Optimal controls for a class of nonlinear evolution systems

We consider the abstract nonlinear evolution equation $dot{z}+ Az =uBz +f$. Viewing $u$ as control, we seek to minimize $J(u)=int_{0}^{T}L(z(t),u(t)),dt$. Under suitable hypotheses, it is shown that there exists an optimal control $overline{u}$ and that it satisfies the appropriate optimality system. An example involving the $p$-Laplacian operator demonstrates the applicability of our results.