Approximate Solution of Optimal Control Problem by Using Linear Programming Technique

Abstract

This paper treats an approximate solution of optimal control problem by means of the linear programming technique. Let the system be linear, then the solution of a set of differential equations which governs the system is given by the variation-of-constants formula. The state variables of the system at a fixed time are described by the definite integral, the integrand of which is a linear form in control variables. Upon use of a suitable integration formula, the integrals are approximately represented by a weighted sum of a finite number of values of the integrand. By introducing auxiliary variables, the performance index which is required to be minimum is expressed as a linear function of the variables subject to constraints. Thus, the minimization of a functional is approximately reduced to the minimization of a linear function of many variables subject to linear constraints. This problem is a linear programming problem, and can be solved by using the simplex method. A feasible basic solution to the linear program is shown also.