A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure
A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure
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DOI：
10.1109/tac.2007.911318
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发表时间：
2008-01-01
影响因子：
6.8
通讯作者：
Sontag, Eduardo D.
中科院分区：
文献类型：
--
作者：
Jovanovic, Mihailo R.;Arcak, Murat;Sontag, Eduardo D.
研究了一类具有循环互连结构的分布式系统。这些系统出现在几个生化应用中，它们可以经历扩散驱动的不稳定性，从而导致空间异质模式的形成。本文确定了一类扩散的加入不具有不稳定效应的循环系统。对于这些系统，当“割线”准则满足时，全局稳定性结果成立。在线性情况下，证明了割线条件是解耦二次Lyapunov函数存在的充分必要条件，将最近的对角稳定性结果推广到偏微分方程。对于非递减耦合的反应扩散方程，建立了原点的全局渐近稳定性。所有导出的结果对线性和非线性正扩散项都成立。隔室系统也显示出类似的结果。
A class of distributed systems with a cyclic interconnection structure is considered. These systems arise in several biochemical applications and they can undergo diffusion-driven instability which leads to a formation of spatially heterogeneous patterns. In this paper, a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified. For these systems global stability results hold if the "secant" criterion is satisfied. In the linear case, it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic Lyapunov function, which extends a recent diagonal stability result to partial differential equations. For reaction-diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established. All of the derived results remain true for both linear and nonlinear positive diffusion terms. Similar results are shown for compartmental systems.