Generalized Solutions of First Order PDEs

In the present chapter we introduce the notion of minimax solution to first-order partial differential equation. The proposed definition is based on the weak invariance property of the graph of a generalized solution with respect to a system of differential inclusions, which will be called characteristic inclusions. This property can be given with the help of apparently different criteria, which are formulated in Sections 2 and 3. The equivalence of these criteria and the equivalence of minimax and viscosity solutions are proven in Section 4.

In the framework of the proposed approach, the existence and uniqueness of minimax solution for a wide class of boundary-value problems and Cauchy problems can be proved. The Cauchy problem for Hamilton-Jacobi equation is examined in this chapter. Proofs of uniqueness and existence theorems are based on the property of weak invariance of minimax solutions with respect to characteristic inclusions. These inclusions are considered in the present section. We formulate also equivalent definitions of minimax solutions of Hamilton-Jacobi equations. Uniqueness and existence theorems are proved in the next sections. It can be seen from the proofs that these theorems actually provide criteria for the stability of solutions with respect to small perturbations of the Hamiltonian and the terminal function.

Investigations on differential games started in the 1950–60s. At first the mathematical models of conflict were dealt with most (see, for example, the well-known monograph of R. Isaacs [92]). In these models the motions of the controlled systems are governed by two antagonistic players. A pursuit-evasion problem is a typical example of an antagonistic differential game. However, problems of such kind are rather exotic. At the same time there are numerous problems in engineering, economics, ecology, etc. in which it is required to construct a feedback control ensuring a certain result in the presence of disturbances. As an illustration we can mention the problems of control of an aircraft landing and takeoff in the presence of the so-called windshear, when the aircraft is subjected to wind bursts. Analysis of differential games can help in elaboration of control algorithms for this and similar problems.

The minimax solution approach can be used for studying various types of first-order PDE’s with boundary and terminal (initial) conditions. In Chapter II, results concerning Cauchy problems for Hamilton-Jacobi equations were presented. In this chapter we consider some other applications of the approach.