Contents
Preface
About the Author
Acknowledgments
The Notation and Preliminary Results
Introduction
1. Basic Concepts
1.1 The Fundamental Definitions
1.2 The Cauchy or Initial-Value Problem
1.3 Some Elementary Integration Methods
1.3.1 The most simple cases
1.3.2 Linear equations
1.3.3 Bernoulli equations
1.3.4 Riccati equations
1.3.5 Homogeneous equations
1.3.6 Exact equations
1.3.7 Reducible to exact equations, integrating factors
1.3.8 Second-order incomplete equations
1.3.9 Second-order linear equations
1.3.9.1 The case of constant coefficients
1.3.9.2 Second-order Euler equations
Exercises
2. The Cauchy Problem: Local Analysis
2.1 Integral Formulation
2.2 Contractions and Fixed Points
2.3 Picard’s Theorem
2.4 Ascoli–Arzelà’s Theorem
2.5 ε-Approximated Solutions, Peano’s Theorem
2.6 Implicit Differential Equations
Exercises
3. Uniqueness
3.1 Gronwall’s Lemma
3.2 Uniqueness of the Solution to the Cauchy Problem
Exercises
4. The Cauchy Problem: Global Analysis
4.1 Existence and Uniqueness of a Maximal Solution
4.2 Characterization of the Maximal Solution
4.3 A Particular Case
Exercises
5. Cauchy Problems and Linear Systems
5.1 The Existence and Uniqueness of a Solution
5.2 Homogeneous Linear Systems
5.3 Non-Homogeneous Linear Systems: Lagrange’s Method
5.4 The Exponential of a Matrix
5.5 Systems with Constant Coefficients
5.6 Linear Equations of Order n
Exercises
6. Boundary-Value Problems for Linear Systems
6.1 First Results
6.2 The Alternative Theorem
6.3 The Case of a Linear Second-Order ODE
Exercises
7. Some Regularity Results
7.1 First Results
7.2 The General Maximal Solution: Continuity
7.3 Proof of Theorem 7.2
7.4 The General Maximal Solution: Differentiability
7.5 Proof of Theorem 7.4
7.6 Continuity and Differentiability with Respect to Parameters
Exercises
8. Stability Results
8.1 Preliminary Concepts
8.2 Stability of Linear Systems
8.3 Comparison and First Approximation
8.4 Liapunov’s Stability Method
Exercises
9. The Method of Characteristics for First-Order Linear and Quasi-Linear Partial Differential Equations
9.1 The Concepts and Some Preliminary Results
9.2 Resolution of the Cauchy Problem
9.3 Proof of Theorem 9.1
Exercises
10. Basic Ideas in Control Theory
10.1 Some Tools and Preliminary Results
10.2 A Very Simple Optimal Control Problem: A Linear ODE, No Constraints
10.3 Other Optimal Control Problems for Equations and Systems
10.3.1 A linear ODS with no constraint
10.3.2 A nonlinear ODE with no constraint
10.3.3 A linear ODS with inequality constraints
10.4 Controllability Problems
Exercises
11. Additional Notes
11.1 Generalities
11.1.1 Historical origin of differential equations
11.1.1.1 The algebraic-resolution period
11.1.1.2 The analytical period
11.1.1.3 Qualitative theory and connection to numerical analysis
11.1.2 Population models
11.1.3 Initial-value problems
11.1.4 Impossibility of the explicit resolution of some elementary ODEs
11.1.5 Linear equations and systems
11.2 Local Analysis of the Cauchy Problem
11.2.1 Other fixed-point theorems
11.2.2 Implicit and inverse functions
11.3 Uniqueness of Solution in a Cauchy Problem
11.4 Global Analysis of the Cauchy Problem
11.4.1 Existence of a maximal solution
11.4.2 Characterization of the maximal solution
11.5 More on Linear Equations and Systems
11.5.1 Lagrange’s formula
11.5.2 Semigroups of operators and applications
11.6 More Results on Boundary-Value Problems
11.6.1 A nonlinear boundary-value problem
11.6.2 Green’s operators
11.6.3 Sturm–Liouville problems
11.7 Regularity of the Solutions to an ODS
11.7.1 Regularity and boundary-value problems
11.8 Stability and Attractivity of the Solutions
11.8.1 The case of a non-autonomous ODS
11.8.2 Global stability, cyclic orbits, and Poincaré–Bendixson’s theorem
11.9 Partial Differential Equations of the First Order
11.9.1 Motivations and applications
11.9.1.1 Traffic problems
11.9.1.2 Inviscid fluids (incompressible or not)
11.9.2 The Cauchy problem for a nonlinear PDE
11.10 Control of Differential Systems
11.10.1 Historical data
11.10.2 Other more complex control problems
11.10.3 Additional comments on control problems
References
Index