论文标题
最佳控制动态系统的合奏
Optimal Control of ensembles of dynamical systems
论文作者
论文摘要
在本文中，我们考虑了仿生控制系统合奏的最佳控制问题。在证明了正在检查的最小化问题的适当性之后，我们建立了一个$γ$ - 融合的结果，使我们能够用一系列有限的增加尺寸的子备件来代替原始（且通常是无限的）合奏。涉及这些子符号的最佳控制问题的解决方案在原始问题的最小化器的$ l^2 $ strong拓扑中提供了近似值。再次使用$γ$ -Convergence参数，我们设法得出了合奏最佳控制问题的最大原则，并获得了终点成本。此外，在有限的子谐振的情况下，我们可以通过数值方案来解决相关成本的最小化。特别是，我们提出了一种算法，该算法由近似成本功能在可允许控制空间引起的梯度场的子空间投影组成。此外，我们考虑了一种基于吡格肽最大原理的迭代方法。最后，我们在$ \ mathbb {r}^2 $中的线性系统集合上测试算法。
In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a $Γ$-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the $L^2$-strong topology of the minimizers of the original problem. Using again a $Γ$-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in $\mathbb{R}^2$.
