Extraction of a computer-certified ODE solver

TL;DR

This paper presents a formal, computer-certified approach to solving initial value problems of ordinary differential equations, ensuring correctness and enabling efficient program extraction for reliable numerical solutions.

Contribution

It introduces a formalization in constructive analysis using Minlog, extracting a verified ODE solver, and discusses potential optimizations and applications in control theory.

Findings

Successfully extracted a verified ODE solver program

Demonstrated efficiency improvements through optimization

Provided numerical experiments validating the approach

Abstract

Reliably determining system trajectories is essential in many analysis and control design approaches. To this end, an initial value problem has to be usually solved via numerical algorithms which rely on a certain software realization. Because software realizations can be error-prone, proof assistants may be used to verify the underlying mathematical concepts and corresponding algorithms. In this work we present a computer-certified formalization of the solution of the initial value problem of ordinary differential equations. The concepts are performed in the framework of constructive analysis and the proofs are written in the \texttt{Minlog} proof system. We show the extraction of a program, which solves an ODE numerically and provide some possible optimization regarding the efficiency. Finally, we provide numerical experiments to demonstrate how programs of a certain high level of abstraction can be obtained efficiently. The presented concepts may also be viewed as a part of preliminary work for the development of formalized nonlinear control theory, hence offering the possibility of computer-assisted controller design and program extraction for the controller implementation.