Chronological algebras: combinatorics and control (Q2771542)

This paper deals with the finite-dimensional control system on the manifold \(M\): NEWLINE\[NEWLINE\dot x=f_0(x)+\sum_{i=1}^{m}u^{i}f_{i}(x),NEWLINE\]NEWLINE where \(f_{i}\) is a family of real analytic vector fields; controls \(u^{i}\) are measurable with values from some finite interval. Following Sussmann the author considers the formal control system \(\dot S=S(X_0+\sum_{i=1}^{m}u^{i}X_{i})\) with initial condition \(S(0)=I\). A solution of this system is given by the Chen-Fliess series \(Ser(T,u)=\sum_{I}\Upsilon^{I}(T,u)X_{I}\), where \(I=(i_1,\ldots,i_{i_{p}}), p\geq0\),\ \(1\leq i_{j}\leq m\); \(X_{I}=X_{i_1}X_{i_2}\ldots X_{i_{p}}\); NEWLINE\[NEWLINE\Upsilon^{I}(T,u)=\int\limits_0^{T}\!\int\limits_0^{t_1}\ldots\int\limits_0^{t_{p-1}} u^{i_{p}}(t_{p})u^{i_{p-1}}(t_{p-1})\ldots u^{i_{1}}(t_{1})\,dt_1\ldots dt_{p}.NEWLINE\]NEWLINE A survey of the main results concerning the product expansion of Chen series is presented. Also a survey of results concerning product expansions obtained in the framework of algebraic combinatorics is presented. The chronological product as well as chronological algebra are introduced and their properties are studied.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00042].