In this article, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky’s normal form allows us to remove the restriction of having to work with diagonalizable system dynamics and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem’s intrinsic symmetries. Numerical experiments help to visualize these properties, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms—this is alleviated by using an evolutionary algorithm.

中文翻译：

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通过 Brunovsky 范式的最佳执行器设计

在这篇文章中，通过使用 Brunovsky 范式，我们提供了一个问题的重新表述，该问题包括找到致动器设计，使具有标量控制的有限维线性系统的可控性成本最小化。此类系统可被视为具有集中控制的空间离散线性偏微分方程。由 Brunovsky 的范式引起的坐标变化使我们能够消除必须使用可对角化系统动力学的限制，并且不需要像过去关于扩散方程或波的文献中那样进行随机化程序。相反，优化问题简化为基础矩阵变化的逆范数的最小化，并允许轻松推导解的存在性，并更清楚地了解问题的一些内在对称性。