Inverse optimal stabilization of 2 × 2 linear hyperbolic partial differential equations

Abstract

We solve an inverse optimal control problem for a class of systems of 2 × 2 linear hyperbolic partial differential equations (PDEs). The derived controller exponentially stabilizes the system in the L2-sense, and also minimizes a cost function that is positive definite in the system states and control signal. Two remarkable features of the inverse optimal controller are 1) it is simply a scaled version of the well-known backstepping controller, and 2) the controller approaches the backstepping controller when the cost of actuation approaches zero. The theory is demonstrated in simulations, where it is shown that the proposed optimal controller minimizes the cost functional. However, the difference in the cost function from applying the backstepping controller instead of the inverse optimal controller is minimal, with only a 5.5% deviation in the simulated scenario. The proposed inverse optimal controller is also compared with an optimal controller designed to minimize a cost functional quadratic in the original states and actuation. Simulation examples show that the cost (with respect to the cost functional of the optimal controller) of using the backstepping or the inverse optimal controller is only slightly worse than the optimal value attained by the optimal controller, with the best value being only 2.7% larger in magnitude.