On solving a linear control problem

Abstract

The problem of a linear regulator is considered. There is a system of linear differential equations with a quadratic control quality criterion. The method of dynamic programming is applied to the solution of the considered linear problem. As is known, the main difficulty in applying this method is to integrate partial differential equations. In this problem, the obtained optimal control function depends on the solution of the Riccati equation. In [1], conditions were obtained under which there is a solution to such optimal control problems with a quadratic quality criterion. These conditions were obtained along with formulas for minimizing control and for optimal trajectory. But all these statements depended on the ability to solve the matrix Riccati equation under certain boundary conditions given at some time point. To construct a solution to the problem under consideration, a system of 2n adjoint differential equations is constructed. After splitting the transition matrix of this system into block ones, it is possible to express the state of the system at the time instant t through the state variable and the adjoint variable at the final time instant t1. A feature of this work is that an example is given, where the solution of the Riccati equation, which determines the optimal solution of the problem, was obtained explicitly.