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Markov Jump Linear System

A Markov jump linear system is a discrete-time dynamical system that switches among finitely many linear modes according to a Markov chain. If the mode at time k is theta_k in {1,...,s} and the corresponding n×n state matrices are A_1,...,A_s, then the state evolves according to

x_(k+1)=A_(theta_k)x_k.
(1)

Writing P=(p_(ij)) for the stochastic matrix with p_(ij)=Pr(theta_(k+1)=j|theta_k=i), define

B=(P^T tensor I_n)(A_1 direct sum ... direct sum A_s),
(2)
T=(P^T tensor I_(n^2))((A_1 tensor A_1) direct sum ... direct sum (A_s tensor A_s)).
(3)

Here direct sum denotes the matrix direct sum, tensor the Kronecker product, and I_r the r×r identity matrix.

The system is mean stable if E[x_k]->0 for every initial state and mode, and it is mean-square stable if E[||x_k||^2]->0, where E denotes an expectation value and ||·|| is a vector norm. It is mean stable iff rho(B)<1, and it is mean-square stable iff rho(T)<1, where rho denotes the spectral radius (Costa and Fragoso 1993).

See also

Control Theory, Dynamical System, Kronecker Product, Markov Chain, Matrix Direct Sum, Spectral Radius, Stochastic Matrix

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References

Becker, L. and Hermanns, H. "Stability Checking of Markov Jump Linear Systems via Probabilistic Temporal Logic (Extended Version)." 23 Jun 2026. https://arxiv.org/abs/2606.24880.Costa, O. L. V. and Fragoso, M. D. "Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters." J. Math. Anal. Appl. 179, 154-178, 1993. https://doi.org/10.1006/jmaa.1993.1341.Costa, O. L. V.; Fragoso, M. D.; and Marques, R. P. Discrete-Time Markov Jump Linear Systems. London: Springer-Verlag, 2005. https://doi.org/10.1007/b138575.

Cite this as:

Weisstein, Eric W. "Markov Jump Linear System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MarkovJumpLinearSystem.html

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