Optimal control of noninstantaneous impulsive differential equations

In this paper, we study optimal control problems for a new class of noninstantaneous impulsive differential equations arising from the dynamics of evolution processes in pharmacotherapy. We construct a suitable control function, which allows us to characterize the structure of controllability by using the terminal time subinterval instead of the global time interval. We apply fixed point approach to show the controllability results that are the foundation of optimal control theory. Next, we study existence of optimal control problems for a certain quadratic functional acting as the performance index. In addition, we design ILC updating laws for deterministic impulsive systems to generate a sequence of control functions to approximate the optimal control function. Further, we extend the deterministic results to random case by designing ILC updating laws with randomly varying trial length. Finally, several examples are given to demonstrate the validity of theoretical results and design methods. Here, we remark that ILC updating algorithm is adopted to find a desired optimal control for impulsive systems, which provides another effective way to solve optimization problem via computer techniques.