Direct optimal control and costate estimation using least square method

In this paper, we present a direct method to solve optimal control problems based on the least square formulation of the state dynamics. In this approach, we approximate the state and control variables in a finite dimensional Hilbert space. We impose the state dynamics as a weighted integral formulation based on the least square method to solve initial value problems. We analyze the resulting nonlinear programming problem to derive a set of conditions under which the costates of the optimal control problem can be estimated from the associated Karush-Kuhn-Tucker multipliers. We present numerical examples to demonstrate the applicability of the present method.