Asymptotics of a Solution to a Singularly Perturbed Time-Optimal Control Problem

Abstract

In the study of singularly perturbed optimal control problems, asymptotic solutions of the boundary value problems resulting from the optimality condition for the control are constructed using the well-known and well-developed method of boundary functions. This approach is effective for problems with smooth controls from an open domain. Problems with a closed bounded domain of control have been less thoroughly investigated. The cases usually considered involve situations where the control is a scalar function or a multidimensional function with values in a convex polyhedron. In the latter case, since the optimal control is a piecewise constant function with values at the vertices of the polyhedron, it is important to describe the asymptotic behavior of switching points of the optimal control. In this paper, we investigate a time-optimal control problem for a singularly perturbed linear autonomous system with smooth geometric constraints on the control in the form of a ball. The main difference between this case and the case of systems with fast and slow variables studied earlier is that the matrix at the fast variables is a multidimensional analog of the second-order Jordan cell with zero eigenvalue and, thus, does not satisfy the standard condition of asymptotic stability. The solvability of the problem is proved. Power asymptotic expansions of the optimal time and optimal control in a small parameter at the derivatives in the equations of the system are constructed and justified.