[Paper Review] On the construction of Lyapunov functions with computer assistance

Kaname Matsue, Tomohiro Hiwaki+1 more|arXiv (Cornell University)|Apr 20, 2016

Control and Stability of Dynamical Systems10 references3 citations

TL;DR

This paper presents a computer-assisted methodology for constructing explicit Lyapunov functions in neighborhoods of hyperbolic fixed points for both continuous and discrete dynamical systems. Using two-stage verification—negative definiteness of an associated matrix (Stage 1) and direct computation of the derivative (Stage 2)—it validates quadratic Lyapunov functions over concrete domains, enabling rigorous analysis of local dynamics and trajectory re-parameterization via Lyapunov values.

ABSTRACT

Computer assisted procedures of Lyapunov functions defined in given neighborhoods of fixed points for flows and maps are discussed. We provide a systematic methodology for constructing explicit ranges where quadratic Lyapunov functions exist in two stages; negative definiteness of associating matrices and direct approach. We note that the former is equivalent to the procedure of cones describing enclosures of the stable and the unstable manifolds of invariant sets, which gives us flexible discussions of asymptotic behavior not only around equilibria for flows but also fixed points for maps. Additionally, our procedure admits a re-parameterization of trajectories in terms of values of Lyapunov functions. Several verification examples are shown for discussions of applicability.

Motivation & Objective

• To develop a systematic, computer-assisted method for validating Lyapunov functions in explicit neighborhoods of hyperbolic fixed points.
• To bridge the gap between abstract existence theorems and concrete, computationally verifiable Lyapunov functions for nonlinear dynamical systems.
• To extend the applicability of Lyapunov functions beyond equilibria by combining two-stage verification procedures.
• To enable re-parameterization of trajectories using Lyapunov function values for improved analysis of asymptotic behavior.
• To provide a unified framework linking Lyapunov functions, cone conditions, and verified numerical computation for dynamical systems.

Proposed method

• Employing interval arithmetic to rigorously verify the negative definiteness of a matrix A(z) associated with the time derivative of a quadratic Lyapunov function along trajectories (Stage 1).
• For domains not containing equilibria, directly computing and verifying dL/dt < 0 along solution orbits using validated numerical integration (Stage 2).
• Using the Jacobian matrix at a fixed point to define the associated matrix A(z), with its negative definiteness implying hyperbolicity and local uniqueness of the equilibrium.
• Extending the method to discrete systems by verifying strict negative definiteness of a matrix B(z) associated with the Lyapunov function increment.
• Applying re-parameterization of trajectories using values of the Lyapunov function to track asymptotic dynamics over finite and infinite time intervals.
• Utilizing verified numerical solvers and domain partitioning to expand the size of Lyapunov domains beyond the immediate neighborhood of fixed points.

Experimental results

Research questions

• RQ1 How can we systematically construct and validate explicit Lyapunov functions for flows and maps around hyperbolic fixed points using computer-assisted methods?
• RQ2 What is the relationship between the negative definiteness of the associated matrix A(z) and the classical cone conditions used in dynamical systems theory?
• RQ3 Can the two-stage verification procedure—matrix definiteness followed by direct derivative computation—be used to extend Lyapunov domains beyond the immediate vicinity of equilibria?
• RQ4 How does the method enable re-parameterization of trajectories in terms of Lyapunov function values for analyzing long-term behavior?
• RQ5 What are the limitations of the method in regions where eigenvalues are close to the imaginary axis (flows) or unit circle (maps)?

Key findings

• The negative definiteness of the matrix A(z) associated with dL/dt is equivalent to the cone condition for stable and unstable manifolds, providing a unified framework for asymptotic analysis.
• Stage 1 verification via matrix definiteness ensures local uniqueness of hyperbolic equilibria and provides a sufficient condition for Lyapunov function validity.
• Stage 2 allows extension of Lyapunov domains even when Stage 1 fails, particularly in regions away from equilibria.
• The method successfully validates Lyapunov functions for both continuous and discrete systems using verified numerical computation with interval arithmetic.
• Re-parameterization of trajectories using Lyapunov function values enables tracking of asymptotic dynamics, including singular behaviors like blow-up.
• The procedure becomes numerically challenging in eigendirections where eigenvalues have real parts near zero (flows) or moduli near one (maps).