A Sweeping Gradient Method for Ordinary Differential Equations with Events

In this paper, we use the calculus of variations to derive a sensitivity analysis for ordinary differential equations with events. This sweeping gradient method (SGM) requires a forward sweep to evaluate the original model and a backwards sweep of the adjoint to compute the sensitivity. The method is applied to canonical optimal control problems with numerical examples, including the sampled linear quadratic regulator and the optimal time-switching and state-switching for minimum-time transfer of the double integrator. We show that the application of the SGM for these examples matches the gradient determined analytically. Numerical examples are produced using gradient-based optimization algorithms. The emphasis of this work is on modeling considerations for the effective application of this method.