Symmetry of Solutions to the Generalized 1-D Optimal Sojourn Time Control Problem

The optimal sojourn time control (OSTC) problem tries to find the feedback control law with reasonable cost that can keep the system state in a certain subset of the state space, called the safe set, for the longest time under random perturbations. The radial symmetry property of its solutions has been proved previously when the state dimension is higher than one. However, the proof does not apply in the one dimensional case. This paper generalizes the OSTC problem by introducing arbitrary weighting functions for both the cost function and the performance measure function, and shows that the solution to the 1D generalized OSTC problem is radially symmetric. Our proof can also be extended to the multi-dimensional case. Numerical simulation is performed to verify the proof