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E-raamat: Stability Theory for Dynamic Equations on Time Scales

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Stability Theory for Dynamic Equations on Time ScalesSuurem pilt
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This monograph is a first in the world to present three approaches for stability analysis of solutions of dynamic equations. The first approach is based on the application of dynamic integral inequalities and the fundamental matrix of solutions of linear approximation of dynamic equations. The second is based on the generalization of the direct Lyapunovs method for equations on time scales, using scalar, vector and matrix-valued auxiliary functions. The third approach is the application of auxiliary functions (scalar, vector, or matrix-valued ones) in combination with differential dynamic inequalities. This is an alternative comparison method, developed for time continuous and time discrete systems. In recent decades, automatic control theory in the study of air- and spacecraft dynamics and in other areas of modern applied mathematics has encountered problems in the analysis of the behavior of solutions of time continuous-discrete linear and/or nonlinear equations of perturbed mot

ion. In the book "Men of Mathematics," 1937, E.T.Bell wrote: "A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both." Mathematical analysis on time scales accomplishes exactly this.  This research has potential applications in such areas as theoretical and applied mechanics, neurodynamics, mathematical biology and finance among others.

Contents.- Preface.- 1 Elements ofTime Scales Analysis.- 2 Method of Dynamic Integral Inequalities.- 3 LyapunovTheory for Dynamic Equations.- 4 Comparison Method.- 5 Applications.- References.

Arvustused

This text published by Birkhauser is relatively short being in a monograph form and presents three approaches for stability analysis and the solution of dynamic equations. The book is more aimed at mathematical control theorists, but it could be important in flight and spacecraft dynamics and other application areas. (ACTC applied control technology consortium, actc-control.com, February, 2017)

1 Elements of Time Scales Analysis
1(24)
1.0 Introduction
1(1)
1.1 Description of a Time Scale
1(3)
1.2 Delta Derivative
4(2)
1.3 Integration
6(8)
1.4 Exponential Function
8(3)
1.5 Matrix Exponential Functions
11(2)
1.6 Variation of Constants Formula
13(2)
1.7 Nabla Derivative
15(4)
1.8 Diamond-Alpha Derivative
19(3)
1.9 Comments and Bibliography
22(3)
2 Method of Dynamic Integral Inequalities
25(60)
2.0 Introduction
25(1)
2.1 Dynamic Integral Inequalities
25(12)
2.1.1 Gronwall inequalities
25(6)
2.1.2 Some nonlinear inequalities
31(6)
2.2 Stability of Linear Dynamic Equations
37(15)
2.2.1 Nonautonomous systems
37(7)
2.2.2 Time-invariant system
44(4)
2.2.3 Elements of Floquet theory
48(4)
2.3 Stability of Nonlinear Dynamic Equations
52(26)
2.3.1 Estimations of solutions
52(5)
2.3.2 Theorems on stability
57(4)
2.3.3 Stability of quasilinear equations
61(12)
2.3.4 Exponential stability
73(2)
2.3.5 Scalar quasilinear equation
75(3)
2.4 Preservation of Stability Under Perturbations
78(6)
2.4.1 Linear systems under parametric perturbations
78(2)
2.4.2 Quasilinear dynamic equations
80(4)
2.5 Comments and Bibliography
84(1)
3 Lyapunov Theory for Dynamic Equations
85(60)
3.0 Introduction
85(1)
3.1 Preliminary Results
86(2)
3.2 Auxiliary Functions for Dynamic Equations
88(4)
3.2.1 Scalar functions
88(1)
3.2.2 Vector functions
89(1)
3.2.3 Matrix-valued functions
90(2)
3.3 Theorems of Stability and Instability
92(22)
3.3.1 General systems of dynamic equations
92(13)
3.3.2 Stability of linear systems
105(9)
3.4 Existence and Construction of Lyapunov Functions
114(11)
3.4.1 Converse theorem
114(2)
3.4.2 Solution of dynamic Lyapunov equation
116(2)
3.4.3 Lyapunov function for linear periodic system
118(7)
3.5 Stability Under Structural Perturbations
125(11)
3.5.1 Description of structural perturbations for dynamic equations
125(3)
3.5.2 Periodic linear system
128(8)
3.6 Polydynamics on Time Scales
136(7)
3.6.1 Problem setting
137(1)
3.6.2 Analysis of polydynamics
138(1)
3.6.3 Conditions for stability and instability
139(4)
3.7 Comments and Bibliography
143(2)
4 Comparison Method
145(40)
4.0 Introduction
145(1)
4.1 Theorems of the Comparison Method
146(4)
4.2 Stability Theorems
150(9)
4.3 Stability of Conditionally Invariant Sets
159(9)
4.4 Stability with Respect to Two Measures
168(6)
4.5 Stability of a Dynamic Graph
174(8)
4.5.1 Description of a dynamic graph
174(2)
4.5.2 Problem of stability
176(2)
4.5.3 Evolution of a dynamic graph
178(1)
4.5.4 Matrix-valued functions and their applications
179(1)
4.5.5 A variant of comparison principle
180(2)
4.6 Comments and Bibliography
182(3)
5 Applications
185(30)
5.0 Introduction
185(1)
5.1 Stability of Neuron Network
185(8)
5.2 Stability of a Complex-Valued Neuron Network
193(6)
5.3 Volterra Model on Time Scale
199(7)
5.3.1 Generalization of Volterra model
200(2)
5.3.2 Stability analysis
202(4)
5.4 Stability of Oscillations
206(8)
5.4.1 Statement of the problem
206(1)
5.4.2 Stability under structural perturbations
206(8)
5.5 Comments and Bibliography
214(1)
References 215(6)
Index 221
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