Statement
For a discrete-time Markov jump linear system
x(k+1) = Γ_{θ(k)} x(k), θ(k) ∈ {1, …, N} a finite Markov chain with transition matrix P = [p_{ij}],
the system is mean square stable (E[x(k) x(k)*] → Q ∈ B(C^n)^+ independent of (x_0, θ_0), with Q = 0 in the homogeneous case) if and only if the spectral radius of the second-moment operator T : H^n → H^n defined by
T_j(V) = Σ_{i=1}^{N} p_{ij} Γ_i V_i Γ_i*
is strictly less than one. Equivalently, r_σ(A_1) < 1 for the augmented matrix A_1 = (P’ ⊗ I_{n²}) · diag(Γ_i ⊗ Γ_i), or there exists a positive-definite solution V > 0 to the coupled Lyapunov equation V − T(V) = S for any S > 0.
Evidence summary
The textbook proof in CFM Theorem 3.9 establishes a six-way equivalence by chaining (a) the operator algebra in Chapter 2 (the H^n framework, T = L*, V = J*), (b) Proposition 3.6’s identification r_σ(T) = r_σ(B) for an appropriately defined Banach-space operator B, (c) Proposition 3.22 (r_σ(A_1) < 1 ⇒ MSS) and Proposition 3.23 (MSS ⇒ r_σ(A_1) < 1), and (d) Corollary 3.21 linking the spectral radius condition to existence of positive-definite Lyapunov solutions. The proof is complete and self-contained in Chapter 3 of CFM and Examples 3.17 / 3.18 falsify the naive per-mode test.
Conditions and scope
- Finite N (number of modes).
- Finite-dimensional state x(k) ∈ C^n.
- Discrete-time, time-invariant transition probabilities p_{ij} (the non-homogeneous case is treated in §3.4 with an analogous criterion via asymptotic wide-sense-stationary stability).
- Homogeneous case (no input); for the input case, MSS is defined as convergence of the first and second moments to (μ, Q) independent of the initial condition.
Counter-evidence
None within the finite-state, finite-dimension setting addressed by CFM. The equivalence MSS ⇔ stochastic stability does break down for countably infinite Markov chains (CFM Remark 3.12, ref [67]); this is a scope restriction, not a counterexample, for the claim as stated.
Linked ideas
(none yet)
Open questions
- Sharp convergence rates for E[x(k) x(k)*] → Q as a function of the spectral gap 1 − r_σ(T)?
- Robust MSS under uncertainty in p_{ij} — addressed by LMI methods in CFM §5.5 but tight necessary-and-sufficient conditions remain open.
- Extension of the operator framework to MJLS with countable or continuous Markov state.