Static PDEs for time-dependent control problems (Q851456)

Summary: We consider two different non-autonomous anisotropic time-optimal control problems. For the min-time-from-boundary problem, we show that the value function is recovered as a viscosity solution of a static Hamilton-Jacobi-Bellman partial differential equation \(H(\nabla u(x), u(x),x)=1\). We demonstrate that the space-marching ordered upwind methods [introduced by \textit{J. A. Sethian} and \textit{A. Vladimirsky} [Proc. Natl. Acad. Sci. USA 98, No. 20, 11069--11074 (2001; Zbl 1002.65112)] for the autonomous control) can be extended to this non-autonomous case. We illustrate this approach with several numerical experiments. For the min-time-to-boundary problem, where no reduction to a static PDE is possible, we show how the space-marching methods can be efficiently used to approximate individual level sets of the time-dependent value function.