Optimal switching control for drug therapy process in cancer chemotherapy

In this paper, the drug therapy problem in cancer chemotherapy is formulated as an optimal control problem of switched systems. In this problem, the modes switch under state-dependent. This is different from the existing optimal control problem of switched systems, in which the modes switch under time-dependent. Thus, the existing optimal control approaches of switched systems with time-dependent switching can not directly used to solve this Problem, and a new numerical computation method is required to develop for solving such problem. Firstly, based on introducing a binary function, relaxing the binary functions and including a penalty term on the relaxation, we obtain an equivalent optimal control problem of constrained nonlinear system. By using the time-scaling transformation, the smoothing technique, and the idea of l1 penalty function method, the equivalent optimal control problem is transformed into a nonlinear parameter optimization problem. Then, a gradient-based continuous filled function algorithm is developed for solving the nonlinear parameter optimization problem. Finally, a numerical example is used to illustrate our method is low time-consuming, has faster convergence speed, and yields a better objective function value than the existing algorithms.