Regularization method for optimally switched and impulsive systems with biomedical applications

Dynamical systems are considered where the control consists of the choice among a finite number fixed vector fields, x˙ = f_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 {τ1, ... ,τm}. 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 parameter ρ, such that if ρ → 0, the original objective is recovered. The regularized control is a smooth function, and its quantization to a predetermined set {u1,...,u_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) = Σ^N_i=1 u_ii∂(t -τ_i). The optimization is over the values u_i and times τ_i. Optimal pulse vaccination as a control of epidemics and the optimal scheduling of chemotherapy in cancer are discussed as applications of optimal timing problems.