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Inspired by the widespread concept of Lyapunov-Krasovskii functionals of complete type, this article proposes an alternative class of functionals, termed Lyapunov-Krasovskii functionals of robust type. Their construction aims at improving…
The fundamental matrix and the delay Lyapunov matrix of linear delay difference equations are introduced. Some properties of the Lyapunov matrix, and the jump discontinuities of its derivative are proven, leading to its construction in the…
This work studies the problem of searching for homogeneous polynomial Lyapunov functions for stable switched linear systems. Specifically, we show an equivalence between polynomial Lyapunov functions for systems of this class and quadratic…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
The present paper addresses the swing equation with additional delayed damping as an example for pendulum-like systems. In this context, it is proved that recurring sub- and supercritical Hopf bifurcations occur if time delay is increased.…
Deep equilibrium (DEQ) models have emerged as a promising class of implicit layer models, which abandon traditional depth by solving for the fixed points of a single nonlinear layer. Despite their success, the stability of the fixed points…
In chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real-time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure…
DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with…
An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proof are presented in details for the case of systems with multiple delays. The state of the art, ongoing…
Recent research in machine learning has given rise to a flourishing literature on the quantification and decomposition of model uncertainty. This information can be very useful during interactions with the learner, such as in active…
Conic optimization has recently emerged as a powerful tool for designing tractable and guaranteed algorithms for non-convex polynomial optimization problems. On the one hand, tractability is crucial for efficiently solving large-scale…
Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free…
This paper presents a counterexample-guided iterative algorithm to compute convex, piecewise linear (polyhedral) Lyapunov functions for uncertain continuous-time linear hybrid systems. Polyhedral Lyapunov functions provide an alternative to…
Partial Differential Equations (PDEs) are fundamental for modeling physical systems, yet solving them in a generic and efficient manner using machine learning-based approaches remains challenging due to limited multi-input and multi-scale…
In order to analyze structure of tangent spaces of a transient orbit, we propose a new algorithm which pulls back vectors in tangent spaces along the orbit by using a calculation method of covariant Lyapunov vectors. As an example, the…
The main goal of this work is to provide a description of transitions from uniform to non-uniform snychronization in diffusions based on large deviation estimates for finite time Lyapunov exponents. These can be characterized in terms of…
In this paper we employ SMT solvers to soundly synthesise Lyapunov functions that assert the stability of a given dynamical model. The search for a Lyapunov function is framed as the satisfiability of a second-order logical formula, asking…
We present an algorithm for the rigorous integration of Delay Differential Equations (DDEs) of the form $x'(t)=f(x(t-\tau),x(t))$. As an application, we give a computer assisted proof of the existence of two attracting periodic orbits…
Training large deep neural networks is resource intensive. This study investigates whether Lyapunov exponents can accelerate this process by aiding in the selection of hyperparameters. To study this I formulate an optimization problem using…
This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's…