On generalizations of the Hamilton-Jacobi equation for hereditary systems
As the author states in the Introduction, the aim of the paper is to use the so-called ``co-invariant derivatives'' introduced, apparently, in [\textit{A. V. Kim}, ``i-Smooth analysis and functional-differential equations'' (Russian), Ural Branch R. A. N., Ekaterinburg (1996)], to extend Subbotin's theory of \textit{minimax solutions} in [\textit{A. I. Subbotin}, ``Generalized solutions of first order PDEs. The dynamical optimization perspective'' (1994; Zbl 0820.35003)], to ``functional'' Hamilton-Jacobi equations of the form: NEWLINE\[NEWLINE \partial_t\varphi(g)+H(g,\nabla \varphi(g))=0,\;g=(t,x_{t_*,t}(.))\in G,\;\varphi(T,x_{t_*,T}(.))=\sigma(x_{t_0,T}(.)) NEWLINE\]NEWLINE where \(t_*\leq t_0<T\) and \(G\) is the set of pairs NEWLINE\[NEWLINEg=(t,x_{t_*,t}(.)),\;t\in [t_0,T],\;x_{t_*,t}(.)\in C([t_*,t];R^n)NEWLINE\]NEWLINE endowed with the ``distance function'' \(\rho(g_1,g_2):= \max\{\rho^*(g_1,g_2),\rho^*(g_2,g_1)\}\), NEWLINE\[NEWLINE \rho^*(g_1,g_2):=\displaystyle \max_{\xi\in[t_*,t_1]}\min_{\eta\in [t_*,t_2]}||(\xi,x^1(\xi))- (\eta,x^2(\eta))||NEWLINE\]NEWLINE if \(g_i=(t_i,x^i_{t_*,t_i}(.))\in G, i=1,2.\) The author introduces first a concept of minimax solution defined in terms of some associated ``generalized characteristic inclusions with delay'' and notes that the existence, uniqueness and continuous dependence on the data of such solutions may be proved in the same way as in the usual case; next, one introduces certain ``extreme directional derivatives'' of a function \(\varphi(.):G\to R\) and one characterizes the minimax solutions in terms of certain differential inequalities that correspond to the so-called ``contingent solutions''; moreover, the author introduces also the concepts of (lower and upper) ``semi-differentials'' of a function \(\varphi(.)\), a corresponding concept of ``viscosity solution'' and proves that a minimax solution is a viscosity solution in this sense. In the last section an illustrative example is presented and a ``guessed'' function is proved to be the unique minimax, contingent and also a viscosity solution in the context of the paper.
scientific article; zbMATH DE number 1671898