SIAM Journal on Control and Optimization, Volume 58, Issue 1, Page 510-528, January 2020.
The Lyapunov exponent corresponding to a set of square matrices $\mathcal{A} = \{A_1, \dots, A_n \}$ and a probability distribution $p$ over $\{1, \dots, n\}$ is $\lambda(\mathcal{A}, p) := \lim_{k \to \infty} \frac{1}{k} \,\mathbb{E} \log \lVert{A_{\sigma_k} \cdots A_{\sigma_2}A_{\sigma_1}\rVert}$, where $\sigma_i$ are independent and identically distributed according to $p$. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate $e^{\lambda(\mathcal{A}, p)}$ of the stochastic linear dynamical system $x_{k+1} = A_{\sigma_k} x_k$. This paper investigates the following “design problem”: Given $\mathcal{A}$, compute the distribution $p$ minimizing $\lambda(\mathcal{A}, p)$. Our main result is that it is $\textsc{NP}$-hard to decide whether there exists a distribution $p$ for which $\lambda(\mathcal{A}, p) < 0$, i.e., it is $\textsc{NP}$-hard to decide whether this dynamical system can be stabilized. This hardness result holds even in the “simple” case where $\mathcal{A}$ contains only rank-one matrices. Somewhat surprisingly, this is in stark contrast to the Joint Spectral Radius---the deterministic counterpart of the Lyapunov exponent---for which the analogous optimization problem over switching rules is known to be exactly computable in polynomial time for rank-one matrices. To prove this hardness result, we first observe via Birkhoff's Ergodic Theorem that the Lyapunov exponent of rank-one matrices admits a simple formula and in fact is a quadratic form in $p$. Hardness of the design problem is shown through a reduction from the Independent Set problem. Along the way, simple examples are given illustrating that $p \mapsto \lambda(\mathcal{A}, p)$ is neither convex nor concave in general. We conclude with extensions to continuous distributions, exchangeable processes, Markov processes, and stationary ergodic processes.

中文翻译：

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一阶矩阵的Lyapunov指数：遍历公式和最优分布的不可约性

SIAM控制与优化杂志，第58卷，第1期，第510-528页，2020年1月。
对应于一组平方矩阵$ \ mathcal {A} = \ {A_1，\ dots，A_n \} $的Lyapunov指数和$ \ {1，\ dots，n \} $上的概率分布$ p $为$ \ lambda（\ mathcal {A}，p）：= \ lim_ {k \ to \ infty} \ frac {1} {k} \，\ mathbb {E} \ log \ lVert {A _ {\ sigma_k} \ cdots A_ {\ sigma_2} A _ {\ sigma_1} \ rVert} $，其中$ \ sigma_i $是独立的，并且根据$ p $分布相同。该数量对于控制理论至关重要，因为它确定了随机线性动力学系统$ x_ {k + 1} = A _ {\的渐近收敛速度$ e ^ {\ lambda（\ mathcal {A}，p）} $ sigma_k} x_k $。本文研究以下“设计问题”：给定$ \ mathcal {A} $，计算使$ \ lambda（\ mathcal {A}，p）$最小的分布$ p $。我们的主要结果是，很难确定是否存在$ \ lambda（\ mathcal {A}，p）<0 $的分布$ p $，即$ \ textsc {NP} $很难确定。 textsc {NP} $-难以确定此动态系统是否可以稳定。即使在$ \ mathcal {A} $仅包含秩一矩阵的“简单”情况下，此硬度结果也成立。出乎意料的是，这与联合光谱半径（Lyapunov指数的确定性对应物）形成了鲜明的对比，联合光谱半径对于切换规则的类似优化问题据称可以在多项式时间内精确计算为秩为1的矩阵。为了证明这种硬度结果，我们首先通过Birkhoff的遍历定理观察到，秩为1的矩阵的Lyapunov指数接受一个简单的公式，实际上是$ p $的二次形式。通过减少独立设置问题来显示设计问题的难度。在此过程中，给出了简单的示例，说明$ p \ mapsto \ lambda（\ mathcal {A}，p）$通常既不是凸面也不是凹面。我们以对连续分布，可交换过程，马尔可夫过程和平稳遍历过程的扩展作为结束。