Introduction to the theory of positional differentional games of systems with aftereffect (based on the $i$-smooth analisys methodology

Although the foundations of the theory of positional differential games of systems with aftereffect described by functional differential equations (FDE) were developed back in the 1970s by N. N. Krasovsky, Yu. S. Osipov and A. V. Kryazhimsky, there are still no works that, similar to [N. N. Krasovsky, A. I. Subbotin. Positional Differential Games. Moscow: Nauka, 1974, 457 p.] (hereinafter referred to as [KS]), would represent a “complete” theory of positional differential games with aftereffect.
The paper presents an approach to constructively transferring all the results of the book [KS] to systems with aftereffect. This approach allows us to present the theory of positional differential games of systems with aftereffect in the same constructive and complete form as for the finite-dimensional case in [KS]. The approach is based on the methodology of $i$-smooth analysis. The obtained results of the theory of positional differential games of systems with aftereffect are completely analogous to the corresponding results of the finite-dimensional Krasovsky–Subbotin theory.