Problems of time-optimal control to infinity (Q923355)

A theorem on the existence of solutions to the following minimum-time control problem \[ \dot x=f(t,x,u),\quad x\in R^ n,\quad u\in U(t,x)\subset R^ m;\quad x(0)=x_ 0,\quad \limsup_{t\to T_{\infty}-0}\| x(t)\| =+\infty,\quad T_{\infty}\to \inf \] is given. As a special case the author considers the control problem for the system \(\dot x=u_ 1xy^{\beta}z^{\gamma}\), \(\dot y=u_ 2x^{\gamma}yz^{\beta}\), \(\dot z=u_ 3x^{\beta}y^{\gamma}z\), \(u_ 1,u_ 2,u_ 3\geq 0\), \(u_ 1+u_ 2+u_ 3=1\), \(\beta >0\geq \gamma >-\beta\), and obtains a characterization of its optimal trajectory.