Multipoint Boundary-Value Problem of Optimal Control for Parabolic Equations with Degeneration - Pukalsky, Ivan; Yashan, Bohdan (Journal of Mathematical Sciences, 2023)
We study the problem of optimal control over the process described by a multipoint problem with oblique derivative for a second-order parabolic equation and consider the cases of internal, starting, and boundary control. The performance criterion is specified as the sum of volume and surface integrals. By using the principle of maximum and a priori estimates, we establish the existence and uniqueness of solutions of a multipoint boundary-value problem with degeneration. The coefficients of the parabolic equation and boundary conditions have power singularities of any order for any variables on a certain set of points. We establish estimates for the solution of the multipoint boundary-value problem and its derivatives in Hölder spaces with power weight. The necessary and sufficient conditions for the existence of the optimal solution of the problem are presented.
Keywords: interpolation inequalities; maximum principle; a priori estimates; degeneration; boundary conditions