平均场Kuramoto模型中速度和非局部相互作用的最优控制
Optimal Control of Velocity and Nonlocal Interactions in the Mean-Field Kuramoto Model
摘要：
在本文中，我们研究了如何自同步属性的一群仓本振荡器可以被控制和利用，以实现目标密度和目标相位相干。在无限数量的振荡器的限制下，代理密度的集体动力学由非局部PDE形式的平均场模型描述，其中非局部性来自同步机制。在这种平均场设置中，我们引入两个空间-不定常控制输入来影响振荡器的密度：对应于个体代理的状态反馈律的角速度场，以及调制代理在空间和时间上的相互作用强度的控制参数，即，关于积分非局部项的乘法控制。我们将密度跟踪问题定义为偏微分方程约束的最优化问题。受控同步和锁相是用经典的极序度量来测量的。在建立了平均场模型的质量守恒性质及其非局部项的界之后，利用拉格朗日方法得到了最优性的一阶必要条件。最优性系统，包括一个非局部偏微分方程的状态动力学方程，相应的非局部伴随动力学，和欧拉方程，迭代求解后，一个标准的优化，然后离散化的方法和一个有效的数值求解器的基础上谱方法。我们证明了我们的方法在模拟中的两个控制输入。
英文摘要：
In this paper, we investigate how the self-synchronization property of a swarm of Kuramoto oscillators can be controlled and exploited to achieve target densities and target phase coherence. In the limit of an infinite number of oscillators, the collective dynamics of the agents' density is described by a mean-field model in the form of a nonlocal PDE, where the nonlocality arises from the synchronization mechanism. In this mean-field setting, we introduce two space-time dependent control inputs to affect the density of the oscillators: an angular velocity field that corresponds to a state feedback law for individual agents, and a control parameter that modulates the strength of agent interactions over space and time, i.e., a multiplicative control with respect to the integral nonlocal term. We frame the density tracking problem as a PDE-constrained optimization problem. The controlled synchronization and phase-locking are measured with classical polar order metrics. After establishing the mass conservation property of the mean-field model and bounds on its nonlocal term, a system of first-order necessary conditions for optimality is recovered using a Lagrangian method. The optimality system, comprising a nonlocal PDE for the state dynamics equation, the respective nonlocal adjoint dynamics, and the Euler equation, is solved iteratively following a standard Optimize-then-Discretize approach and an efficient numerical solver based on spectral methods. We demonstrate our approach for each of the two control inputs in simulation.
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