Tianyuan Mathematical Center in Northeast China will open the short Course “Control and optimization for non-local and fractional differential equations” between August1 and August 12. It is held by Prof. Dr. Enrique Zuazua, Friedrich-Alexander University (FAU), Erlangen-Nürnberg (Germany) and Prof. Dr. Umberto Biccari，Fundación Deusto, Bilbao, Basque Country (Spain). 

一、About the short course

1、Instructors：

Prof. Dr. Enrique Zuazua, Friedrich-Alexander University (FAU), Erlangen-Nürnberg (Germany)

Prof. Dr. Umberto Biccari, Fundación Deusto, Bilbao, Basque Country (Spain)

2、Course name: Control and optimization for non-local and fractional differential equations

3、Period：August 1, 2022 -- August 12, 2022

4、Prerequisite knowledge：Ordinary Differential Equations (ODE), Partial Differential Equations (PDE) and numerical analysis.

5、Textbooks：

[1] J. M. Coron, Control and Nonlinearity, Mathjematical Surveys and Monographs, 143, AMS, 2009.

[2] 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., Elsevier Science, 2006, pp. 527-621.

[3] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo. Theory and applications of fractional differential equations, volume 204 of North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 2006.

[4] U. Biccari, M. Warma and E. Zuazua, Control and Numerical approximation of Fractional Diffusion Equations. Handbook of Numerical Analysis, Vol. 23 (2022), pp. 1-58.

[5] E. Isaacson, and H. B. Keller. Analysis of numerical methods. Courier Corporation, 2012.

[6] R. Glowinski, J.-L. Lions and J. He. Exact and approximate controllability for distributed parameter systems: a numerical approach. Cambridge University Press, 2008.

[7] B. Geshkovski and E. Zuazua, Turnpike in optimal control nof PDEs, ResNets, and beyond, preprint (2022)

二、Content and Schedule

In this series of lectures, we will discuss several topics related with the modeling, analysis, numerical simulation and control of non-local differential equations (NLDE)  arising in various areas of science and technology.

This course will be structured in 5 parts: 

• PART I: Introduction to control theory. We will give a short introduction to control theory for ODE and PDE, presenting the notion of controllability and its dual version, the so-called observability problem.

• PART II: Fractional in time models. We will iscuss non-local in time models and their control properties. We will start by presenting ODE and PDE with memory, and introducing the moving control strategy to obtain the controllability of such systems. Later on, we will consider fractional-in-time ODE and PDE.

• PART III: Fractional in space models. We will discuss non-local in space models and their control properties. We will start by presenting the fractional Laplace operator and its employment in PDE. Later on, we will analyze the controllability problem for this class of models.

• PART IV: Numerical approximation of PDE and numerical control. We will discuss the numerical approximation of (local and non-local) PDE through finite difference and finite element methods. Secondly, we will address the numerical approximation of controls for local and non-local PDE, relating the controllability property to optimal control theory and optimization.

• PART V: Complementary topics. We will discuss a couple of complementary topics related with the contents of this course: the so-called turnpike property, and PDE with non-local integral kernels.

Week 1：August 1, August 2, August 4, August 5

Week 2：August 7, August 8, August 10, August 11 

三、About the Lecturer

1 Prof. Dr. Enrique Zuazua, Chair in Applied Analysis, Alexander von Humboldt-Professorship. Expert in Applied Mathematics: Partial Differential Equations, Systems Control, Numerical Analysis and Machine Learning.

2、Prof. Dr. Umberto Biccari，Associated researcher in the Chair of Computational Mathematics at Fundación Deusto. Expert in Applied Mathematics: Partial Differential Equations, Non-local models, Control theory and Numerical Analysis. 

四、Contact us

   If you have any question, please contact us.

   Liu,Yang, Email: tianyuanmath@jlu.edu.cn

  Address: Room 315, Math building, Jilin University