scholarly journals Lyapunov Characterization for the Stability of Stochastic Control Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Fakhreddin Abedi ◽  
Wah June Leong ◽  
Mohammad Abedi
Keyword(s):  
Control Systems ◽  
Stochastic Control ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Stability ◽  
Necessary And Sufficient

Lyapunov-like characterization for the problem of input-to-state stability in the probability of nonautonomous stochastic control systems is established. We extend the well-known Artstein-Sontag theorem to derive the necessary and sufficient conditions for the input-to-state stabilization of stochastic control systems. Illustrating example is provided.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

scholarly journals Multiresolution analysis and wavelets on Vilenkin groups

2008 ◽  
Vol 21 (3) ◽  
pp. 309-325 ◽  
Author(s):  
Yury Farkov
Keyword(s):  
Multiresolution Analysis ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Partition Of Unity ◽  
Necessary And Sufficient Conditions ◽  
Refinable Functions ◽  
Vilenkin Groups ◽  
Orthogonal Wavelets ◽  
The Stability ◽  
Necessary And Sufficient

This paper gives a review of multiresolution analysis and compactly sup- ported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of 'integer shifts' of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L2-spaces on Vilenkin groups. Several examples are provided to illustrate these results. .

ASYMPTOTIC HYPERSTABILITY OF A CLASS OF NEURAL NETWORKS

1999 ◽  
Vol 09 (02) ◽  
pp. 95-98 ◽  
Author(s):  
ANKE MEYER-BÄSE
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lyapunov Methods ◽  
Network Parameters ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Stability Results

This paper is concerned with the asymptotic hyperstability of recurrent neural networks. We derive based on the stability results necessary and sufficient conditions for the network parameters. The results we achieve are more general than those based on Lyapunov methods, since they provide milder constraints on the connection weights than the conventional results and do not suppose symmetry of the weights.

On the rate of growth of the overshoot and the maximum partial sum

1998 ◽  
Vol 30 (1) ◽  
pp. 181-196 ◽  
Author(s):  
P. S. Griffin ◽  
R. A. Maller
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Strong Sense ◽  
Partial Sum ◽  
Order Of Magnitude ◽  
The Stability ◽  
Dominance Properties ◽  
Necessary And Sufficient ◽  
First Time

Let Tr be the first time at which a random walk Sn escapes from the strip [-r,r], and let |STr|-r be the overshoot of the boundary of the strip. We investigate the order of magnitude of the overshoot, as r → ∞, by providing necessary and sufficient conditions for the ‘stability’ of |STr|, by which we mean that |STr|/r converges to 1, either in probability (weakly) or almost surely (strongly), as r → ∞. These also turn out to be equivalent to requiring only the boundedness of |STr|/r, rather than its convergence to 1, either in the weak or strong sense, as r → ∞. The almost sure characterisation turns out to be extremely simple to state and to apply: we have |STr|/r → 1 a.s. if and only if EX2 < ∞ and EX = 0 or 0 < |EX| ≤ E|X| < ∞. Proving this requires establishing the equivalence of the stability of STr with certain dominance properties of the maximum partial sum Sn* = max{|Sj|: 1 ≤ j ≤ n} over its maximal increment.

Robust Stability of Sequential Multi-input Multi-output Quantitative Feedback Theory Designs

2004 ◽  
Vol 127 (2) ◽  
pp. 250-256 ◽  
Author(s):  
Murray L. Kerr ◽  
Suhada Jayasuriya ◽  
Samuel F. Asokanthan
Keyword(s):  
Control Systems ◽  
Closed Loop ◽  
Sufficient Conditions ◽  
Internal Stability ◽  
Quantitative Feedback Theory ◽  
Recursive Design ◽  
Stability Theorems ◽  
Improved Design ◽  
The Stability ◽  
Necessary And Sufficient

This paper re-examines the stability of multi-input multi-output (MIMO) control systems designed using sequential MIMO quantitative feedback theory (QFT). In order to establish the results, recursive design equations for the SISO equivalent plants employed in a sequential MIMO QFT design are established. The equations apply to sequential MIMO QFT designs in both the direct plant domain, which employs the elements of plant in the design, and the inverse plant domain, which employs the elements of the plant inverse in the design. Stability theorems that employ necessary and sufficient conditions for robust closed-loop internal stability are developed for sequential MIMO QFT designs in both domains. The theorems and design equations facilitate less conservative designs and improved design transparency.

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