An ODE Comparison Theorem With Application in the Optimal Exit Time Control Problem

The optimal exit time control (OETC) problem tries to find the feedback control law with a reasonable cost that can keep the system state inside a certain subset of the state space, called the safe set, for the longest time under random perturbations. The symmetry property of its solutions has been proved previously when the state dimension is higher than one. In this note, a comparison theorem is established that compares the solutions to two ODEs arising in the 1-D OETC problem. The symmetry of solutions to the 1-D OETC problem is proved using this comparison theorem. An example is presented to show how the symmetry result can help to solve the OETC problem analytically in certain cases.