An existence theorem for a nonlinear time-optimization problem (Q2704031)

The paper deals with the smooth control system NEWLINE\[NEWLINE\dot{x} = f(x,u), u \in U, x \in M,\tag{*}NEWLINE\]NEWLINE where \(f(u), u \in R^{m}\), is the set of smooth vector fields on \(M\) smoothly depending on the parameter \(u \in U\), \(U\) is a compact convex polygon in \(R^{m}\). The assumptions taken here imply that any admissible control function \(u(t)\) is bounded and measurable and it has a flow \(p_{t}(u)\), \(t \in R\), on the manifold \(M\). The paper deals with the time-optimal problem of the system (*) in the following way: given the starting state \(x_{0}\) and the target state \(x_{1}\), find an admissible control \(u(t), t \in R\), such that the trajectory \(x(t)=x(t;x_0,u(\cdot))=x_{0} \circ p_{t}(u), t \in R\), of the system (*) reaches the target state \(x_{1}(T)\) in minimum time \(T < \infty\). The paper gives an explicit condition for the convexity of the vectogram, which is one of the principle conditions on which there is based the theorem saying that the time-optimal control function is of ``\textit{bang-bang}'' type with finite number of switchings.