Identifying Lyapunov functionals with a PINN : application to a hyperbolic transport equation
Abstract
The identification of a Lyapunov operator P certifying the exponential stability of a linear infinite-dimensional system is, in general, as hard as solving the dynamical system itself. When the modal decomposition of the generator is unknown, the Lyapunov equation must be solved directly, which amounts to solving a high-dimensional operator equation. In this paper, we show that the unavoidable numerical residual of any approximate Lyapunov kernel can be reinterpreted, without any loss of rigor, as a perturbation of the right-hand side Q of the Lyapunov equation. This reformulation turns an approximated numerical object into an identified object that satisfies an exact algebraic relation, thereby allowing the classical converse Lyapunov theorem to be applied directly to certify -or to disprove -the µ-exponential stability of the original system through finite-dimensional positivity tests. We then propose to leverage this certification framework using Physics-Informed Neural Networks (PINNs) as parametric solvers. By taking the system's varying parameters directly as inputs, the PINN avoids the mesh-generation bottleneck and enables an efficient, warm-started adaptive search for critical stability boundaries. The proposed methodology is illustrated on a hyperbolic transport equation with a boundary coupling.
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BibTeX
@misc{bajodek2026identifying,
title = {{Identifying Lyapunov functionals with a PINN : application to a hyperbolic transport equation}},
author = {Mathieu Bajodek and Johan Peralez and Hassan Megdoul and Claire Valentin},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05766324v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Identifying Lyapunov functionals with a PINN : application to a hyperbolic transport equation
AU - Mathieu Bajodek
AU - Johan Peralez
AU - Hassan Megdoul
AU - Claire Valentin
PY - 2026
DA - 2026/09/27
PB - HAL
UR - https://hal.science/hal-05766324v1
AB - The identification of a Lyapunov operator P certifying the exponential stability of a linear infinite-dimensional system is, in general, as hard as solving the dynamical system itself. When the modal decomposition of the generator is unknown, the Lyapunov equation must be solved directly, which amounts to solving a high-dimensional operator equation. In this paper, we show that the unavoidable numerical residual of any approximate Lyapunov kernel can be reinterpreted, without any loss of rigor, as a perturbation of the right-hand side Q of the Lyapunov equation. This reformulation turns an approximated numerical object into an identified object that satisfies an exact algebraic relation, thereby allowing the classical converse Lyapunov theorem to be applied directly to certify -or to disprove -the µ-exponential stability of the original system through finite-dimensional positivity tests. We then propose to leverage this certification framework using Physics-Informed Neural Networks (PINNs) as parametric solvers. By taking the system's varying parameters directly as inputs, the PINN avoids the mesh-generation bottleneck and enables an efficient, warm-started adaptive search for critical stability boundaries. The proposed methodology is illustrated on a hyperbolic transport equation with a boundary coupling.
ER -