Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

Abstract

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.