NCMW - Control Theory for Differential Equations (2022)

Speakers and Syllabus

 Name of the possible Speakers with affiliations

1. Dr Saugata Bandyopadhyay, IISER Kolkata
2. Dr Rajib Dutta, IISER Kolkata
3. Dr. Shirshendu Chowdhury, IISER Kolkata
4. Prof. Mythily Ramaswamy , ICTS-TIFR, Bangalore
5. Dr. Debanjana Mitra, IIT Bombay.
6. Dr. Debayan Maity , TIFR-CAM, Bangalore

Possible Tutorial Assistants :

Syllabus:

First week: (Existence and Stability of ODE in R^N)

1. Existence and stability for Linear ODE:Linear ODE, Exponential of Matrix, Global existence Theorem for linear ODE,(from [7], [8] )Linear stability (eigenvalue criteria from [9]),
2. Existence and stability for Non-Linear ODE:Nonlinear ODE, Picard iteration, existence uniqueness theorem ([7] , [8]), stability of nonlinear ODE, Lyapunov’s Method([7]), LaSalle's invariance principle([4]
   Second week: (Controllability and Stabilizability in R^N for linear and nonlinear ODE)

3.  Controllability and Stabilizability for Linear ODE:Finite dimensional linear system, Various concepts of controllability. Reachable states. Kalman rank condition, Popov-Belevitch-Hautus test. The moment method and HUM control. Concepts of observability, duality between controllability and observability, cost of the control. ( [1],[2],[5],[6],[9]). Stabilizability for linear system, Popov-Belevitch-Hautus test, Feedback stabilization, pole shifting, Gramian, Complete stabilization, relation between null controllability and stabilizability ([2], [5], [9]).
4. Controllability and Stabilizability for Non- Linear ODE:Controllability for nonlinear systems in finite dimension: Fixed point method, Return method,Quasi-static deformation method and Power series expansion method, Iterated Lie bracket stechnique (Lie algebra Rank condition) from [2] and phantom tracking method (from [3] as application of Lyapunov method and Lasalle's invariance principle).

References:

1. Franck Boyer. Controllability of linear parabolic equations and systems, 2020. Lecture Notes, https://hal.archives-ouvertes.fr/hal-02470625.
2. Jean-Michel Coron, Control and nonlinearity. Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.
3. Jean-Michel Coron, Phantom Tracking method, homogeneity and rapid stabilization. Math. Control Relat. Fields3(2013), no. 3,303-322.
4. Jean-Michel Coron, Ivonne Rivas, Quadratic approximation and time-varying feedback laws. SIAM J. Control Optim.55 (2017), no. 6, 3726-3749.
5. Hassan K. Khalil, Nonlinear systems. Macmillan Publishing Company, New York, 1992
6. Weijiu Liu, Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation, Mathématiques & Applications (Berlin) [Mathematics & Applications], 66. Springer-Verlag, Berlin, 2010.
7. Sorin Micu and Enrique Zuazua, An Introduction to the Controllability of Partial Differential Equations. Available at http://www.uam.es/personal_pdi/ciencias/ezuazua/informweb/argel.pdf.
8. Lawrence Perko, Differential Equations and Dynamical Systems.Third edition. Texts in Applied Mathematics, 7.
9. Springer-Verlag, New York, 2001.9. George Simmons, Differential Equations with Applications and Historical Notes. Third edition, Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2017
10. Jerzy Zabczyk, Mathematical control theory. An introduction. Modern Birkhäuser Classics. , Inc., Boton, MA, 1995.

Time Table