Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

Drăgan, Vasile; Morozan, Toader

doi:10.1007/s00020-005-1371-7

Cited by 22 publications

(14 citation statements)

References 26 publications

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“…Replacing the backward difference equations with algebraic equations and further applying Theorem 3.5 in Dragan and Morozan (2006) lead to Corollary 2.5.…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

Robust stabilisation control for discrete-time networked control systems

Chen

2010

International Journal of Control

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“…Replacing the backward difference equations with algebraic equations and further applying Theorem 3.5 in Dragan and Morozan (2006) lead to Corollary 2.5.…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

Robust stabilisation control for discrete-time networked control systems

Chen

2010

International Journal of Control

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“…The results of this section are special cases of those stated in a more general framework in [6], which is why we present them here without proofs.…”

Section: Lyapunov-type Criteriamentioning

confidence: 93%

“…Direct from the above Definition 2.3 and Theorem 3.4 in [6] applied to Lyapunov-type operators t we obtain: Under this condition the zero state equilibrium of (4.3) is SESMS. Indeed if (4.4) holds then the corresponding system (4.2) associated to (…”

Section: The General Casementioning

confidence: 93%

“…From Theorem 3.5 in [6] one obtains that if the coefficients of (4.1) are periodic with period , the unique bounded and positive solution of (4.1) is periodic with the same period . Also, if the system of inequalities (4.2) has a bounded and uniform positive solution then it has a periodic solution.…”

Section: The Periodic Casementioning

confidence: 94%

“…Set N n = n ⊕ n ⊕ · · · ⊕ n . One can see that N n is a real ordered Hilbert space (see [6], Example 2.5(iii)). The usual inner product on N n is…”

Section: Lyapunov-type Operatorsmentioning

confidence: 97%

See 2 more Smart Citations

Exponential Stability in Mean Square for a General Class of Discrete-Time Linear Stochastic Systems

Drăgan

Morozan

2008

Stochastic Analysis and Applications

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The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. The case of the linear systems whose coefficients depend both to present state and the previous state of the Markov chain is considered. Three different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other two definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain.The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions. 496 Dragan and MorozanThe results developed in this article may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems in the presence of a delay in the transmission of the data.

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Global solutions of a class of discrete‐time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control

Ungureanu

Drăgan

Morozan

2012

Optim Control Appl Methods

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SUMMARYIn this paper, we consider the problem of existence of certain global solutions for general discrete-time backward nonlinear equations, defined on infinite dimensional ordered Banach spaces. This class of nonlinear equations includes as special cases many of the discrete-time Riccati equations arising both in deterministic and stochastic optimal control problems. On the basis of a linear matrix inequalities approach, we give necessary and sufficient conditions for the existence of maximal, stabilizing, and minimal solutions of the considered discrete-time backward nonlinear equations. As an application, we discuss the existence of stabilizing solutions for discrete-time Riccati equations of stochastic control and filtering on Hilbert spaces. The tools provided by this paper show that a wide class of nonlinear equations can be treated in a uniform manner.

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Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

Cited by 22 publications

References 26 publications

Robust stabilisation control for discrete-time networked control systems

Robust stabilisation control for discrete-time networked control systems

Exponential Stability in Mean Square for a General Class of Discrete-Time Linear Stochastic Systems

Global solutions of a class of discrete‐time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control

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