Stability of vector differential delay equations Michael I. Gilʹ

Format:

Book

Author/Creator:

Gilʹ, M. I. (Mikhail Iosifovich)

Series:

Frontiers in mathematics

Language:

English

Subjects (All):

Delay differential equations.

Physical Description:

1 online resource

Place of Publication:

Basel Birkhäuser London Springer [distributor] 2013

Summary:

Differential equations with delay naturally arise in various applications, such as control systems, viscoelasticity, mechanics, nuclear reactors, distributed networks, heat flows, neural networks, combustion, interaction of species, microbiology, learning models, epidemiology, physiology, and many others. This book systematically investigates the stability of linear as well as nonlinear vector differential equations with delay and equations with causal mappings. It presents explicit conditions for exponential, absolute and input-to-state stabilities. These stability conditions are mainly formulated in terms of the determinants and eigenvalues of auxiliary matrices dependent on a parameter; the suggested approach allows us to apply the well-known results of the theory of matrices. In addition, solution estimates for the considered equations are established which provide the bounds for regions of attraction of steady states. The main methodology presented in the book is based on a combined usage of the recent norm estimates for matrix-valued functions and the following methods and results: the generalized Bohl-Perron principle and the integral version of the generalized Bohl-Perron principle; the freezing method; the positivity of fundamental solutions. A significant part of the book is devoted to the Aizerman-Myshkis problem and generalized Hill theory of periodic systems. The book is intended not only for specialists in the theory of functional differential equations and control theory, but also for anyone with a sound mathematical background interested in their various applications

Contents:

Preliminaries Some Results of the Matrix Theory General Linear Systems Time-invariant Linear Systems with Delay Properties of Characteristic Values Equations Close to Autonomous and Ordinary Differential Ones Periodic Systems Linear Equations with Oscillating Coefficients Linear Equations with Slowly Varying Coefficients Nonlinear Vector Equations Scalar Nonlinear Equations Forced Oscillations in Vector Semi-Linear Equations Steady States of Differential Delay Equations Multiplicative Representations of Solutions

Machine generated contents note: 1. Preliminaries

1.1. Banach and Hilbert spaces

1.2. Examples of normed spaces

1.3. Linear operators

1.4. Ordered spaces and Banach lattices

1.5. abstract Gronwall lemma

1.6. Integral inequalities

1.7. Generalized norms

1.8. Causal mappings

1.9. Compact operators in a Hilbert space

1.10. Regularized determinants

1.11. Perturbations of determinants

1.12. Matrix functions of bounded variations

1.13. Comments

2. Some Results of the Matrix Theory

2.1. Notations

2.2. Representations of matrix functions

2.3. Norm estimates for resolvents

2.4. Spectrum perturbations

2.5. Norm estimates for matrix functions

2.6. Absolute values of entries of matrix functions

2.7. Diagonalizable matrices

2.8. Matrix exponential

2.9. Matrices with non-negative off-diagonals

2.10. Comments

3. General Linear Systems

3.1. Description of the problem

3.2. Existence of solutions

3.3. Fundamental solutions

3.4. generalized Bohl-Perron principle

3.5. Lp-version of the Bohl-Perron principle

3.6. Equations with infinite delays

3.7. Proof of Theorem 3.6.1

3.8. Equations with continuous infinite delay

3.9. Comments

4. Time-invariant Linear Systems with Delay

4.1. Statement of the problem

4.2. Application of the Laplace transform

4.3. Norms of characteristic matrix functions

4.4. Norms of fundamental solutions of time-invariant systems

4.5. Systems with scalar delay-distributions

4.6. Scalar first-order autonomous equations

4.7. Systems with one distributed delay

4.8. Estimates via determinants

4.9. Stability of diagonally dominant systems

4.10. Comments

5. Properties of Characteristic Values

5.1. Sums of moduli of characteristic values

5.2. Identities for characteristic values

5.3. Multiplicative representations of characteristic functions

5.4. Perturbations of characteristic values

5.5. Perturbations of characteristic determinants

5.6. Approximations by polynomial pencils

5.7. Convex functions of characteristic values

5.8. Comments

6. Equations Close to Autonomous and Ordinary Differential Ones

6.1. Equations "close" to ordinary differential ones

6.2. Equations with small delays

6.3. Nonautomomous systems "close" to autonomous ones

6.4. Equations with constant coefficients and variable delays

6.5. Proof of Theorem 6.4.1

6.6. fundamental solution of equation (4.1)

6.7. Proof of Theorem 6.6.1

6.8. Comments

7. Periodic Systems

7.1. Preliminary results

7.2. main result

7.3. Norm estimates for block matrices

7.4. Equations with one distributed delay

7.5. Applications of regularized determinants

7.6. Comments

8. Linear Equations with Oscillating Coefficients

8.1. Vector equations with oscillating coefficients

8.2. Proof of Theorem 8.1.1

8.3. Scalar equations with several delays

8.4. Proof of Theorem 8.3.1

8.5. Comments

9. Linear Equations with Slowly Varying Coefficients

9.1. "freezing" method

9.2. Proof of Theorem 9.1.1

9.3. Perturbations of certain ordinary differential equations

9.4. Proof of Theorems 9.3.1

9.5. Comments

10. Nonlinear Vector Equations

10.1. Definitions and preliminaries

10.2. Stability of quasilinear equations

10.3. Absolute Lp-stability

10.4. Mappings defined on Ω (g) [∩] L2

10.5. Exponential stability

10.6. Nonlinear equations "close" to ordinary differential ones

10.7. Applications of the generalized norm

10.8. Systems with positive fundamental solutions

10.9. Nicholson-type system

10.10. Input-to-state stability of general systems

10.11. Input-to-state stability of systems with one delay in linear parts

10.12. Comments

11. Scalar Nonlinear Equations

11.1. Preliminary results

11.2. Absolute stability

11.3. Aizerman-Myshkis problem

11.4. Proofs of Lemmas 11.3.2 and 11.3.4

11.5. First-order nonlinear non-autonomous equations

11.6. Comparison of Green's functions to second-order equations

11.7. Comments

12. Forced Oscillations in Vector Semi-Linear Equations

12.1. Introduction and statement of the main result

12.2. Proof of Theorem 12.1.1

12.3. Applications of matrix functions

12.4. Comments

13. Steady States of Differential Delay Equations

13.1. Systems of semilinear equations

13.2. Essentially nonlinear systems

13.3. Nontrivial steady states

13.4. Positive steady states

13.5. Systems with differentiable entries

13.6. Comments

14. Multiplicative Representations of Solutions

14.1. Preliminary results

14.2. Volterra equations

14.3. Differential delay equations

14.4. Comments

Appendix A General Form of Causal Operators

Appendix B Infinite Block Matrices

B.1. Definitions

B.2. Properties of π-Volterra operators

B.3. Resolvents of π-triangular operators

B.4. Perturbations of block triangular matrices

B.5. Block matrices close to triangular ones

B.6. Diagonally dominant block matrices

B.7. Examples

Notes:

Includes bibliographical references and index

Print version record

Other Format:

Print version Gilʹ, M.I. (Mikhail Iosifovich). Stability of vector differential delay equations

ISBN:

9783034805773

3034805772

OCLC:

828148836

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