Optimal control problems with final observation governed by explosive parabolic equations

We study optimal control problems with final observation. The governing parabolic equations or systems involve superlinear nonlinearities, and their solutions may blow up infinite time. Our proof of the existence, regularity, and optimality conditions for an optimal pair is based on uniform a priori estimates for the approximating solutions. Our conditions on the growth of the nonlinearity are essentially optimal. In particular, we also solve a long-standing open problem of Lions concerning singular systems.