By O.L.V. Costa, M.D. Fragoso, R.P. Marques

Safety severe and high-integrity platforms, similar to commercial vegetation and monetary structures could be topic to abrupt adjustments - for example as a result of part or interconnection failure, and unexpected atmosphere alterations and so forth.

Combining chance and operator thought, Discrete-Time Markov bounce Linear structures presents a unified and rigorous remedy of contemporary effects for the regulate conception of discrete leap linear platforms, that are utilized in those parts of program.

The booklet is designed for specialists in linear platforms with Markovian bounce parameters, yet can also be the 1st publication for experts in stochastic regulate to offer stochastic regulate difficulties for which an particular answer is feasible - making the ebook compatible for direction use.

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7). 7) and equilibrium point xe if 1. ) is continuous, 2. φ(xe ) < φ(x) for every x ∈ Γ such that x = xe , 3. ∆φ(x) = φ(f (x)) − φ(x) ≤ 0 for all x ∈ Γ . With this we can proceed to the Lyapunov Theorem. A proof of this result can be found in [165]. 13 (Lyapunov Theorem). 7) and xe , then the equilibrium point is stable in the sense of Lyapunov. Moreover, if ∆φ(x) < 0 for all x = xe , then it is asymptotically stable. Furthermore if φ is defined on the entire state space and φ(x) goes to infinity as any component of x gets arbitrarily large in magnitude then the equilibrium point xe is globally asymptotically stable.

2 some operators that are closely related to the Markovian property of the augmented state (x(t), θ(t)), greatly simplifying the solution for the mean square stability and other problems that will be analyzed in the next chapters. As a consequence, we can adopt an analytical view toward mean square stability, using the operator theory in Banach spaces provided in Chapter 2 as a primary tool. The outcome is a clean and sound theory ready for application. As mentioned in [134], among the advantages of using the MSS concept and the results derived here, are: (1) the fact that it is easy to test for; (2) it implies stability of the expected dynamics; (3) it yields almost sure asymptotic stability of the zero-input state space trajectories.

3. For any S ∈ Hn+ , S > 0, there exists a unique V ∈ Hn+ , V > 0, such that V − T (V ) = S. 14) 4. For some V ∈ Hn+ , V > 0, we have V − T (V ) > 0. 3 MSS: The Homogeneous Case 37 5. For some β ≥ 1, 0 < ζ < 1, we have for all x0 ∈ C0n and all θ0 ∈ Θ0 , 2 2 2 E( x(k) ) ≤ βζ k x0 , k = 0, 1, . . 6. For all x0 ∈ C0n and all θ0 ∈ Θ0 ∞ 2 E( x(k) ) < ∞. 15) by L, V or J . 10. 1) is equivalent to Q(k) → 0 and µ(k) → 0 as k → ∞. 6 that rσ (B) < 1. 9 it follows that Q(k) → 0, q(k) → 0, and Q(k) → 0, µ(k) → 0, as k → ∞.

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