This article introduces a formation shape control algorithm, in the optimal control framework, for steering an initial population of agents to a desired configuration via employing the Gromov-Wasserstein distance. The underlying dynamical system is assumed to be a constrained linear system and the objective function is a sum of quadratic control-dependent stage cost and a Gromov-Wasserstein terminal cost. The inclusion of the Gromov-Wasserstein cost transforms the resulting optimal control problem into a well-known NP-hard problem, making it both numerically demanding and difficult to solve with high accuracy. Towards that end, we employ a recent semi-definite relaxation-driven technique to tackle the Gromov-Wasserstein distance. A numerical example is provided to illustrate our results.

翻译：本文在最优控制框架下，提出一种编队构型控制算法，通过采用Gromov-Wasserstein距离将初始智能体群引导至期望构型。假设底层动力学系统为受约束线性系统，目标函数由控制相关的二次阶段成本与Gromov-Wasserstein终端成本之和构成。引入Gromov-Wasserstein成本将最优控制问题转化为经典的NP难问题，导致数值计算要求高且难以获得高精度解。为此，我们采用最新的半定松弛技术来处理Gromov-Wasserstein距离。文中提供了数值算例以验证所提方法的有效性。