Variational optimality condition in the control problem of hyperbolic equations with dynamic boundary conditions

The paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a quadratic cost functional and boundary condi- tions determined from controlled bilinear ordinary differential equations. These ordi- nary differential equations are linear for state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology, social demographics, and population dynamics. General optimal control methods are used, normally, for these problems because of bilinear ordinary differential equations. The problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact second-order increment formu- las for the cost functional. This treatment allows using some efficient optimal control methods for the problem.