# M01 â€“ PARISâ€SACLAY 18/01/2016 â€ 22/01/2016

## Control by Partial Differential Equation modelling

**Enrique Zuazua**

BCAM & IKERBASQUE

Basque Foundation for Science, Bilbao, Spain Â

http://www.ikerbasque.net/enrique.zuazua

enrique.zuazua@gmail.com

### Summary of the course

In this series of lectures we shall discuss several topics related with the modelling, analysis, numerical simulation and control of Partial Differential Equations (PDE) arising in various contexts of Science and Technology. In these lectures we shall introduce some of the most relevant work that has been done in the subject in recent years, paying special attention to the fundamental methodological aspects, and pointing towards some potential future perspectives of research.We shall first describe and document some of the most relevant applications to Sciences, Engineering and Technology in which these problems arise from an historical viewpoint. After a short introduction to the finite-dimensional theory, we shall then describe the basic theory for the wave and heat equation, to later address some important multi-physics models. Then we shall address the problem of the numerical approximation of control problems. To begin with we shall show that the control and the numerical approximation process do not commute so that, in general, when controlling a finite-dimensional approximation of the continuous model, one does not actually compute an approximation of the control one is looking for. We shall see what are the possible remedies to these pathologies: space discretizations, numerical damping, filtering of high frequencies, multi-grid algorithms, etc.. The latter fact is of great impact from a modelling point view since in practical applications numerical approximation schemes may also be used (and they are often used that way) as discrete models. We shall also present some interesting concepts and results on switching, sparse, averaged and bang-bang control, and the turnpike property ensuring that, most often, in long-tie horizons, optimal controls and trajectories are close to the steady-state ones. To conclude, we shall present a list of open problems and directions of possible future research.

### Outline

1.- Historical introduction2.- Introduction to finite-dimensional control

3.- Wave propagation

4.- Heat diffusion

5.- Some relevant models of multi-physics nature.

6.- Numerical approximation of control problems

7.- Averaged control on parameter depending systems

8.- Switching, sparse and bang-bang control

9.- The turnpike property

10.- Pespectives and open problems

### References

J. M. Coron, Control and Nonlinearity,Â AMS, 2009.S Ervedoza, E. Z. On the numerical approximation of exact controls for waves, SpringerBriefs in Math., 2013, XVII.

E. TrÃ©lat. ContrÃ´le optimal: thÃ©orie& applications. Vuibert. Collection "MathÃ©matiques ConcrÃ¨tes", 2005.

E TrÃ©lat, E. Zuazua. Turnpike property in finite-dimensional nonlinear optimal control, JDE, 218 (2015),Â 81-114.

E. Zuazua, Controllability and Observability of Partial Differential Equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Equations, vol. 3, C. M. Dafermos and E. Feireisl eds.,Â Elsevier Science, 2006, pp. 527-621.

E. Zuazua, Switching control, J. Eur. Math. Soc., 2011, 13, 85-117.

E. Zuazua, Averaged controllability, Automatica, 50 (2014) 3077â€“3087.