Solution of the Continuous Time Bilinear Quadratic Regulator Problem by Krotov's Method

TL;DR

This paper applies Krotov's method to solve a continuous-time bilinear quadratic regulator problem, providing an algorithmic approach and demonstrating its effectiveness through a numerical example.

Contribution

It introduces a novel application of Krotov's method to bilinear quadratic regulator problems with a detailed algorithm and numerical validation.

Findings

Successful formulation of the improving sequence for the control problem

Development of an algorithmic solution approach

Numerical example demonstrating the method's effectiveness

Abstract

This work contributes to the field of optimal control of bilinear systems. It concerns a continuous time, finite dimensional, bilinear state equation with a quadratic performance index to be minimized. The state equation is non-autonomous and comprises a deterministic, a-priori known excitation. The control trajectory is constrained to an admissible set without a specific structure. The performance index is a functional, quadratic in the state variables and control signals. Krotov's method is used for solving this problem by means of an improving sequence. To this end, the required sequence of an improving functions is formulated. Finally, the solution is encapsulated in an algorithm form and a numerical example of structural control problem is provided.