Mathematical models and optimal control of the filtration and deformation processes

The article presents a review author's works on study of optimal control problems for semilinear Sobolev type models with $s$-monotone and $p$-coercive operators. Theorems of existence and uniqueness of weak generalized solution to the Cauchy or the Showalter–Sidorov problem for a class of degenerate non-classical models of mathematical physics are stated. The theory is based on the phase space and the Galerkin–Petrov methods. The developed scheme of a numerical method allows one to find an approximate solution to the Cauchy or Showalter–Sidorov problems for considered models. An abstract scheme for study of the optimal control problem for this class of models is constructed. On the basis of abstract results the existence of optimal control of processes of filtration and deformation are obtained. The necessary conditions for optimal control are provided.