Generalized Solutions of First Order PDEs The Dynamical Optimization Perspective

This text presents an approach to partial differential equations that can be considered as a non-classical method of characteristics, according to which the generalized solution (the minimax solution) is assumed to be flow invariant with respect to the so-called characteristic inclusions. The research on minimax solutions employs methods of the theory of differential games, dynamical optimization and nonsmooth analysis. At the same time, this research has contributed to the development of these new branches of mathematics. The book is intended as a self-contained exposition of the theory of minimax solutions. It includes existence and uniqueness results, examples of modelling and applications to the theory of control and differential games.

I Generalized Characteristics of First-Order PDE’s.- II Cauchy Problems for Hamilton-Jacobi Equations.- III Differential Games.- IV Boundary-Value Problems for First-Order PDE’s.- A1 Justification of the Classical Method of Characteristics.- A2 Multifunctions.- A3 Semicontinuous Functions.- A4 Convex Functions.- A5 Contingent Tangent Cones, Directional Derivatives, Subdifferentials.- A6 On a Property of Subdifferentials.- A7 Differential Inclusions.- A8 Criteria for Weak Invariance.