| Titre : | Optimal control of stochastic systems with memory under noisy observations |
| Auteurs : | Khouloud Makhlouf, Auteur ; Nacira Agram, Directeur de thèse ; Boulakhras Gherbal, Directeur de thèse |
| Type de document : | Thése doctorat |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2023 |
| Format : | 1 vol. (81 p.) / ill., couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Résumé : |
This thesis aims to study a new type of stochastic partial differential equations (SPDEs) with space interactions. By space interactions, we mean that the dynamics of the system at time t and position in space x also depend on the space-mean of values at neighbouring points. In the first part, we introduce linear SPDEs. Then we prove the existence and uniqueness results (mild solution) for nonlinear SPDEs under linear growth and Lipschitz conditions on the coefficients. In the second part of this thesis, using results from Noisy Observation (nonlinear filtering), we transformed this noisy observation stochastic differential equation (SDE) control problem into full observation stochastic partial differential equations (SPDEs), and then we prove a sufficient and necessary maximum principle for the optimal control of SPDEs. In the third part of this thesis, we prove the existence and uniqueness of strong, smooth solutions of a class of stochastic partial differential equations with space interactions., and we show that, under some conditions,we usewhite noise theory to prove a positivity theorem for a class of SPDEs with space interactions. The solutions are positive for all times if the initial values are. Then we study the general optimization problem for such a system. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an example of optimal vaccination strategy for epidemics modelled as stochastic partial differential equations (SPDEs) with space interactions. |
| Sommaire : |
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Symbols and Abbreviations 1 General introduction 1 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Preliminaries 7 1.1 Elements from Stochastic Calculus: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The Itô formula: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Cauchy problems: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Elements of Semigroup Theory: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Factorization formula: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Useful results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Stochastic Partial Differential Equations 22 2.1 Linear SPDEs: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Nonlinear SPDEs: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Existence and uniqueness for nonlinear SPDEs: . . . . . . . . . . . . . . . . . 24 3 Partial (Noisy) Observation Optimal Control 37 3.1 Problem formulation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Stochastic maximumprinciple for SPDEs: . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 A sufficientMaximum Principle: . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.2 A NecessaryMaximum Principle: . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Controls which are independent of x . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 Stochastic Partial Differential Equations with space interactions and application to population modelling 54 4.1 Solutions of SPDEs with space interactions, and positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 The non-homogeneous stochastic heat equation and positivity . . . . . . . . . . . . 60 4.3 The optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.1 The Hamiltonian and the adjoint BSPDE . . . . . . . . . . . . . . . . . . . . . 65 4.3.2 A sufficient maximumprinciple approach (I) . . . . . . . . . . . . . . . . . . 70 4.3.3 A necessary maximumprinciple approach (I) . . . . . . . . . . . . . . . . . . 73 4.3.4 Controls which are independent of x . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Application to vaccine optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 conclusion 80 References 81 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/6323 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/149 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
