Shape optimization with Stokes constraints over the set of axisymmetric domains

International audienceIn this paper, we are interested in the study of shape optimizations problems with Stokes constraints within the class of axisymmetric domains represented by the graph of a function. Existence results with weak assumptions on the regularity of the graph are provided. We strongly use these assumptions to get some topological properties. We formulate the (shape) optimization problem using different constraints formulations: uniform bound constraints on the function and its derivative and/or volume (global) constraint. Writing the first order optimality conditions allows to provide quasi-explicit solutions in some particular cases and to give some hints for the treatment of the generic problem. Furthermore, we extend the (negative) result of [16] dealing with the non optimality of the cylinder

The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schr\"odinger equation

Prox-DBRO-VR: A Unified Analysis on Decentralized Byzantine-Resilient Composite Stochastic Optimization with Variance Reduction and Non-Asymptotic Convergence Rates

Decentralized Byzantine-resilient stochastic gradient algorithms resolve efficiently large-scale optimization problems in adverse conditions, such as malfunctioning agents, software bugs, and cyber attacks. This paper targets on handling a class of generic composite optimization problems over multi-agent cyberphysical systems (CPSs), with the existence of an unknown number of Byzantine agents. Based on the proximal mapping method, two variance-reduced (VR) techniques, and a norm-penalized approximation strategy, we propose a decentralized Byzantine-resilient and proximal-gradient algorithmic framework, dubbed Prox-DBRO-VR, which achieves an optimization and control goal using only local computations and communications. To reduce asymptotically the variance generated by evaluating the noisy stochastic gradients, we incorporate two localized variance-reduced techniques (SAGA and LSVRG) into Prox-DBRO-VR, to design Prox-DBRO-SAGA and Prox-DBRO-LSVRG. Via analyzing the contraction relationships among the gradient-learning error, robust consensus condition, and optimal gap, the theoretical result demonstrates that both Prox-DBRO-SAGA and Prox-DBRO-LSVRG, with a well-designed constant (resp., decaying) step-size, converge linearly (resp., sub-linearly) inside an error ball around the optimal solution to the optimization problem under standard assumptions. The trade-offs between the convergence accuracy and the number of Byzantine agents in both linear and sub-linear cases are characterized. In simulation, the effectiveness and practicability of the proposed algorithms are manifested via resolving a sparse machine-learning problem over multi-agent CPSs under various Byzantine attacks.Comment: 14 pages, 0 figure

Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions

We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset $\Omega$ of $\R^n$. The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a $L^1$ constraint on densities, the so-called {\it Rellich functions} maximize this functional.Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that the $L^\infty$-norm of {\it Rellich functions} may be large, depending on the shape of $\Omega$, we analyze the effect of adding pointwise constraints when maximizing the same functional. We investigate the optimality of {\it bang-bang} functions and {\it Rellich densities} for this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximation.Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations

Density-feedback control in traffic and transport far from equilibrium

A bottleneck situation in one-lane traffic-flow is typically modelled with a constant demand of entering cars. However, in practice this demand may depend on the density of cars in the bottleneck. The present paper studies a simple bimodal realization of this mechanism to which we refer to as density-feedback control (DFC): If the actual density in the bottleneck is above a certain threshold, the reservoir density of possibly entering cars is reduced to a different constant value. By numerical solution of the discretized viscid Burgers equation a rich stationary phase diagram is found. In order to maximize the flow, which is the goal of typical traffic-management strategies, we find the optimal choice of the threshold. Analytical results are verified by computer simulations of the microscopic TASEP with DFC.Comment: 7 pages, 5 figure