Investigation of boundary control and final observation in mathematical model of motion speed potentials distribution of filtered liquid free surface

In this paper, we study the problem of boundary control and final observation for one degenerate mathematical model of motion speed potentials distribution of filtered liquid free surface with the Showalter–Sidorov initial condition. The mathematical model is based on the degenerate Boussinesq equation with an inhomogeneous Dirichlet condition. This model belongs to the class of semilinear Sobolev-type models in which the nonlinear operator is $p$-coercive and $s$-monotone. In the paper, the problem of boundary control and final observation for a semilinear Sobolev-type model is considered and conditions for the existence of a control-state pair of the problem are found. In applied studies of a research problem, it is allowed to find such a potentials distributionof filtered liquid free surface, at which the system transitions from the initial condition to a given final state within a certain period of time $T$.