Contribution of the Dynamic Programming Method to Solving the Dolichobrachistochrone Differential game / Aicha Ghanem

Public ISBD Titre : Contribution of the Dynamic Programming Method to Solving the Dolichobrachistochrone Differential game Type de document : document électronique Auteurs : Aicha Ghanem, Auteur ; Touffik Bouremani, Directeur de thèse Editeur : Setif:UFA Année de publication : 2025 Importance : 1 vol (92 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Differential game Differential inclusion Feedback strategies Dynamic programming Hamiltonian flow Value function Verification theorem Index. décimale : 510 - Mathématique Résumé : In this thesis, we focus on applying St. Mirică's Dynamic Programming method to solve the Dolichobrachistochrone differential game introduced by R. Isaacs. We propose feedback strategies as a new contribution that offers adaptability, efficiency, and simplicity while reducing algorithmic complexity. This approach employs a refined Cauchy characteristics method to handle stratified Hamilton-Jacobi equations while ensuring the existence of the value function. The optimality of the feedback strategies is rigorously validated using the Verification Theorem for locally Lipschitz value functions and further supported by established numerical tests. Note de contenu : Sommaire I CONCEPTS AND AUXILIARY RESULTS FROM NON-SMOOTH ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Analysis of differentiable mappings on smooth manifolds of Rn . . . . . 12 1.2.1 Differentiable mappings on manifolds of Rn . . . . . . . . . . . 14 1.2.2 Tangent space of a differentiable manifolds . . . . . . . . . . . . 14 1.3 Stratified sets and mappings . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Tangent cones and generalized derivatives . . . . . . . . . . . . . . . . 19 1.5 Contingent derivatives of marginal functions . . . . . . . . . . . . . . . 21 1.6 Necessary conditions for monotonicity . . . . . . . . . . . . . . . . . . . 25 1.7 Smooth Hamiltonian and characteristic flows . . . . . . . . . . . . . . . 27 1.8 Cauchy’s method of characteristics . . . . . . . . . . . . . . . . . . . . 29 II AUTONOMOUS DIFFERENTIAL GAMES . . . . . . . . . . . . . . 32 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Formulation of a Differential Game . . . . . . . . . . . . . . . . . . . . 32 2.3 Admissible feedback strategies and their relative optimality . . . . . . . 34 2.4 Verification theorems for admissible feedback strategies . . . . . . . . . 38 2.5 The general algorithm of Dynamic Programming . . . . . . . . . . . . 42 III ON THE SOLUTION OF DOLICHOBRACHISTOCHRONE DIFFERENTIAL GAME VIA DYNAMIC PROGRAMMING APPROACH 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Position of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.1 Dynamic programming formulation . . . . . . . . . . . . . . . . 51 3.2.2 The Hamiltonian and set of transversality terminal points . . . 51 3.3 The generalized Hamiltonian and characteristic flows . . . . . . . . . . 53 3.3.1 Hamiltonian field on the singular stratum Z0 . . . . . . . . . . . 55 3.4 Construction of the Hamiltonian flow . . . . . . . . . . . . . . . . . . . 56 3.4.1 The Hamiltonian flow ending on the stratum Z+ . . . . . . . . 56 3.4.2 The Hamiltonian system on the stratum Z− . . . . . . . . . . . 60 3.4.3 Continuation of trajectories on the stratum Z− . . . . . . . . . 62 3.4.4 Other admissible trajectories . . . . . . . . . . . . . . . . . . . . 67 3.5 Value function and optimal feedback strategies . . . . . . . . . . . . . . 72 3.5.1 Invertibility of the trajectories X+(., .) . . . . . . . . . . . . . . 73 3.5.2 Invertibility of the trajectories X−(., .) . . . . . . . . . . . . . . 78 3.5.3 Invertibility of the trajectories X(., .) . . . . . . . . . . . . . . . 