Seminario Prof. Sergei Kuksin
Ergodicity , mixing, controllability and KAM
Abstract:
To prove the ergodicity for Hamiltonian systems of big or infinite dimension is a notoriously complicated problem of high importance. But what we often have in physics are not Hamiltonian systems, but systems of the form
Hamiltonian system + small dissipation + small random forcing,
where the random forcing may be very degenerate (i.e. it affects only a few modes). For such systems an analogy of the ergodicity is called the mixing.
In my talk I will remind the definition of the mixing and explain how the (optimal) control and KAM-theory provide powerful tools to prove the mixing for the systems above.
The talk is based on joint works with Armen Shirikyan and Vahagn Nersesyan, arXiv:1802.03250v2, and with Huilin Zhang, arXiv:1812.11706v2--
Seminario Prof. Jean-Paul Gauthier
Hypoelliptic diffusion, Chu duality and human vision
Abstract:
In neuroscience, there is a model of the primary visual cortex of mammals V1 as
a sub-Riemannian structure over the group SE(2) of motions of the plane. The
Hypoelliptic diffusion associated with this metric is used for the purpose of
image completion or image reconstruction.
In my talk, I shall present the theory, together with a semi-discrete
improvement of the model more in accordance with the discrete structure of V1,
over the group SE(2,N) of dicrete rotations and all translations.
The group under consideration being maximally almost periodic, and therefore
subject to Chu duality, there is a much simpler harmonic analysis on it.
It results in nice and efficient algorithms both for image completion and pattern
recognition.
A preliminary version of the full work may be found at
href= https://arxiv.org/pdf/1704.03069.pdf
Seminario Prof. Valerii Obukhovskii
General boundary value problems for fractional order degenerate control systems.
Abstract: We consider nonlocal boundary value problems for a feedback control system governed by a fractional degenerate (Sobolev type) semilinear differential inclusion in a Banach space.
To solve this problem, we introduce a multivalued integral operator whose fixed points determine its solutions and study the properties of this operator.
Seminario Prof.ssa Francesca Carlotta Chittaro
Asymptotic ensemble stabilizability of the Bloch equation.
See the abstract
Seminario Prof. Andrey Agrachev
Geometry of Constrained optimization: the Indices of Morse and Maslov.
Abstract:
In this talk, I am going to explain how to connect classical Lagrange multipliers rule with elementary symplectic geometry. Central observation is a universal formula that relates the Morse index of the second variation to the Maslov index of a curve in the Lagrange Grassmannian. No preliminary knowledge of symplectic geometry is required.
Seminario Prof. Andrey Agrachev
OPTIMAL CONTROL AND A GENERALIZED HAMILTONIAN SYSTEM.
Abstract: We study time-optimal problems for n-dimentional systems controlled by a k-dimensional control with values in a ball. We assume that k is smaller than n.
Pontryagin Maximum Principle essentially reduces the study of optimal solutions to the Hamiltonian system with a discontinuous right-hand side. We show that under certain generic conditions solutions of the system are piecewise smooth and effectively compute the left and right limits of their derivative at the nonsmooth points. We also show that the Cauchy problem is well-posed in C^0-topology.
Seminario Prof. Velimir Jurdjevic
INTEGRABLE HAMILTONIAN SYSTEMS ON LIE GROUPS, see the abstract