Quasi-reversibility methods of optimal control for ill-posed final value diffusion equations

Abstract

We consider optimal control problems associated to generally non-well posed Cauchy problems in a general framework. Firstly, we approximate the ill-posed problem with a family of well-posed one and show that solutions of the latter one converge to solutions of the former one. Secondly, we investigate the minimization problem associated with the approximated state equation. We prove the existence and uniqueness of minimizers that we characterize with the optimality systems. Finally, we show that minimizers of the approximated problems converge to the minimizers of the optimal control subjected to the ill-posed state equation that we characterize with a singular optimality system. This characterization is obtained as the limit of the optimality systems of the approximated minimization problem. We use the techniques of quasi-reversibility developed by Lattès and Lions in 1969. Our general framework includes classical elliptic second order operators with Dirichlet and Robin conditions, as well as the fractional Laplace operator with the Dirichlet exterior condition.