**Prelectura de tesis**

**Título de la tésis**: Long time control with applications

**Candidato a doctor**: Dario Pighin

**Director de tesis**: Enrique Zuazua Iriondo

Jueves 16 de Enero 2020 a las 11.30 hr.

Módulo 17, Sala 520

**Abstract**:

This talk is concerned with the study of some control problems in a large time horizon.

The first part of the talk is devoted to controllability of Partial Differential Equations under state and/or control constraints. We address the controllability under positivity constraints of semilinear heat equations and linear wave equations, by employing a ‘‘stair-case argument’’. We prove further the positivity of the minimal controllability time under positivity constraints, by applying a new method, based on the choice of a particular test function in the definition of weak solutions to evolution equations. Hence, despite the infinite velocity of propagation for parabolic equations, a waiting time phenomenon occurs in the constrained case.

The second part of the talk is devoted to the study of stability properties of optimal control problems over long time horizons. Under appropriate assumptions, the optima of a time-evolution control problem simplifies as the time horizon T goes to infinity, namely converge to the corresponding steady optima. When this occurs, we say the control problem enjoys the turnpike property. Some theoretical results in (nonlinear) PDE control will presented. We present an example of steady optimal problem which admits (at least) two solutions. An industrial application to rotors imbalance suppression will be given.