Observability of a string-beams network with many beams

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-07-11 DOI:10.1051/cocv/2023054

Anna Chiara Lai, P. Loreti, M. Mehrenberger

引用次数: 0

Abstract

We prove the direct and inverse observability inequality for a network connecting one string with infinitely many beams, at a common point, in the case where the lengths of the beams are all equal. The observation is at the exterior node of the string and at the exterior nodes of all the beams except one. The proof is based on a careful analysis of the asymptotic behavior of the underlying eigenvalues and eigenfunctions, and on the use of a Ingham type theorem with weakened gap condition [6]. On the one hand, the proof of the crucial gap condition already observed in the case where there is only one beam [1] is new and based on elementary monotonicity arguments. On the other hand, we are able to handle both the complication arising with the appearance of eigenvalues with unbounded multiplicity, due to the many beams case, and the terms coming from the weakened gap condition, arising when at least 2 beams are present. AMS Subject Classification Primary: 93B07, 74K10; Secondary: 42A16, 35M10, 35A25.

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具有多束的弦-梁网络的可观测性

我们证明了在一个公共点上连接一个具有无限多束的弦的网络，在这些束的长度都相等的情况下，其正、逆可观察性不等式。观察是在弦的外部节点和所有光束的外部节点，除了一个。该证明是基于对潜在的特征值和特征函数的渐近行为的仔细分析，以及使用带有弱间隙条件的Ingham型定理[6]。一方面，在只有一根梁[1]的情况下已经观察到的关键间隙条件的证明是新的，并且是基于初等单调性参数的。另一方面，我们能够处理由于多光束情况下具有无界多重性的特征值的出现所引起的复杂性，以及来自弱间隙条件的项，当至少存在2束时产生。AMS学科分类初级:93B07, 74K10;二级:42A16、35M10、35A25。

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来源期刊

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

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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.