SINGULAR CHARACTERISTICS OF FIRST ORDER PDEs IN OPTIMAL CONTROL AND DIFFERENTIAL GAMES

Abstract

The interconnection between the maximum principle [13] and dynamic programming [3], the two main methods in the theory of optimal control and differential games, is now understood much better due to the theory of viscosity solutions of firstand second-order nonlinear PDEs [2, 8, 14]. The value functions in optimal control and differential games have been proved to be generalized (viscosity) solutions of the corresponding Hamilton–Jacobi–Bellman–Isaacs (HJBI) equations. The classical method of characteristics (MC), which reduces the solution of a PDE to the integration of an ODE (characteristic) system, is one of the attractive and powerful tools for solving nonlinear first-order PDEs arising in control theory and mathematical physics [5]. Characteristics are of special interest in control theory, since they represent optimal paths [10]. This is clear in regarding the regular characteristics. The nonsmoothness of the generalized (e.g., viscosity) solution and/or of the Hamiltonian (left-hand-side function of the PDE), i.e., the presence of singular surfaces, is often referred to as an obstacle to the implementation of the MC. On the other hand, the construction of singular surfaces and lines is an essential and interesting part of the solution of a problem in control theory. The method of singular characteristics (MSC) shows that one can overcome this obstacle by using the same notion of characteristics suitably modified [11,12]. In this paper, a new notion of singular characteristics (SC) is suggested, which, together with the classical (regular) ones, form generalized characteristics. SC are effective for the construction of singular lines, surfaces, and manifolds of a nonsmooth solution (to firstor second-order PDEs), which carry essential information on the corresponding control or physical problem or phenomenon. Singular characteristics were found owing to investigation of singular paths in differential games and optimal control [4,9,10]. Regular paths in these domains are known to be governed by a Hamiltonian ODE system, the characteristic system for the HJBI-equation. In many cases, singular paths are described by similar equations by using the so-called singular controls. The attempt to eliminate singular controls from these equations has led to the discovery of singular characteristics, which appear to be inherent not only to game or control problems but also to the general nonlinear first-order PDEs. Owing to the notion of SC, many singular lines and surfaces known in control theory have received their invariant description in terms of general (abstract) PDEs. The possible interconnection between singular paths and characteristics (in some generalized sense) was mentioned in [7]. This general mathematical insight into the nature of a singularity, as a rule, simplifies the solution procedure of a game or control problem. Correspondingly, the experience accumulated during the construction of solutions to the latter problems is highly useful for the understanding the structure of singularities of the viscosity solutions to general PDEs. The considerations of the present paper are given in general mathematical terms rather than in terms specific for control theory. The ODE systems of singular characteristics are derived for several types of well-known singular surfaces in optimal control and differential games. The solutions of particular problems are presented.