Mean-field sparse Jurdjevic-Quinn control
Mean-field sparse Jurdjevic-Quinn control
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DOI：
10.1142/s0218202517400140
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发表时间：
2017-06-30
影响因子：
3.5
通讯作者：
Trelat, Emmanuel
中科院分区：
文献类型：
--
作者：
Caponigro, Marco;Piccoli, Benedetto;Trelat, Emmanuel
本文考虑了具有非局部速度的非线性输运方程，描述了测度的时间演化。当考虑有限维系统的平均场极限时，这种方程经常出现。我们解决的问题，控制这些方程的一个时变有界的控制作用本地化的小勒贝格测度的时变控制子集。我们首先定义耗散性的非线性输运方程的李导数的李雅普诺夫函数依赖于措施。然后，假设不受控制的系统是耗散的，我们提供了一个明确的建设的控制律引导系统的不变子水平的李雅普诺夫函数。根据李雅普诺夫函数的李导数设计控制函数和控制域。在这个意义上的建设可以被看作是一个无穷维的模拟著名的Jurdjevic-Quinn过程。此外，控制律呈现稀疏性的意义上说，支持的控制是小的。最后，我们表明，我们的结果适用于一大类动力学方程建模多智能体动力学。
We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic-Quinn procedure. Moreover, the control law presents sparsity properties in the sense that the support of the control is small. Finally, we show that our result applies to a large class of kinetic equations modeling multi-agent dynamics.