Stabilization of the wave equation through nonlinear Dirichlet actuation

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-08-29 DOI:10.1051/cocv/2022077

Nicolas Vanspranghe, Francesco Ferrante, C. Prieur

引用次数: 0

Abstract

In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space L 2 (Ω) × H −1 (Ω). It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in H 1 (Ω) × L 2 (Ω) of the wave equation with Neumann boundary conditions.

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非线性狄利克雷驱动下波动方程的镇定

本文研究了具有Dirichlet边界条件的高维波动方程的非线性(特别是饱和)镇定问题。波浪动力学受耗散非线性速度反馈影响，在最优能量空间l2 (Ω) × H−1 (Ω)上产生强连续收缩半群。首先证明了在上述拓扑结构下闭环方程的任意解收敛于零。其次，在反馈非线性在零附近线性增长的条件下，建立了初始数据光滑解的多项式能量衰减率。这构成了与具有诺伊曼边界条件的波动方程h1 (Ω) × l2 (Ω)的非线性稳定有关的著名结果的新的狄利克雷对立物。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

自引率

7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.