1,510 research outputs found
Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces
The aim of this paper is twofold. On the one hand, we discuss the notions of
strong chain recurrence and strong chain transitivity for flows on metric
spaces, together with their characterizations in terms of rigidity properties
of Lipschitz Lyapunov functions. This part extends to flows some recent results
for homeomorphisms of Fathi and Pageault. On the other hand, we use these
characterisations to revisit the proof of a theorem of Paternain, Polterovich
and Siburg concerning the inner rigidity of a Lagrangian submanifold
contained in an optical hypersurface of a cotangent bundle, under the
assumption that the dynamics on is strongly chain recurrent. We also
prove an outer rigidity result for such a Lagrangian submanifold ,
under the stronger assumption that the dynamics on is strongly chain
transitive.Comment: 26 pages, 2 figure
Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory
We consider Lagrangian submanifolds lying on a fiberwise strictly convex
hypersurface in some cotangent bundle or, respectively, in the domain bounded
by such a hypersurface.
We establish a new boundary rigidity phenomenon, saying that certain
Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the
same Liouville class) into the interior domain.
Moreover, we study the "non-removable intersection set" between the
Lagrangian and the hypersurface, and show that it contains a set with specific
dynamical behavior, known as Aubry set in Aubry-Mather theory.Comment: The main new point of this revised and substantially enlarged
version, with G.P. Paternain as new co-author, is the relation between
non-removable intersections and Aubry-Mather theor
Propagation of singularities for weak KAM solutions and barrier functions
This paper studies the structure of the singular set (points of
nondifferentiability) of viscosity solutions to Hamilton-Jacobi equations
associated with general mechanical systems on the n-torus. First, using the
level set method, we characterize the propagation of singularities along
generalized characteristics. Then, we obtain a local propagation result for
singularities of weak KAM solutions in the supercritical case. Finally, we
apply such a result to study the propagation of singularities for barrier
functions
A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems
In this paper, we prove a Pontryagin Maximum Principle for constrained
optimal control problems in the Wasserstein space of probability measures. The
dynamics, is described by a transport equation with non-local velocities and is
subject to end-point and running state constraints. Building on our previous
work, we combine the classical method of needle-variations from geometric
control theory and the metric differential structure of the Wasserstein spaces
to obtain a maximum principle stated in the so-called Gamkrelidze form.Comment: 35 page
Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems
We study a growth maximization problem for a continuous time positive linear
system with switches. This is motivated by a problem of mathematical biology
(modeling growth-fragmentation processes and the PMCA protocol). We show that
the growth rate is determined by the non-linear eigenvalue of a max-plus
analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the
ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the
solutions or subsolutions of which yield Barabanov and extremal norms,
respectively. We exploit contraction properties of order preserving flows, with
respect to Hilbert's projective metric, to show that the non-linear eigenvector
of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low
dimensional examples are presented, showing that the optimal control can lead
to a limit cycle.Comment: 8 page
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