129 research outputs found
Challenges in Optimal Control of Nonlinear PDE-Systems
The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered
On quantitative hypocoercivity estimates based on Harris-type theorems
This review concerns recent results on the quantitative study of convergence
towards the stationary state for spatially inhomogeneous kinetic equations. We
focus on analytical results obtained by means of certain probabilistic
techniques from the ergodic theory of Markov processes. These techniques are
sometimes referred to as Harris-type theorems. They provide constructive proofs
for convergence results in the (or total variation) setting for a large
class of initial data. The convergence rates can be made explicit (both for
geometric and sub-geometric rates) by tracking the constants appearing in the
hypotheses. Harris-type theorems are particularly well-adapted for equations
exhibiting non-explicit and non-equilibrium steady states since they do not
require prior information on the existence of stationary states. This allows
for significant improvements of some already-existing results by relaxing
assumptions and providing explicit convergence rates. We aim to present
Harris-type theorems, providing a guideline on how to apply these techniques to
the kinetic equations at hand. We discuss recent quantitative results obtained
for kinetic equations in gas theory and mathematical biology, giving some
perspectives on potential extensions to nonlinear equations.Comment: 40 pages, typos are corrected, new references are added and structure
of the paper has change
Odd elastohydrodynamics: non-reciprocal living material in a viscous fluid
Motility is a fundamental feature of living matter, encompassing single cells
and collective behavior. Such living systems are characterized by
non-conservativity of energy and a large diversity of spatio-temporal patterns.
Thus, fundamental physical principles to formulate their behavior are not yet
fully understood. This study explores a violation of Newton's third law in
motile active agents, by considering non-reciprocal mechanical interactions
known as odd elasticity. By extending the description of odd elasticity to a
nonlinear regime, we present a general framework for the swimming dynamics of
active elastic materials in low-Reynolds-number fluids, such as wave-like
patterns observed in eukaryotic cilia and flagella. We investigate the
non-local interactions within a swimmer using generalized material elasticity
and apply these concepts to biological flagellar motion. Through simple
solvable models and the analysis of {\it Chlamydomonas} flagella waveforms and
experimental data for human sperm, we demonstrate the wide applicability of a
non-local and non-reciprocal description of internal interactions within living
materials in viscous fluids, offering a unified framework for active and living
matter physics.Comment: 18 pages, 9 figure
Allee optimal control of a system in ecology
International audienceThe Allee threshold of an ecological system distinguishes the sign of population growth either towards extinction or to carrying capacity. In practice human interventions can tune the Allee threshold for instance thanks to the sterile male technique and the mating disruption. In this paper we address various control objectives for a system described by a diffusion-reaction equation regulating the Allee threshold, viewed as a real parameter determining the unstable equilibrium of the bistable nonlinear reaction term. We prove that this system is the mean field limit of an interacting system of particles in which individual behaviours are driven by stochastic laws. Numerical simulations of the stochastic process show that population propagations are governed by wave-like solutions corresponding to traveling solutions of the macroscopic reaction-diffusion system. An optimal control problem for the macroscopic model is then introduced with the objective of steering the system to a target traveling wave. The relevance of this problem is motivated by the fact that traveling wave solutions model the fact that bounded space domains reach asymptotically an equilibrium configuration. Using well known analytical results and stability properties of traveling waves, we show that well-chosen piecewise constant controls allow to reach the target approximately in sufficiently long time. We then develop a direct computational method and show its efficiency for computing such controls in various numerical simulations. Finally we show the efficiency of the obtained macroscopic optimal controls in the microscopic system of interacting particles and we discuss their advantage when addressing situations that are out of reach for the analytical methods. We conclude the article with some open problems and directions for future research
[Book of abstracts]
USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos
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