LpL^p-LqL^q Maximal Regularity for some Operators Associated with Linearized Incompressible Fluid-Rigid Body Problems

We study an unbounded operator arising naturally after linearizing the system modelling the motion of a rigid body in a viscous incompressible fluid. We show that this operator is $\mathcal{R}$ sectorial in $L^q$ for every $q\in (1,\infty)$, thus it has the maximal $L^p$-$L^q$ regularity property. Moreover, we show that the generated semigroup is exponentially stable with respect to the $L^q$ norm. Finally, we use the results to prove the global existence for small initial data, in an $L^p$-$L^q$ setting, for the original nonlinear problem

The mathematics of aquatic locomotion

Aquatic locomotion is a self-propelled motion through a liquid medium. It can be of biological nature (fish, microorganisms,. . .) or performed by robotic swimmers. This snapshot aims to introduce the reader to some of the challenges raised by the mathematical modelling of aquatic locomotion, even in seemingly very simple cases

Perturbations of time optimal control problems for a class of abstract parabolic systems

In this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge, in an appropriate sense, to a given strictly negative operator. Our main application to PDEs systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients, as we encounter in homogenization theory. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the solutions of the time optimal control problems for the systems with oscillating coefficients converge, in the usual norms, to the solution of the corresponding problem for the homogenized system. In order to prove our main theorem, we provide several new results, which could be of a broader interest, on time and norm optimal control problems

Unconditionnally stable scheme for Riccati equation

We present a numerical scheme for the resolution of matrix Riccati equation used in control problems. The scheme is unconditionnally stable and the solution is definite positive at each time step of the resolution. We prove the convergence in the scalar case and present several numerical experiments for classical test cases.Comment: 11 page

Controllability and positivity constraints in population dynamics with age structuring and diffusion

This Accepted Manuscript will be available for reuse under a CC BY-NC-ND licence after 24 months of embargo periodIn this article, we study the null controllability of a linear system coming from a population dynamics model with age structuring and spatial diffusion (of Lotka–McKendrick type). The control is localized in the space variable as well as with respect to the age. The first novelty we bring in is that the age interval in which the control needs to be active can be arbitrarily small and does not need to contain a neighbourhood of 0. The second one is that we prove that the whole population can be steered into zero in a uniform time, without, as in the existing literature, excluding some interval of low ages. Moreover, we improve the existing estimates of the controllability time and we show that our estimates are sharp, at least when the control is active for very low ages. Finally, we show that the system can be steered between two positive steady states by controls preserving the positivity of the state trajectory. The method of proof, combining final-state observability estimates with the use of characteristics and with L∞ estimates of the associated semigroup, avoids the explicit use of parabolic Carleman estimatesThe research of Enrique Zuazua was supported by the Advanced Grant DyCon (Dynamical Control) of the European Research Council Executive Agency (ERC), the MTM2014-52347 and MTM2017-92996 Grants of the MINECO (Spain) and the ICON project of the French ANR-16-ACHN-001

Fast and strongly localized observation for the Schrödinger equation

International audienceWe study the exact observability of systems governed by the Schrödinger equation in a rectangle with homogeneous Dirichlet (respectively Neumann) boundary conditions and with Neumann (respectively Dirichlet) boundary observation. Generalizing previous results of Ramdani, Takahashi, Tenenbaum and Tucsnak, we prove that these systems are exactly observable in arbitrarily small time. Moreover, we show that the above results hold even if the observation regions have arbitrarily small measures. More precisely, we prove that in the case of homogeneous Neumann boundary conditions with Dirichlet boundary observation, the exact observability property holds for every observation region whith non empty interior. In the case of homogeneous Dirichlet boundary conditions with Neumann boundary observation, we show that the exact observability property holds if and only if the observation region has an open intersection with an edge of each direction. Moreover, we give explicit estimates for the blow-up rate of the observability constants as the time and (or) the size of the observation region tend to zero. The main ingredients of the proofs are an effective version of a theorem of Beurling and Kahane on non harmonic Fourier series and an estimate for the number of lattice points in the neighbourhood of an ellipse

A uniformly stable finite difference space semi-discretization for the internal stabilization of the plate equation in a square

International audienceWe propose a finite difference space semi-discretization of the stabilized Bernoulli-Euler plate equation in a square. The scheme studied yields a uniform exponential decay rate with respect to the mesh size