Optimal control by multiple shooting and weighted tchebycheff penalty-based scalarization

Abstract

Numerical direct multiple shooting (MS) methods have shown to be important and efficient tools to solve optimal control problems (OCP). The use of an MS method to solve the OCP gives rise to a finite-dimensional optimization problem with a set of "continuity constraints" that should be satisfied together with the other algebraic states and control constraints of the OCP. Using non-negative functions to measure the violation of the "continuity constraints" and of the algebraic constraints separately, the finite-dimensional problem is reformulated as a multi-objective problem with three objectives to be optimized. This paper explores the use of a multi-objective approach, the weighted Tchebycheff scalarization method, to minimize the objective functional and satisfy all the constraint conditions of the OCP. During implementation, a penalty term is added to the Tchebycheff aggregated objective function aiming to force and accelerate the convergence of the constraint violations to zero. The effectiveness of the new methodology is illustrated with the experiments carried out with six OCP.