Numerical solution of a nonlinear time-optimal control problem

Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable, it is proved that the minimizing sequence converges to an $\varepsilon$-optimal solution after a finite number of iterations. A pair $\{T,u(\cdot)\}$ is called an $\varepsilon$-optimal solution if $|T-T_{\mathrm{opt}}|\le\varepsilon$, where $T_{\mathrm{opt}}$ is the optimal time required for moving the system from the initial state to the origin and $u$ is an admissible control that moves the system to an $\varepsilon$-neighborhood of the origin over the time $T$.