original one but with matrices $A_0$ (the H-limit of $A_\varepsilon$) and $B^\sharp$ (which is a perturbation of the H-limit $B_0$ of $B_\varepsilon$). We also study some particular cases. This paper extends former results obtained by Kesavan and Vanninathan in the periodic case.