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Consider a general linear Hamiltonian system

The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it The formulas relating the growth

The Hamiltonian \\int_X(\\lvert{\\partial_t u}\\rvert^2 + \\lvert{\\nabla u}\\rvert^2 + \\mathbf{m}^2\\lvert{u}\\rvert^2)\\,dx, defined on functions on \\mathbb{R}\\times X, where X is a compact manifold, has critical points which are solutions of the linear Klein-Gordon equation. The author considers perturbations of this Hamiltonian, given by polynomial expressions depending on first order derivatives of u. The associated PDE is then a quasi-linear Klein-Gordon equation. The author shows that, when X is the sphere, and when the mass parameter \\mathbf{m} is outside an exceptional subset of zero measure, smooth Cauchy data of small size \\epsilon give rise to almost global solutions, i.e. solutions defined on a time interval of length c_N\\epsilon^{-N} for any N. Previous results were limited either to the semi-linear case (when the perturbation of the Hamiltonian depends only on u) or to the one dimensional problem. The proof is based on a quasi-linear version of the Birkhoff normal forms method, relying on convenient generalizations of para-differential calculus.

We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in

Let

This work concerns the application of physics‐informed neural networks to the modeling and control of complex robotic systems. Achieving this goal requires extending physics‐informed neural networks to handle nonconservative effects. These learned models are proposed to combine with model‐based controllers originally developed with first‐principle models in mind. By combining standard and new techniques, precise control performance can be achieved while proving theoretical stability bounds. These validations include real‐world experiments of motion prediction with a soft robot and trajectory tracking with a Franka Emika Panda manipulator. The application of physics‐informed neural networks in modeling and controlling intricate robotic systems is explored. This work accounts for nonconservative effects, combining learning models with first‐principles controllers. The synergy of model‐based and learning‐based techniques ensures precise control and establishes theoretical stability bounds. Validation encompasses real‐world experiments predicting soft robot motion and executing trajectory tracking with a Franka Emika Panda manipulator.

Different representations of linear dissipative Hamiltonian and port-Hamiltonian differential–algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between different representations are presented. Characterizations are also derived when a general DAE system can be transformed into one of these structured representations. Approaches for computing the structural information and the described transformations are derived that can be directly implemented as numerical methods. The results are demonstrated with a large number of examples.