## UE Modeling and control of PDE

Informations aux utilisateurs

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Diplômes intégrant cet élément pédagogique :

### Descriptif

This set of courses proposes an overview of recent techniques for the identification, observation, simulation and control of distributed parameter systems. This class of systems is widely used in physics and considered in many applications (such as in environment dynamics, airflow control, structural mechanics, and adaptive optics) having a large or an infinite number of degrees of freedom. A Partial Differential Equation (PDE) usually models them. Their mathematical study asks for a special care to analyze the dynamics behavior and to describe their control properties. Different aspects of this description are considered in this Teaching Unit, by emphasizing the practical methods allowing for some real applications.

This Teaching Unit is composed by three different courses:

#### Analysis and control (13.5 h)

Lesson Topic
Some recalls in the analysis of PDE
Differential calculus; derivation of a PDE; classification of infinite dimensional systems.
Semigroup theory
Strongly continuous semigroups; contraction semigroups.
Control and Observation of some particular PDEs
Transport equation; heat equation.
Stability and Stabilization
Definitions; Lyapunov functions.

#### Modeling and Inverse problems (13.5h)

Lesson Topic
Discretization methods for the numerical approximation of PDEs
basics of finite difference and finite element methods; stability analysis for evolution equations.
Identification and inverse problems
basics of optimization algorithms; derivation of the adjoint of a discretized model; some practical aspects of the derivation of a numerical model.
Link with the linear statistical estimation

#### Distributed optimization  (13.5h)

Lesson Topic
Open-loop optimal control of PDE
Adjoint-based method for some particular PDEs: a parabolic and a hyperbolic PDE case studies; a short introduction to numerical methods for the solution of open-loop infinite-dimensional optimal control problems.
Optimal control of PDE with state-feedback
The Linear Quadratic Regulator; solution via the operator Riccati equation; two case studies.
Robust control of PDE with state-feedback
A game-theoretic approach: the Hinfinity optimal regulator; solution via the associated operator Riccati equation; one case study.

Prerequisites: basic mathematical background, control theory of finite dimensional systems (control and observation theory for linear ODEs, in particular optimal LQ regulation)

### Informations complémentaires

Lieu(x) : Grenoble
Langue(s) : Anglais