Optimal linear–quadratic control of asymptotically stabilizable systems using approximations

In this paper we study approximations to the infinite-horizon quadratic optimal control problem for linear systems that may be only asymptotically stabilizable. For linear systems, this issue only arises with infinite-dimensional systems. We provide sufficient conditions which guarantee when approximations to the optimal feedback result in the cost converging to the optimal cost. One technique for approximate solution of the optimal control problem is to use Newton–Kleinman iterations for the associated Riccati equation. Some new results in this direction are provided. Several important classes of systems, lightly damped second-order systems and a platoon-type system, are shown to be optimizable. Also, finding an initial stabilizing control for the Newton–Kleinman iteration can be non-trivial. The initial iterate for these classes is described.