On the structure of locally Lipschitz minimax solutions of the Hamilton–Jacobi–Bellman equation in terms of classical characteristics

Necessary and sufficient conditions for the minimax solution to the Cauchy problem for the Hamilton–Jacobi–Bellman equation are obtained as viability conditions for classical characteristics inside the graph of the minimax solution. Using this property, a representative formula for a one-dimensional conservation law in terms of classical characteristics is derived. An estimate of the numerical integration of the characteristic system is presented and errors of numerical realizations of representative formulas are determined for the conservation law and its potential equal to the minimax solution of the Hamilton–Jacobi–Bellman equation.