An Internal Transition Layer in a Linear Optimal Control Problem

Abstract

Internal nonuniform processes characterized by alternating rapid and slow motions have relatively recently become a subject of singular perturbation theory. Here the priority is with the Vasil’eva–Butuzov school, whose members’ investigations resulted in a new trend in differential equations, which is known as the theory of contrast structures and has so far experienced dramatic progress (e.g., see [1–3]). Much later, different methods came into use in the study of contrast structures. In the mid-90s, it was noted in [4] that internal transition layers can appear not only in nonlinear systems (as a result of the influence of nonlinearities of a special form), but also in linear problems. For example, the contrast structures in the problem considered in [4] are due to the instability of the spectrum of the limit operator of the original system; moreover, depending on the instability character, one encounters various types of contrast solutions, including spikeand step-like structures considered earlier in [1–3]. The notion of spectrum instability itself was introduced by Lomov in the development of the regularization method [5]; he also considered various types of singularly perturbed problems with discrete noninvertibility of the limit operator, which are characterized by the spectrum instability at separate points of a given time interval [5, 6]. This kind of spectrum instability induces spike-type contrast structures in solutions. The possible existence of contrast structures in solutions of optimal control problems has proved to be quite unexpected. A linear singularly perturbed system in which the appearance of contrast structures is caused not by the behavior of the spectrum of the limit operator but by the presence of a damping factor in the performance functional to be minimized was considered for the first time in [7]. The simplest case in which the damping factor induces spike-like contrast structures in solutions was analyzed. The subsequent investigation of this phenomenon from the viewpoint of a more general situation (where, in a control system, there may appear step-like contrast structures and so on) is carried out in the present paper.