Turnpike Phenomenon in Metric Spaces

Description

Preface Contents 1 Introduction 1.1 Turnpike Property for Variational Problems 1.2 Notation 1.3 General Dynamical Systems with a Lyapunov Function 1.4 Continuous-Time Nonautonomous Problems on Half-Axis 2 Differential Inclusions 2.1 Banach Space-Valued Functions 2.2 Set-Valued Mappings and a Convergence Result 2.3 Dynamical Systems 2.4 Auxiliary Results 2.5 Proof of Proposition 2.7 2.6 Proof of Proposition 2.8 2.7 Proof of Theorem 2.11 2.8 An Example 3 Discrete-Time Dynamical Systems 3.1 Preliminaries and Main Results 3.2 Proof of Proposition 3.1 3.3 Proofs of Propositions 3.2 and 3.3 3.4 Proof of Theorem 3.5 3.5 Proof of Theorem 3.6 3.6 Proof of Theorem 3.7 4 Continuous-Time Dynamical Systems 4.1 Preliminaries and Main Results 4.2 Proof of Proposition 4.1 4.3 Proof of Proposition 4.2 4.4 Proof of Theorem 4.5 4.5 Proof of Theorem 4.6 4.6 Proof of Theorem 4.7 5 General Dynamical Systems with a Lyapunov Function 5.1 Preliminaries and Two Turnpike Results 5.2 Two Auxiliary Results 5.3 Proof of Proposition 5.2 5.4 Proof of Proposition 5.3 5.5 Proof of Theorem 5.4 5.6 An Auxiliary Result for Theorem 5.5 5.7 Proof of Theorem 5.5 5.8 Proofs of Propositions 5.6 and 5.7 5.9 A Weak Turnpike Result 5.10 Turnpike Results 5.11 Auxiliary Results 5.12 Proof of Theorem 5.15 5.13 Auxiliary Results for Theorem 5.16 5.14 Proofs of Theorems 5.16 and 5.17 5.15 Extensions of Theorems 5.16 and 5.17 5.16 An Auxiliary Result for Theorem 5.24 5.17 Proof of Theorem 5.24 5.18 Generalizations of Theorems 5.16 and 5.17 5.19 Proof of Theorem 5.28 5.20 Continuity of the Function π 5.21 Examples 6 Discrete-Time Nonautonomous Problems on Half-Axis 6.1 Preliminaries and Boundedness Results 6.2 Turnpike Properties 6.3 Perturbed Problems 6.4 Examples 6.5 The Space of Objective Functions and the Stability Result 6.6 Auxiliary Results 6.7 Proof of Theorem 6.22 6.8 Problems with Discount 6.9 Auxiliary Results for Theorem 6.25 6.10 Proof of Theorem 6.25 6.11 Genericity Results for Discrete-Time Problems 6.12 Proof of Theorem 6.29 6.13 Examples of the Space M 6.14 Extensions of the Generic Results 6.15 Smooth Integrands 7 Infinite Dimensional Control 7.1 Unbounded Operators 7.2 C0 Semigroup 7.3 Evolution Equations 7.4 C0 Groups 7.5 Admissible Control Operators 7.6 Examples 8 Continuous-Time Nonautonomous Problems on Half-Axis 8.1 Preliminaries 8.2 The First Class of Problems 8.3 The Second Class of Problems 8.4 A0 and the First and Second Classes of Problems 8.5 The Third Class of Problems 8.6 Boundedness Results 8.7 Turnpike Results 8.8 Lower Semicontinuity Property 8.9 Perturbed Problems 8.10 Auxiliary Results for Theorems 8.5 and Its Proof 8.11 Proof of Theorem 8.6 8.12 Proof of Theorem 8.10 8.13 Proof of Theorem 8.25 8.14 Proof of Theorem 8.26 8.15 An Auxiliary Result 8.16 Proof of Proposition 8.28 8.17 Auxiliary Results for Theorem 8.29 8.18 Proof of Theorem 8.29 8.19 Proof of Theorem 8.33 8.20 The Strong Turnpike Property 8.21 Auxiliary Results for Theorem 8.39 8.22 Proof of Theorem 8.39 8.23 Examples 9 Stability and Genericity Results 9.1 Preliminaries 9.2 Stability of TP 9.3 Auxiliary Results for Theorem 9.2 9.4 Proof of Theorem 9.2 9.5 Problems with Discount 9.6 Auxiliary Results for Theorem 9.5 9.7 Proof of Theorem 9.5 9.8 Genericity 9.9 Proof of Theorem 9.9 9.10 A Turnpike Result 9.11 Spaces od Smooths Integrands 9.12 Assumption (A5) for the First Class of Problems 9.13 Assumption (A5) for the Second Class of Problems 9.14 Assumption (A5) for the Third Class of Problems References Index