Optimal Control Problems Subject to Multiple Characteristic

Time Constraints

by

TEO Kok Lay

Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom, Kowloon Hong Kong

Abstract

In some real world optimal control problems, their constraints depend on the state vector specified at two or more discrete time-points, such as the constraint

g = x(τ_{1}) − x(τ_{2}) ≥ 0,

where 0 < τ_{1} < τ_{2}. Here, τ_{1} and τ_{2} are called the characteristic times, and these constraints are called the multiple characteristic time constraints (which are abbreviated as MCT constraints). This class of optimal control problems arise naturally in drug administration in cancer chemotherapy, where the optimal control model for tumour growth is derived. The tumour size is to be forced to decrease at, or faster than, a fixed rate, giving rise the set of MCT constraints

N(τ_{i+1}) ≤ εN(τ_{i}), for i = 1, …, M,

where the parameter ε is a positive real number less than 1, and {τ_{i}}_{i=1}^{M} is a set of fixed time-points.

MCT constraints also arise when the constraint transcription technique is applied to reduce a large number of constraints to a single equivalent constraint.

In this talk, we shall present a computational method based on the control parametrization technique for solving optimal control problems involving MCT constraints. Some convergence results are established, and gradient formulae of the MCT cost functional and the MTC constraint functionals are derived.