The well-posedness of Riccati equations plays a central role in the study of the optimal control problem with quadratic functionals for linear partial differential equations (PDEs). Indeed, it allows the synthesis of the optimal control by solving the Riccati equation corresponding to the minimization problem, and then of the closed-loop equation. In this lecture I…

I will discuss the existence and orbital stability of standing-wave solutions (i.e., with a specific time-dependence) with minimal energy (so-called ground states) to a non-linear Schrödinger equation where the L² norm is prescribed. I will focus on the simpler case where the energy is bounded below and show a novel approach that simplifies the proof.  Zoom Meeting: Link  

We study the effect of total variation regularization on PDE-constrained optimization problems, where the control input functions may only attain finitely many integer values. The regularization helps to avoid undesirable effects such as chattering behavior. In particular, the weak-* compactness of the feasible set in the space of functions of bounded variation allows to derive…

Control systems are a class of dynamical systems that contain forcing terms. When control systems are used in engineering applications, the forcing terms can represent forces that can be applied to the systems. Then the feedback control problem consists of finding formulas for the forcing terms, which are functions that can depend on the state…

We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls.…

Classification is a fundamental task in data science and machine learning, and in the past ten years there have been significant improvements on classification tasks (e.g. via deep learning). However, recently there have been a number of works demonstrating that these improved algorithms can be “fooled” using specially constructed adversarial examples. In turn, there has been increased…

We consider a long-time behavior of a stochastically forced nonlinear oscillator. In a long-time limit the force converges to fractional Brownian motion, a process that has memory. In contrast, we show that the  limit of the nonlinear oscillator driven by this force converges to diffusion driven by standard (not fractional) Brownian motion, and thus retains…

We will discuss some old and new results concerning the long-time behavior of solutions to the two-dimensional incompressible Euler equations. Specifically, we discuss whether steady states can be isolated, wandering for solutions starting nearby certain steady states, singularity formation at infinite time, and finally some results/conjectures on the infinite-time limit near and far from equilibrium.…

The so-called $l_0$ pseudonorm counts the number of nonzero components of a vector. It is standard in sparse optimization problems. However, as it is a discontinuous and nonconvex function, the l0 pseudonorm cannot be satisfactorily handled with the Fenchel conjugacy. In this talk, we review a series of recent results on a class of Capra…

During the preparation of a foundational chapter on manifolds of mappings for a book on geometric continuum mechanics I found out that the following object behaves surprisingly well as source of a manifold of mappings: — A Whitney manifold germ M˜ ⊃ M consists of an open manifold M˜ together with a closed subset M…

In this talk, I will first present a very simple quantitative form of the Young-Fenchel inequality.  I will then discuss some applications: a short proof of the Brøndsted-Rockafellar in Hilbert spaces and a primal-dual attainment for perturbed convex minimization problems. I will finally explain how this inequality (or some generalizations) can be used for quantitative…

his talk presents various characterizations of the subdifferential of the pointwise supremum of an arbitrary family of convex functions, as well as some featured applications. Starting by the maximum generality framework, we move after to particular contexts in which some continuity and compacity assumptions are either imposed or inforced via processes of compactification of the…

In this talk I will present recent results with In-Jeong from Seoul national university where we study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on $\mathbb{S}$. We show that this system is locally well-posed in $L^p, p\geq 1$ as well as for atomic measures, that is logarithmic…

This article is devoted to a general class of one dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that…

Motivated by motion of myxobacteria, we review several kinetic approaches for modeling motion of self-propelled, interacting rods. We will focus on collisional models of Boltzmann type and discuss the derivation of the governing equations, the range of their validity, and present some analytical and numerical results. We will show that collisional models have a natural…

We are concerned with the rigorous justification of the  so-called hard congestion limit from a compressible system with singular pressure towards a mixed  compressible-incompressible system modeling partially congested dynamics, in the framework of BV solutions. We will consider small BV perturbations of reference solutions constituted by (possibly interacting) large interfaces, and we will  analyze the dynamics of…

In this talk we present some recent result about the long time behavior of the solutions for some diffusion processes on a metric graph.  We study  evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat…

The talk is about oscillatory integral operators on manifolds.  Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.

In this talk, I will talk about some recent well-posedness and stability results for several fluid models in 2D. More precisely, I will discuss the global well-posedness for the 2D Boussinesq equations with fractional dissipation. For the Oldroyd-B model, we show that small smooth data lead to global and stable solutions. When Navier-Stokes is coupled…

In this talk, we discuss second-order optimality conditions for optimal control problems. This analysis is very important when we study the stability of the solution to the control problem with respect to small perturbations of the data. It is also crucial for proving superlinear or quadratic convergence of numerical algorithms for solving the problem, as…