NCMW - Control Theory for Partial Differential Equation
Speakers and Syllabus
Pre-requisites: The audience should be familiar with measure and integration, functional analysis, Sobolev spaces, and well-posedness of linear ordinary and partial differential equations.
Target Audience: Ph.D. students, postdocs, scientists in research labs, young faculty members in educational institutions.
Syllabus:
| Name of the Speaker and their Affiliation | No of Hours | Syllabus to be covered in modules of at least hrs ( Tentative) |
|
Dr Mrinmay Biswas |
5 (7.5 hrs) |
Semigroup Theory and Time dependent Sobolev Spaces: A brief review of time-dependent Sobolev spaces. Motivation for semigroup , Strongly continuous semigroups and infinitesimal generators, adjoint semigroups and their generators. Semigroup of contractions : theorems of Lumer-Philips, Stone and Hille-Yosida. Analytic semigroup. Non-homogeneous linear evolution equations : various concepts of solutions. Application to existence and regularity of solutions for Transport, Heat and Wave, KDV equations |
| Dr Rajib Dutta IISER Kolkata |
5 (7.5 hrs) |
Weak Solutions , Compactness and Energy Methods for Linear PDE: Motivation for Banach valued functions and compactness theorems for time dependent Sobolev spaces. Weak solutions for evolution equations, Galerkin approximation for solution of Heat equation, energy method for weak solutions of heat and wave equations. Very week (/Transposition solution) for Transport, KDV, Heat and Wave equation. |
| Dr Shirshendu Chowdhury IISER Kolkata |
5 (6.5 hrs) |
Brief Recall of Controllability and Stabilizability for Linear ODE (Covered in the last NCM Workshop in IISER Kolkata) Abstract linear control system and application to hyperbolic equations, Admissible control and observation operators. Various notions of controllability and observability. Duality between and control and observation. Application to Transport, Wave and KDV equations. Spectral methods (Ingham inequalities and non harmonic Fourier series), Compactness uniqueness Method. |
| Dr Debanjana Mitra IIT Bombay |
5 (7.5 hrs) |
Exact Controllability of Wave Equation: Method of multiplier,extension methods, Holmgren’s uniqueness theorem, Hörmander’s Theorem, GCC, Transmutation approach etc.Stabilization of Wave Equation: Wonham’s theorem. |
| Dr Debayan Maity TIFR-CAM, Bangalore |
5 (7.5 hrs) |
Null Controllability of Heat equations via Spectral methods (The moment method, Biorthogonal family of exponentials, non- harmonic Fourier series), The Lebeau-Robbiano approach (back-stepping type) and The Fursikov-Imanuvilov strategy (Carleman estimates) , Fundamental solution methods, flatness approach, back-stepping design, Approximate Controllability of Heat Equations. |
| Dr Dharmatti Sheetal, IISER TVM | 5 (6.5 hrs) |
Optimal control for Heat equation, Regularity of control via optimality system.Optimal control for Wave equation, Regularity of control via optimality system |
| Prof Mythily Ramaswamy ICTS-TIFR, Bangalore | 5 (6 hrs) |
Feedback Stabilization for Heat and KDV equation, Finite dimensional feedback, Ricatti based feedback, Back-stepping Feedback. Introduction to Controllability and Stabilizability for Non- Linear PDE |
References:
- Alain Bensoussan,; Giuseppe Da Prato,; Michel C. Delfour, Sanjoy K. Mitter,Representation and control of infinite dimensional systems. Second edition. Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 2007.
- Jean-Michel Coron, Control and nonlinearity. Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.
- A. V. Fursikov, O. Yu. Imanuvilov Controllability of evolution equations. Lecture Notes Series,34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
- Vilmos Komornik, Paola Loreti, Fourier series in control theory. Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.
- Vilmos Komornik;. Exact controllability and stabilization. The multiplier method. RAM:Research in Applied Mathematics.Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.
- M. Krstic, A. Smyshlyaev,.: Boundary control of PDEs: A course on back-stepping designs,vol. 16, SIAM, Philadelphia, 2008
- 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.
- Jean-Pierre RAYMOND, Optimal Control of Partial Differential Equations, Ficus Lecture Notes https://www.math.univ-toulouse.fr/~raymond/book-ficus.pdf
- Fredi Tröltzsch, Optimal control of partial differential equations. Theory, methods and applications.
- Marius Tucsnak and George Weiss. Observation and control for operator semigroups.Birkhauser Verlag, Basel, 2009.
- Jerzy Zabczyk, Mathematical control theory. An introduction. Modern Birkhäuser Classics. ,Inc., Boston, MA, 1995.
- Enrique Zuazua, Controllability of Partial Differential Equations, Lecture Notes, https://dcn.nat.fau.eu/wp-content/uploads/Topics-PDE-Control.pdf
Names of Possible Tutors with their Affiliation and Abbreviations
| Sr. No. | Name | Affiliation | Abbreviation |
| 1 | Samprita Das Roy | IISER Kolkata | SDR |
| 2 | Subrata Majumdar | IIT Bombay | SM |
| 3 | Jiten Kumbhakar | IISER Kolkata | JK |
| 4 | Wasim Akram | IIT Bombay | WA |
| 5 | Sakil Ahamed | IIT Bombay | SA |
| 6 | Manish Kumar | IISER Kolkata | MK |
| 7 | Manika Bag | IISER TVM | MB |
| 8 | Ritabrata Jana | IISER TVM | RJ |
Time Table
| Date | Lecture 1 9.30 – 11.00 |
Tea 11.00 – 11. 30 |
Lecture 2 11.30 – 1.00 |
Lunch 1.00 – 2.30 |
Lecture 3 2:30 – 3. 30 |
Tea 3.30 - 4. 00 |
Tutorial/ 4.00-5.15 |
Snacks 5.30 |
| 1st week | (Speaker + two tutors) | |||||||
| Mon 04/12/2023 | MB | T E A | RD | L U N C H | SC | T E A | MB + SDR+SM | S N A C K S |
| Tue 05/12/2023 | MB | RD | SC | RD+ JK+WA | ||||
| Wed 06/12/2023 | MB | RD | SC | SC +SA+MK | ||||
| Thur 07/12/2023 | MB | RD | SC | MB + SDR+SM | ||||
| Fri 08/12/2023 | MB | RD | SC | RD+ JK+WA | ||||
| Sat 09/12/2023 | SC | DMi | SD | SC +SA+MK | ||||
| 2nd week | ||||||||
| Mon 11/12/2023 | MR | T E A | DMi | L U N C H | SD | T E A | SD+ SDR+SM | S N A C K S |
| Tue 12/12/2023 | MR | DMi | SD | DMi+ +JK+WA | ||||
| Wed 13/12/2023 | DM | DMi | MR | DM+ +SA+MK | ||||
| Thur 14/12/2023 | DM | DMi | MR | MR+ SDR+SM | ||||
| Fri 15/12/2023 | DM | SD | MR | DMi+ +JK+WA | ||||
| Sat 16/12/2023 | DM | SD | DM | DM+ +SA+MK |