An Iterative Procedure for Optimal Control of Bilinear Systems

TL;DR

This paper introduces an iterative method combining homotopy perturbation and Pontryagin's principle to efficiently solve bilinear quadratic optimal control problems, providing convergent series solutions.

Contribution

The paper proposes a novel iterative procedure that transforms the bilinear optimal control problem into a sequence of linear problems using HPM, with proven convergence conditions.

Findings

Method effectively computes optimal control and trajectory.

Converges under specified conditions, ensuring solution accuracy.

Demonstrated through an illustrative example.

Abstract

This paper presents a new and straightforward procedure for solving bilinear quadratic optimal control problem. In this method, first the original optimal control problem is transformed into a nonlinear twopoint boundary value problem (TPBVP) via the Pontryagin's maximum principle. Then, the nonlinear TPBVP is transformed into a sequence of linear time-invariant TPBVPs using the homotopy perturbation method (HPM) and introducing a convex homotopy in topologic space. Solving the latter problems through an iterative process yields the optimal control law and optimal trajectory in the form of infinite series. Finally, sufficient condition for convergence of these series is proved by a theorem. Simplicity and efficiency of the proposed method is shown through an illustrative example.