Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Editors
Contributors
1. Elements of Time Scales Calculus
1.1. History and Objectives
1.2. Time Scales
1.3. Delta Differentiation
1.4. Delta Integration
1.5. Dynamic Equations
1.6. Nabla Calculus Essentials
2. First-Order Functional Dynamic Equations
2.1. Functional Dynamic Equations–Basic Concepts, Existence and Uniqueness Theorems
2.1.1. Classification of Functional Dynamic Equations
2.1.2. The Picard–Lindelöf Theorem
2.1.3. Existence and Uniqueness Theorems
2.1.4. Continuous Dependence on Initial Data
2.2. Uniform Stability
2.3. Uniformly Asymptotical Stability
2.4. Global Stability
2.5. Asymptotic Stability
2.6. Exponential Stability
2.7. Positive Solutions
2.8. Iterated Oscillation Criteria for First-Order Functional Dynamic Equations
2.9. Oscillations of the Solutions of First-Order Functional Dynamic Equations with Several Delays
2.10. Nonoscillations of First-Order Functional Dynamic Equations with Several Delays
3. Foundations of Linear Control Theory on Time Scales
3.1. Introduction
3.2. Linear Deterministic Systems
3.2.1. Controllability and Observability
3.2.2. Linear Time Varying
3.2.3. Switched Systems
3.3. Stability Analysis
3.3.1. Definitions
3.3.2. Realizations, Stability, and Stabilizability
3.3.3. Lyapunov Methods
3.4. Optimization
3.4.1. Calculus of Variations
3.4.2. Linear Quadratic Regulator
3.4.3. Linear Quadratic Tracking (LQT)
3.5. Kalman Filter
3.6. Stochastic Time Scales
4. Optimal Control Theory for Dynamic Equations
4.1. Optimal Control Problems
4.2. Preliminaries
4.2.1. Δ-Measurable Sets
4.2.2. Δ-Integrable Functions
4.2.3. Absolutely Continuous Functions
4.2.4. Set-Valued Functions and Measurability
4.2.5. Control Processes for (P2)
4.2.6. Δ-Measurable Selection
4.3. Main Results
4.4. Proofs of the Main Results
4.5. Conclusions
5. Controllability of Dynamic Equations
5.1. Introduction
5.2. Controllability of Linear Systems
5.3. Controllability of Dynamic Equations with Memory
5.4. Examples
5.5. Controllability of a Semilinear Neutral Dynamic Equation with Impulses and Nonlocal Conditions
5.6. An Example
5.7. Conclusion and Final Remark
6. Delayed Dynamic Equations with sp-Terms
6.1. Introduction
6.2. General Definitions
6.2.1. Doubly Weighted Pseudo Almost Periodic Functions on Time Scales
6.2.2. Stepanov-like Almost Periodic Functions
6.3. Doubly Weighted Stepanov-like Pseudo Almost Periodic Functions on Time Scales
6.4. Doubly Weighted Pseudo Almost Periodic Solution on Time Scales
6.5. Numerical Example
6.6. Conclusion
7. Integro-Dynamic System with Stepanov-like Coefficients
7.1. Introduction
7.2. Preliminaries
7.3. Existence and Uniqueness
7.4. Stability of Solution
7.5. Example
7.6. Conclusion
8. Terminal Value Problems for Discrete Fractional Relaxation Equations
8.1. Introduction
8.2. Preliminaries
8.3. Construction of the Green Functions
8.4. Existence of Solutions
8.5. Uniqueness of Solutions
8.6. Conclusion
9. Diamond-Alpha Hardy–Copson Type Dynamic Inequalities-I
9.1. Introduction
9.2. Literature Review
9.3. Preliminaries
9.4. Diamond-Alpha Hardy–Copson Type Dynamic Inequalities
9.5. Conclusion
10. Diamond-Alpha Hardy–Copson Type Dynamic Inequalities-II
10.1. Introduction
10.2. Literature Review
10.3. Preliminaries
10.4. Complementary Diamond-Alpha Hardy–Copson Type Dynamic Inequalities
10.5. Conclusion
11. Fishing Model with Feedback Control on Time Scales
11.1. Introduction
11.2. Preliminaries
11.3. Persistence
11.4. Almost Periodic Solutions and Stability Analysis
11.5. Numerical Simulations
11.6. Conclusion and Future Work
12. Some Geometric Properties of Dual Space on Time Scales
12.1. Introduction
12.2. The Dual Numbers, Dual Vectors and Dual Space on the Time Scales
12.2.1. The Inner Product and Norm of the Dual Numbers on the Time Scale
12.2.2. Module-D on the Time scales
12.3. Dual Directional Derivative on the Time Scale
12.4. Dual Vector Field on the Time Scales
12.5. The Taylor Expansion of Dual Analytic Function on the Time Scales
12.6. Dual Derivative Mapping on the Time Scales
13. Serret–Frenet Frame of a Curve Parametrized by Time Scales: A Brief Survey
13.1. Introduction
13.2. The Discrete Frenet Frame
13.3. The Frenet Frame of a Curve Parametrized by Time Scales
14. Applications of Time Scales in Nature: A Brief Survey
14.1. Motivating Examples
14.1.1. El Nino Effect
14.1.2. Growth of a Plant Species
14.1.3. Tumor Growth Model on Time Scales
14.2. A COVID-19 Model on Time Scales
14.2.1. Introduction and Preliminaries
14.2.2. A Nonautonomous Model for COVID-19 on Time Scales
14.2.3. Endurance and Extinction of COVID-19 Infection
14.2.4. Illustrative Examples
14.3. Delayed Predator–Prey System on Time Scales
14.3.1. Introduction and Preliminaries
14.3.2. Existence of Periodic Solutions
14.3.3. An Illustration
14.4. Existence of Periodic Solutions for an Ecological Model on Time Scales
14.4.1. Introduction and Preliminaries
14.4.2. Existence Result
14.5. Summary
15. Applications of Time Scales in Economics: A Brief Survey
15.1. HMMS Models on Time Scales
15.1.1. Introduction and Preliminaries
15.1.2. Statement of HMMS Model
15.1.3. Analysis of the HMMS Model
15.2. Dynamic Optimization Problems on Multiple Time Scales
15.2.1. Introduction and Preliminaries
15.2.2. Dynamic Maximization Utility Problem
15.2.3. Consumption Paths on Various Time Scales
15.3. Applications of Calculus of Variations in Behavioral Economics
15.3.1. Introduction
15.3.2. The Cake-Eating Problem
15.3.3. The Household Problem
15.4. Qualitative Analysis of a Solow Model on Time Scales
15.4.1. The Solow Model on Time Scales
15.4.2. Improved Solow Model on Time Scales
15.5. Summary
Index