Regularization Method for Optimally Switched and Impulsive Systems with Biomedical Applications

摘要

Dynamical systems are considered where the control consists of the choice among a finite number fixed vector fields, x = f{sub}i (x), where i∈{1,...,N} = A. To avoid pathological cases, it is assumed that the usual Lie-algebraic conditions guaranteeing reachability are satisfied. The control is parameterized by the number of switches, m, a "word" of length m+1 with alphabet A and a sequence of switching times {τ{sub}1,..., τ{sub}m }. Two regularization schemes are introduced and motivated, replacing the given problem by a smooth control problem, which can be solved by standard numerical optimal control methods. This solution depends on a regularization parameterρ, such that ifρ→0, the original objective is recovered. The regularized control is a smooth function, and its quantization to a predetermined set {u{sub}1,...,u{sub}n} gives the approximation of the switching times and the "word". Substituting the word in the original problem, allows an iterative refinement towards the optimal switching times. Impulsive control determines a timing problem. Here a fixed affine system x = f(x) + g(x)u is considered where the control takes the singular form u(t) =∑{sub}(i=1){sup}N u{sub}iδ(t -τ{sub}i), The optimization is over the values u{sub}i and times n. Optimal pulse vaccination as a control of epidemics and the optimal scheduling of chemotherapy in cancer are discussed as applications of optimal timing problems.