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• W3099205471 abstract "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." @default.
• W3099205471 created "2020-11-23" @default.
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• W3099205471 date "2020-01-01" @default.
• W3099205471 modified "2023-10-16" @default.
• W3099205471 title "Lyapunov Exponent of Rank-One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution" @default.
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• W3099205471 doi "https://doi.org/10.1137/19m1264072" @default.
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