UE Modeling and control of PDE

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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)