83 Côte titre : DM/0211 Contribution of the Dynamic Programming Method to Solving the Dolichobrachistochrone Differential game [document électronique] / Aicha Ghanem, Auteur ; Touffik Bouremani, Directeur de thèse . - [S.l.] : Setif:UFA, 2025 . - 1 vol (92 f.) ; 29 cm. Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Differential game Differential inclusion Feedback strategies Dynamic programming Hamiltonian flow Value function Verification theorem Index. décimale : 510 - Mathématique Résumé : In this thesis, we focus on applying St. Mirică's Dynamic Programming method to solve the Dolichobrachistochrone differential game introduced by R. Isaacs. We propose feedback strategies as a new contribution that offers adaptability, efficiency, and simplicity while reducing algorithmic complexity. This approach employs a refined Cauchy characteristics method to handle stratified Hamilton-Jacobi equations while ensuring the existence of the value function. The optimality of the feedback strategies is rigorously validated using the Verification Theorem for locally Lipschitz value functions and further supported by established numerical tests. Note de contenu : Sommaire I CONCEPTS AND AUXILIARY RESULTS FROM NON-SMOOTH ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Analysis of differentiable mappings on smooth manifolds of Rn . . . . . 12 1.2.1 Differentiable mappings on manifolds of Rn . . . . . . . . . . . 14 1.2.2 Tangent space of a differentiable manifolds . . . . . . . . . . . . 14 1.3 Stratified sets and mappings . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Tangent cones and generalized derivatives . . . . . . . . . . . . . . . . 19 1.5 Contingent derivatives of marginal functions . . . . . . . . . . . . . . . 21 1.6 Necessary conditions for monotonicity . . . . . . . . . . . . . . . . . . . 25 1.7 Smooth Hamiltonian and characteristic flows . . . . . . . . . . . . . . . 27 1.8 Cauchy’s method of characteristics . . . . . . . . . . . . . . . . . . . . 29 II AUTONOMOUS DIFFERENTIAL GAMES . . . . . . . . . . . . . . 32 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Formulation of a Differential Game . . . . . . . . . . . . . . . . . . . . 32 2.3 Admissible feedback strategies and their relative optimality . . . . . . . 34 2.4 Verification theorems for admissible feedback strategies . . . . . . . . . 38 2.5 The general algorithm of Dynamic Programming . . . . . . . . . . . . 42 III ON THE SOLUTION OF DOLICHOBRACHISTOCHRONE DIFFERENTIAL GAME VIA DYNAMIC PROGRAMMING APPROACH 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Position of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.1 Dynamic programming formulation . . . . . . . . . . . . . . . . 51 3.2.2 The Hamiltonian and set of transversality terminal points . . . 51 3.3 The generalized Hamiltonian and characteristic flows . . . . . . . . . . 53 3.3.1 Hamiltonian field on the singular stratum Z0 . . . . . . . . . . . 55 3.4 Construction of the Hamiltonian flow . . . . . . . . . . . . . . . . . . . 56 3.4.1 The Hamiltonian flow ending on the stratum Z+ . . . . . . . . 56 3.4.2 The Hamiltonian system on the stratum Z− . . . . . . . . . . . 60 3.4.3 Continuation of trajectories on the stratum Z− . . . . . . . . . 62 3.4.4 Other admissible trajectories . . . . . . . . . . . . . . . . . . . . 67 3.5 Value function and optimal feedback strategies . . . . . . . . . . . . . . 72 3.5.1 Invertibility of the trajectories X+(., .) . . . . . . . . . . . . . . 73 3.5.2 Invertibility of the trajectories X−(., .) . . . . . . . . . . . . . . 78 3.5.3 Invertibility of the trajectories X(., .) . . . . . . . . . . . . . . . 83 Côte titre : DM/0211

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Code-barres	Cote	Support	Localisation	Section	Disponibilité
DM/0211	DM/0211	Thèse	Bibliothèque des sciences	Anglais	Disponible Disponible

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