Optimal switching sequence for switched linear systems
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We study a discrete optimization problem over a dynamical system that consists of several linear subsystems. In particular, we aim to find a sequence of K matrices, each chosen from a given finite set of square matrices, to maximize a convex function over the product of the K matrices and a given vector. This problem models many applications in operations research and control. We show that this problem is NP-hard given a pair of stochastic matrices or binary matrices. We propose a polynomial-time exact algorithm for the problem when the given set of matrices has the oligo-vertex property, a concept we introduce to characterize the numbers of extreme points of certain polytopes related to the given matrices. We derive a set of sufficient conditions for a pair of matrices to have the oligo-vertex property. We show that a pair of 2 × 2 binary matrices has this property, and conjecture that any pair of 2 × 2 real matrices has this property.
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