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Absorption in Time-Varying Markov Chains Graph-Based Conditions

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Document pages: 6 pages

Abstract: We investigate absorption, i.e., almost sure convergence to an absorbingstate, in time-varying (non-homogeneous) discrete-time Markov chains withfinite state space. We consider systems that can switch among a finite set oftransition matrices, which we call the modes. Our analysis is focused on twoproperties: 1) almost sure convergence to an absorbing state under anyswitching, and 2) almost sure convergence to a desired set of absorbing statesvia a proper switching policy. We derive necessary and sufficient conditionsbased on the structures of the transition graphs of modes. More specifically,we show that a switching policy that ensures almost sure convergence to adesired set of absorbing states from any initial state exists if and only ifthose absorbing states are reachable from any state on the union of simplifiedtransition graphs. We then show three sufficient conditions for absorptionunder arbitrary switching. While the first two conditions depend on theacyclicity (weak acyclicity) of the union (intersection) of simplifiedtransition graphs, the third condition is based on the distances of each stateto the absorbing states in all the modes. These graph theoretic conditions canverify the stability and stabilizability of absorbing states based only on thefeasibility of transitions in each mode.

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