Affine relaxations for the solutions of constrained parametric ordinary differential equations

Abstract

Summary This work presents a numerical method for evaluating affine relaxations of the solutions of parametric ordinary differential equations. This method is derived from a general theory for the construction of a polyhedral outer approximation of the reachable set (“polyhedral bounds”) of a constrained dynamic system subject to uncertain time‐varying inputs and initial conditions. This theory is an extension of differential inequality‐based comparison theorems. The new affine relaxation method is capable of incorporating information from simultaneously constructed interval bounds as well as other constraints on the states; not only does this improve the quality of the relaxations but it also yields numerical advantages that speed up the computation of the relaxations. Examples demonstrate that tight affine relaxations can be computed efficiently with this method.

Introduction to the Special Issue on Global and Robust Optimization of Dynamic Systems

“…The paper by Harwood and Barton presents a novel approach to computing affine relaxations on the solutions of constrained parametric ordinary differential equations (ODEs). They developed an extension of differential inequality–based comparison theorems, whereby a polyhedral outer approximation of the reachable set of the constrained ODE systems is constructed under uncertain time‐varying inputs and initial conditions.…”