Optimal parameter selection problem of the state dependent impulsive differential equations

The optimal parameter selection problem for the state dependent impulsive equations is a challenging problem since the impulse number and impulse moments are dependent on the control parameters implicitly. In the simple continuous case, the state variables are continuous with respect to the control parameter when the impulse number is fixed. We prove the existence of the optimal solution and present the gradient of the cost functional such that the optimal problem can be solved by some gradient based optimization methods. For the difficult discontinuous case, i.e., the unfixed impulse number, we construct an approximation problem, and prove that its optimal value converges to that of the original optimal problem by the definition of epi-convergence. Then the corresponding gradient and solving algorithm are given. Finally, two numerical examples are presented to show the efficiency of the algorithm. (C) 2019 Elsevier Ltd. All rights reserved.