A comparison principle for state-constrained differential inequalities and its application to time-optimal control

In this paper, we present a comparison principle that characterizes the maximal solutions of state-constrained differential inequalities in terms of solutions of certain differential equations with discontinuous right-hand sides. For the sake of completeness, we show through some set-valued analysis that the differential equations determining the maximal solutions have the unique solutions in the Carathe´odory sense, in spite of discontinuity of their right-hand sides. We apply our comparison principle to the explicit characterization of the solution to a time-optimal control problem for a class of state-constrained second-order systems which includes the dynamic equations of robotic manipulators with geometric path constraints as well as single-degree-of-freedom mechanical systems with friction. Specifically, we show that the time-optimal trajectory is uniquely determined by two curves that can be constructed by solving two scalar ordinary differential equations with continuous right-hand sides. Hence, the time-optimal trajectory can be found in a computationally efficient way through the direct use of the well-known Euler or Runge-Kutta methods. Another interesting feature is that our method to solve the time-optimal control problem works even when there exist an infinite number of switching points. Finally, some simulation results using a two-degrees-of-freedom (DOF) robotic manipulator are presented to demonstrate the practical use of our complete characterization of the time-optimal solution.