Dynamic Equations on Time Scales and Applications

This book presents the theory of dynamic equations on time scales and applications, providing an overview of recent developments in the foundations of the field as well as its applications. It discusses the recent results related to the qualitative properties of solutions like existence and uniqueness, stability, continuous dependence, controllability, oscillations, etc. • Presents cutting-edge research trends of dynamic equations and recent advances in contemporary research on the topic of time scales • Connects several new areas of dynamic equations on time scales with applications in different fields • Includes mathematical explanation from the perspective of existing knowledge of dynamic equations on time scales • Offers several new recently developed results, which are useful for the mathematical modeling of various phenomena • Useful for several interdisciplinary fields like economics, biology, and population dynamics from the perspective of new trends The text is for postgraduate students, professionals, and academic researchers working in the fields of Applied Mathematics. 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