On a differential game in a distributed system

We consider the game problem of approach for a system whose dynamics is described by a partial differential equation not of Kovalevskaya type, that is not solved in time derivative. In a Hilbert function space, the equation with boundary conditions are written in an abstract form as a differential operator equation. Using the method of resolving functionals, we obtain sufficient conditions for the approach of a dynamical vector of system to a cylindrical terminal set. Results are exemplified by means of a model problem concerning filtering fluids in fractured-porous rocks.