Machine learning-based techniques have recently been adapted to solve control problems. The Neural Lyapunov Control (NLC) method is one such example. This approch combines Artificial Networks with Satisfiability Modulo Theories (SMT) solvers synthesise stabilising laws and prove their formal correctness. formers are trained over a dataset of state-space samples generate candidate functions, whi...

1. Lyapunov Exponents for Differential Equations 2. Abstract Theory of Lyapunov Exponents 3. Regularity of Lyapunov Exponents Associated with Differential Equations 4. Lyapunov Stability Theory 5. The Oseledets Decomposition 6. Dynamical Systems with Nonzero Lyapunov Exponents. Multiplicative Ergodic Theorem 7. Nonuniform Hyperbolicity. Regular Sets 8. Examples of Nonuniformly Hyperbolic System...

We consider one-step cocycles of 2ˆ 2 matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist, and are characterized by their support. Under an ...

Lyapunov exponents are an important concept to describe qualitative properties of dynamical systems. For instance, chaotic systems can be caracterized with the positivity of the largest Lyapunov exponent. In this paper, we use the Iwasawa decomposition of the semisimple Lie group Sp(n,R) and the enlargement of the phase space to give a theoretical computation of Lyapunov exponents of almost per...

Learning and synthesizing stabilizing controllers for unknown nonlinear control systems is a challenging problem real-world industrial applications. Koopman operator theory allows one to analyze through the lens of linear bilinear systems. The key idea these methods lies in transformation coordinates system into observables, which are that allow representation original (control system) as highe...

A new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees freedom is introduced. We thus obtain explicit real entire Hamiltonians on <mml:mrow class="MJX-TeXAto...

We prove that if a homogeneous, continuously differentiable vector field is asymptotically stable, then it admits a Lyapunov function which is the ratio of two polynomials (i.e., a rational function). We further show that when the vector field is polynomial, the Lyapunov inequalities on both the rational function and its derivative have sums of squares certificates and hence such a Lyapunov fun...

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton-Jacobi-Bellman partial differential inequality of 2nd order. We give local and global versions of the First and Second Lyapunov Theorems assuming the e...