Minimax solutions to the Hamilton-Jacobi equations (Q5948401)

This article is a very short overview of the main aspects of author's theory in [\textit{A. I. Subbotin}, ``Generalized solutions of first-order PDEs. The dynamic optimization perspective'' (1994; Zbl 0820.35003)], of the so-called ``minimax solutions'' of first-order PDEs of the form: \[ F(x,u(x),Du(x))=0, \;x\in G\subset \mathbb{R}^n, \] in view of well-known applications in optimal control and differential games. Recalling the basic equivalent definitions of the concept of minimax solution, the author points out its origin in the concepts of \(u-stable\) and \(v-stable\) functions in [\textit{N. N. Krassovskij} and \textit{A. I. Subbotin}, ``Positional differential games'' (Russian) (1974; Zbl 0298.90067)] and the equivalence with the better known concept of ``viscosity solution'' introduced in [\textit{M. G. Crandall} and \textit{P.-L. Lions}, Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)]. Finally, the author summarizes some of the more important and more recent contributions to the theory and applications of minimax solutions obtained by himself and his collaborators. One should note perhaps that, as mentioned at the beginning of the article, ``this paper was prepared, by A. I. Subbotin himself but was presented by his wife, N. N. Subbotina, at a plenary talk at the Pontryagin Conference, because of his unfortunate passing''.