Procedure for regularization of bilinear optimal control problems based on a finite-dimensional model

Abstract

An optimization problem of a linear system of ordinary differential equations on a set of piecewise continuous scalar controls with two-sided restrictions is considered. The cost functional contains the bilinear part (control, state) and a control square with a parameter, which plays the role of a regularization term. An approximate solution of the optimal control problem is carried out on a subset of piecewise constant controls with a non-uniform grid of possible switching points. As a result of the proposed parametrization, reduction to the finite-dimensional problem of quadratic programming was carried out with the parameter in the objective function and the simplest restrictions. In the case of a strictly convex objective function, the finite-dimensional problem can be solved in a finite number of iterations by the method of special points. For strictly concave objective functions, the corresponding problem is solved by simple or specialized brute force methods. In an arbitrary case, parameter conditions and switching points are found at which the objective function becomes convex or concave. At the same time, the corresponding problems of mathematical programming allow a global solution in a finite number of iterations. Thus, the proposed approach allows to approximate the original non-convex variation problem with a finite-dimensional model that allows to find a global solution in a finite number of iterations.