Optimal impulsive control

Abstract

This chapter describes methods for optimizing cancer therapies that consist of a sequence of impulsive actions. Since the manipulated variable is not an “ordinary” time function but a distribution (generalized function), some background on this topic is included, together with the basic facts that justify the approach taken to cancer therapy that yields the results presented in Chap. 5. Therefore, this chapter begins with a review of basic facts about distributions and then addresses the interaction between distributions and dynamical systems. Solutions of differential equations driven by impulsive inputs do not exist in the classical sense, i.e., as continuous functions that satisfy the differential equation at all times. Therefore, this topic is briefly addressed to justify the approach followed that explores the particular structure of the model, in which the manipulated variable enters linearly and the number of impulses is fixed a priori. Another issue is the conditions for control optimality, where again, the situation is much different when the admissible class of control functions includes impulsive sequences. Although there are necessary conditions that generalize Pontryagin’s maximum principle for impulsive optimal control, the approach followed here uses a direct method that reduces the problem to a finite-dimensional optimization problem.