机器人群体对流扩散场的间接最优控制
Indirect Optimal Control of Advection-Diffusion Fields through Robotic Swarms
摘要：
在本文中，我们考虑的问题，最佳引导的大规模群的水下航行器的任务是间接控制的对流扩散环境场。车辆的微观动力学模型由一个带漂移的随机微分方程控制。漂移项模拟了航行器的自推进速度和水流的速度场。在平均场条件下，宏观飞行器动力学方程由线性抛物型平流-扩散方程形式的Kolmogorov向前方程控制。环境场由平流-扩散方程控制，其中平流项由流体速度场定义。该车辆配备了板载致动器，使集群作为一个分布式源在环境领域，调制的标量控制参数，确定本地源强度。在这种情况下，我们制定了一个最优控制的问题来计算车辆的速度和致动器的强度场，驱动环境场在指定的时间内所需的分布。换句话说，我们设计了最优向量和标量驱动场，通过群产生的分布式源间接控制环境场。在证明了最优控制问题解的存在性之后，我们用有限元法对该问题进行了离散和求解。FEM离散化自然地提供了一个算子，该算子表示控制进入车辆群和环境场的动力学的双线性方式。最后，我们通过数值模拟表明，我们的控制策略的有效性，在调节环境场为零或所需的分布中存在的双回旋流场。
英文摘要：
In this paper, we consider the problem of optimally guiding a large-scale swarm of underwater vehicles that is tasked with the indirect control of an advection-diffusion environmental field. The microscopic vehicle dynamics are governed by a stochastic differential equation with drift. The drift terms model the self-propelled velocity of the vehicle and the velocity field of the currents. In the mean-field setting, the macroscopic vehicle dynamics are governed by a Kolmogorov forward equation in the form of a linear parabolic advection-diffusion equation. The environmental field is governed by an advection-diffusion equation in which the advection term is defined by the fluid velocity field. The vehicles are equipped with on-board actuators that enable the swarm to act as a distributed source in the environmental field, modulated by a scalar control parameter that determines the local source intensity. In this setting, we formulate an optimal control problem to compute the vehicle velocity and actuator intensity fields that drive the environmental field to a desired distribution within a specified amount of time. In other words, we design optimal vector and scalar actuation fields to indirectly control the environmental field through a distributed source, produced by the swarm. After proving an existence result for the solution of the optimal control problem, we discretize and solve the problem using the Finite Element Method (FEM). The FEM discretization naturally provides an operator that represents the bilinear way in which the controls enter into the dynamics of the vehicle swarm and the environmental field. Finally, we show through numerical simulations the effectiveness of our control strategy in regulating the environmental field to zero or to a desired distribution in the presence of a double-gyre flow field.
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