Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDES

A terminal-state tracking optimal control problem for linear hyperbolic equations with distributed control is studied in this paper. An analytic solution formula for the optimal control problem is derived in the form of eigenseries. We show that the optimal solution satisfies the approximate controllability property. An explicit solution formula for the exact controllability problem is also expressed by the eigenseries formula when the target state and the controlled state have matching boundary conditions. We demonstrate by numerical simulations that the optimal solutions expressed by the series formula approach the target functions.