High-order necessary conditions of optimality for nonlinear control systems
Consider the control system \(\dot x=f_{\tau}(x)+g(x)\cdot u\), where \(f_{\tau}\) is a time-dependent vector field on a manifold \(M\), \(g=(g_1,\ldots,g_r)\) a collection of vector fields on \(M\) and \(u=(u_1,\ldots,u_r)\) a vector valued function of the time parameter. The main result of the paper under review is a set of necessary conditions for time-optimality of solutions in terms of the Lie brackets of the \(g_j\)'s. More precisely, the author first gives a Pontryagin type maximum principle and then uses a Hall-basis of the free Lie algebra in \(r\) indeterminates to rewrite the conditions including the variational cone as a condition involving the Lie brackets.
necessary conditions for time-optimality
scientific article; zbMATH DE number 4210920
scientific article