## Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution

##### Author(s)

Altschuler, Jason M; Parrilo, Pablo A
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The Lyapunov exponent corresponding to a set of square matrices A = {A 1 , ... , A n } and a probability distribution p over {1, ... , n} is λ(A, p) := lim k→∞ 1/k E log ||A σk ⋯ A σ2 A σ1 ||, where σ i are i.i.d. according to p. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate e λ(A,p) of the stochastic linear dynamical system x k+1 = A σk x k . This paper investigates the following “design problem”: given A, compute the distribution p minimizing λ(A, p). Our main result is that it is NP-hard to decide whether there exists a distribution p for which λ(A, p) <; 0, i.e. it is NP-hard to decide whether this dynamical system can be stabilized. This hardness result holds even in the “simple” case where A contains only rank-one matrices. Somewhat surprisingly, this is in stark contrast to the Joint Spectral Radius - the deterministic kindred 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 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 → λ(A, p) is neither convex nor concave in general, and a connection is made to the fact that the Martin distance on the (1, n) Grassmannian is not a metric.

##### Date issued

2020-03##### Department

Massachusetts Institute of Technology. Laboratory for Information and Decision Systems##### Journal

2019 IEEE 58TH CONFERENCE ON DECISION AND CONTROL (CDC)

##### Publisher

Institute of Electrical and Electronics Engineers (IEEE)

##### Citation

Altschuler, Jason M and Pablo A Parrilo. "Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution." 2019 IEEE 58TH CONFERENCE ON DECISION AND CONTROL, December 2019, Nice, France, Institute of Electrical and Electronics Engineers, March 2020. © 2019 IEEE

Version: Original manuscript

##### ISBN

978-1-7281-1398-2

##### ISSN

2576-2370

978-1-7281-1399-9