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In a Hilbert space, we consider the linear boundary value optimal control problem based on a linear dynamics. The initial and terminal values of trajectories on the left and right ends of the time interval are given implicitly as the solutions of linear programming problems. We reduce the original problem to finding a saddle points of the Lagrangian. The boundary value differential system was solved by an iterative method of extragradient type. The convergence of the method was proved.