Optimal Control Problem with State Constraint for Hyperbolic-Type PDE

Abstract

AbstractThe optimal control problem with a terminal-type objective function is solved. The controlled process is described by a hyperbolic PDE with initial and boundary conditions of the second type. The special control is selected so that the state constraint is executed throughout the entire action: the integral of the spatial coordinate of the square of the solution is equal to one. The optimal control problem is considered. Note that this problem is purely theoretical, but has good practical prospects. It is proved that for such a particular equation, its solution can be represented through the solution of a standard linear initial-boundary value problem of hyperbolic type, which in turn allows us to apply the method of separated variables. The transition from the original problem to the problem of optimization by the Fourier coefficients of the solution of a linear problem is shown. The solution of the initial-boundary value problem is reduced to the system of differential equations of the second order. Further we demonstrate a transition to a finite shortened system of first-order differential equations. The solution can be obtained arbitrarily close to the solution of an infinite-dimensional system by increasing dimensionality of the shortened system. A description of the algorithm for iterating over the control coefficients for finding a solution to the optimal control problem for a shortened system and constructing a minimizing sequence are given. The value of the objective function for a shortened system by increasing dimension can be obtained arbitrarily close to the original optimal value.KeywordsIntegro-differential PDEHyperbolic equationState constraintInfinite-dimensional system of ODEsShortened system