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Stability of Weakly Connected Nonlinear Systems

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Language:  English
The book consists of five chapters in which the mathematical foundations of the analysis of the stability of systems with a small parameter are given as well as methods of their investigation.
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The book consists of five chapters in which the mathematical foundations of the analysis of the stability of systems with a small parameter are given as well as methods of their investigation. The proposed methods are based on nonlinear integral inequalities, the direct Lyapunov method, using scalar, vector and matrix-valued functions, as well as the method of averaging of nonlinear mechanics. Here we also obtain sufficient conditions for various types of boundedness and stability of motion of nonlinear weakly connected systems, including systems of equations in a Banach space.

About the authors

A.A.Martynyuk is Professor and academician of the National Academy of Sciences of Ukraine Head of Stability of Processes Department at the S.P.Timoshenko Institute of Mechanics of NAS of Ukraine. The author or coauthor of more than 350 journal publications and 26 of books published in Russian, English and Chinese. He is founder and Editor of the International Journal of Nonlinear Dynamics and Systems Theory and the International Series of Scientific Monographs: Stability, Oscillations and Optimization of Systems published by the Cambridge Scientific Publishers (United Kingdom). He received D.Sc.degree (1973) in physics and mathematics from Institute of Mathematics NAS of Ukraine, Kiev.

L.N. Chernetskaya is a senior research fellow of the Department of Processes Stability of the Institute of Mechanics of the National Academy of Sciences of Ukraine. The author (co-author) of more than 100 scientific works in the field of theories of stability and motion control. Thesis defended in 1986 in the Institute of Mechanics of the National Academy of Sciences of Ukraine.

V.A. Martynyuk is a specialist in the field of nonlinear dynamics of weekly connected systems. He is the author of several scientific works published in Journal of Mathematical Analysis and Applications (USA) and International Applied Mechanics (Ukraine).

  1. Preliminaries 
    1. Introductory Remarks 
    2. Fundamental Inequalities 
    3. Stability in the Sense of Lyapunov 
    4. Comparison Principle 
    5. Stability of Systems with a Small Parameter 
    6. Comments and References 
  2. Analysis of the Boundedness of Motion 
    1. Introductory Remarks 
    2. Statement of the Problem 
    3. μ-Boundedness with Respect to Two Measures 
    4. Boundedness and the Comparison Technique 
    5. Boundedness with Respect to a Part of Variables 
    6. Algebraic Conditions of μ-Boundedness 
    7. Applications 
    8. Comments and References 
  3. Analysis of the Stability of Motion 
    1. Introductory Remarks 
    2. Statement of the Problem 
    3. Stability with Respect to Two Measures 
    4. Equistability Via Scalar Comparison Equations 
    5. Dynamic Behavior of an Individual Subsystem 
    6. Asymptotic Behavior 
    7. Polystability of Motion 
    8. Applications 
    9. Comments and References 
  4. Stability of Weakly Perturbed Systems 
    1. Introductory Remarks 
    2. Averaging and Stability 
    3. Stability on a Finite Time Interval 
    4. Methods of Application of Auxiliary Systems 
    5. Systems with Nonasymptotically Stable Subsystems 
    6. Stability with Respect to a Part of Variables 
    7. Applications 
    8. Comments and References 
  5. Stability of Systems in Banach Spaces 
    1. Introductory Remarks 
    2. Preliminary Results 
    3. Statement of the Problem 
    4. Generalized Direct Lyapunov Method 
    5. μ-Stability of Motion of Weakly Connected Systems 
    6. Stability Analysis of a Two-Component System 
    7. Comments and References 

home.book.about_authors

V.A. Martynyuk

V.A. Martynyuk is a specialist in the field of nonlinear dynamics of weekly connected systems. He is the author of several scientific works published in Journal of Mathematical Analysis and Applications (USA) and International Applied Mechanics (Ukraine).

A.

A. A. Martynyuk

Martynyuk is organizer and head of the Department of Processes Stability at the S.P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine (NAS of Ukraine) since 1978.  

He developed the method of Lyapunov matrix functions as applied to uncertain systems, systems with fuzzy information, and dynamic equations on time scales, non-classical theories of stability of motion, chaotic dynamics, and mathematical ecology. 

As the editor of a series of scientific monographs "Stability and Control: Theory, Methods and Applications" (1995-2002) at the publishing house Gordon and Breach (UK) and Taylor and Francis (USA) he organized the work of scientists from many countries of the world to create the series consist with 22 volumes, which opened new opportunities for young scientists to continue research in many areas of research in the qualitative theory of equations. 

He is currently focusing on the publication of a series of scientific monographs "Stability, Oscillation and Optimization of Systems" at the publishing house Cambridge Scientific Publishers (UK). 

Martynyuk is the founder and editor-in-chief of the International Scientific Journal "Nonlinear Dynamics and Systems Theory", which is well known throughout the world. 

He is the author (co-author) of a number of monographs in Russian, English and Chinese. 

He is a doctor of physical and mathematical sciences, professor, academician of the NAS of Ukraine, laureate of the State Prize of Ukraine in the field of science and technology. 

Martynyuk is based in Ukraine, but he actively communicates with experts from all over the world that are interested in stability theory and motion control in the study of real-world phenomena.

L.N. Chernetskaya

L.N. Chernetskaya is a senior research fellow of the Department of Processes Stability of the Institute of Mechanics of the National Academy of Sciences of Ukraine. The author (co-author) of more than 100 scientific works in the field of theories of stability and motion control. Thesis defended in 1986 in the Institute of Mechanics of the National Academy of Sciences of Ukraine.