SIAM Journal on Control and Optimization, Volume 58, Issue 5, Page 2790-2820, January 2020.
The NP-hardness of the optimal decentralized control (ODC) problem is reflected in the properties of its feasible set. We study the complexity of the ODC problem through an analysis of the set of stabilizing static decentralized controllers and show that there is no polynomial upper bound on its number of connected components. In particular, it is proved that this number is exponential in the order of the system for a class of problems. Since every point in each of these components is the unique solution of the ODC problem for some quadratic objective functional, the results of this work imply that, without prior knowledge for initialization, local search algorithms cannot solve the ODC problem to global optimality for all decentralized control structures. In an effort to understand the connection between the geometric properties of the feasible set of the ODC problem and the control structure, we further identify decentralized structures that admit tractable connectivity properties, using a combination of the Routh--Hurwitz criterion and Lyapunov stability theory.

中文翻译：

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稳定静态分散控制器集的连接性

SIAM控制与优化杂志，第58卷，第5期，第2790-2820页，2020年1月。
最佳分散控制（ODC）问题的NP硬度反映在其可行集的属性中。我们通过分析一组稳定的静态分散控制器来研究ODC问题的复杂性，并表明在其连接组件的数量上没有多项式上限。特别地，对于一类问题，证明该数字在系统顺序中是指数的。由于这些组件中每个组件的每个点都是某些二次目标函数的ODC问题的唯一解决方案，因此这项工作的结果意味着，如果没有先验的初始化知识，本地搜索算法就无法将ODC问题求解为所有分散式的全局最优性控制结构。