A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints

Abstract

In this article, structure-exploiting optimisation algorithms of the sequential quadratic programming (SQP) type are considered for optimal control problems with control and state constraints. Our approach is demonstrated for a 1D mathematical model of a vehicle transporting a fluid container. The model involves a fully coupled system of ordinary differential equations (ODE) and nonlinear hyperbolic first-order partial differential equations (PDE), although the ideas for exploiting the particular structure may be applied to more general optimal control problems as well. The time-optimal control problem is solved numerically by a full discretisation approach. The corresponding nonlinear optimisation problem is solved by an SQP method that uses exact first and second derivative information. The quadratic subproblems are solved using an active-set strategy. In addition, two approaches are examined that exploit the specific structure of the problem: (A) a direct method for the KKT system, and (B) an iterative method based on combining the limited-memory BFGS method with the preconditioned conjugate gradient method. Method (A) is faster for our model problem, but can be limited by the problem size. Method (B) opens the door for a potential extension of the truck-container model to three space dimensions.