On the Solutions of Optimal Control Problems Constrained by Ordinary Differential Equations with Vector-Matrix Coefficients Using FICO Xpress Mosel

Abstract

This study addresses a general class of quadratic optimal control problems (OCPs) constrained by ordinary differential equations (ODEs) with vector-matrix coefficients. Due to the intractability of analytical solutions for complex dynamic systems, the focus is on developing and comparing efficient numerical methods. An analytical framework is first established by applying first-order optimality conditions to the Hamiltonian, yielding a system of first-order ODEs. The associated Riccati differential equation is then solved using a state transformation approach. For numerical solutions, the objective functional is discretized using Simpson's 1331​ rule, and the system dynamics are approximated using a fifth-order implicit integration scheme. The discretized problem is reformulated as an unconstrained optimization problem via the Augmented Lagrangian Method and solved using both the CGM and FICO Xpress Mosel. Comparative results reveal that FICO Xpress Mosel provides faster convergence and greater numerical stability, especially for high-dimensional problems. These findings underscore the effectiveness of commercial solvers like FICO Xpress Mosel in solving large-scale quadratic OCPs with enhanced accuracy and efficiency.