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

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