A note on convexity in smooth nonlinear systems
The paper presents an existence theorem for a two-point time-optimal control problem associated with a smooth control system \({\dot x}=f(x,u)\), \(x \in M\), \(u \in U\), where \(M\) is an \(n\)-dimensional \(C^{\infty}\)-manifold that is regularly embedded into \(R^d\); \(U\) is a compact convex polyhedron in \(R^m\); \(\{ f(\cdot,u): u \in R^m \}\) is a family of \(C^{\infty}\) vector fields on \(M\) that smoothly depend on \(u\). The result completes a previous bang-bang theorem by the author which states the bang-bang structure of the solution for this problem provided that the optimal control exists.
scientific article; zbMATH DE number 1526514