Sparse stabilization of dynamical systems driven by attraction and avoidance forces
Sparse stabilization of dynamical systems driven by attraction and avoidance forces
复制标题
由吸引力和回避力驱动的动力系统的稀疏稳定性
DOI：
10.3934/nhm.2014.9.1
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Fornasier
中科院分区：
文献类型：
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作者：
M. Bongini;M. Fornasier
条件自组织和模式形成是在生物、社会和经济背景下出现的相关现象，近年来在数学建模中受到越来越多的关注。与最佳政府战略有关的一个重要问题是，如何设计外部的节俭干预措施，旨在强制执行系统，使其与特定模式趋同。这与其他模型相反，在其他模型中，系统的参与者被允许自由互动，并被认为是自主的，无论是通过游戏规则还是通过嵌入的分散反馈控制规则，收敛到模式。
在本文中，我们解决了为受相互吸引力、排斥力和摩擦力影响的运动粒子系统设计最优集中反馈控制的问题。
在一定的吸引力和排斥力条件下，如果系统的总能量（由其动能和势能部分之和组成）低于某个临界阈值，则已知此类系统
在空间上、时间上一致地自主收敛到保持约束、避免碰撞的稳定构形。如果能量高于这样的临界水平，那么空间相干性可能会丢失。
我们发现，在后一种情况下失去的自组织，一个仍然可以引导系统返回到稳定的能量水平的反馈控制定义为最小化的某个功能与l_1 $-范数惩罚和约束。此外，我们表明，在这类控制的最优策略必然是稀疏的，即，控制每次作用于至多一个代理。这是另一个显著的例子，说明了同亲系统，即，系统中的代理人往往受到附近代理人的影响比远代理人更大，自然倾向于稀疏稳定，解释了政府在社会中的吝啬干预的有效性。
Conditional self-organization and pattern-formation are relevant phenomena arising in biological, social, and economical contexts, and received a growing attention in recent years in mathematical modeling. An important issue related to optimal government strategies is how to design external parsimonious interventions, aiming at enforcing systems to converge to specific patterns. This is in contrast to other models where the players of the systems are allowed to interact freely and are supposed autonomously, either by game rules or by embedded decentralized feedback control rules, to converge to patterns.
In this paper we tackle the problem of designing optimal centralized feedback controls for systems of moving particles, subject to mutual attraction and repulsion forces, and friction.
Under certain conditions on the attraction and repulsion forces, if the total energy of the system, composed of the sum of its kinetic and potential parts, is below a certain critical threshold, then such systems are known to
converge autonomously to the stable configuration of keeping confined and collision avoiding in space, uniformly in time. If the energy is above such a critical level, then the space coherence can be lost.
We show that in the latter situation of lost self-organization, one can nevertheless steer the system to return to stable energy levels by feedback controls defined as the minimizers of a certain functional with $l_1$-norm penalty and constraints. Additionally we show that the optimal strategy in this class of controls is necessarily sparse, i.e., the control acts on at most one agent at each time. This is another remarkable example of how homophilious systems, i.e., systems where agents tend to be strongly more influenced by near agents than far ones, are naturally prone to sparse stabilization, explaining the effectiveness of parsimonious interventions of governments in societies.