Control Theory for Partial Differential Equation

Convener(s)

 
Name: Dr Dharmatti Sheetal  Dr. Shirshendu Chowdhury
Mailing Address:  Indian Institute of Science Education and Research Thiruvananthapuram (IISER TVM),
School of Mathematics, Maruthamala PO, Vithura, Thiruvananthapuram – 695551. Kerala, India 		Indian Institute of Science and Research – Kolkata,
Department of Mathematics and Statistics (DMS) Mohanpur, Nadia 741246, West Bengal, India
Email: sheetal at iisertvm.ac.in  shirshendu at iiserkol.ac.in

This NCMW aims to introduce the participants to the control theory for systems governed by Partial differential equations (PDE) . The motivation comes from the need to understand control of partial differential equations modeling various physical phenomena. The controllability problem, generally speaking, is to move a dynamical system by means of a suitable control (input), from one state to the desired state in finite time, whereas stabilizability refers to the system being brought to rest or an equilibrium state as time tends to infinity. Moreover, in an optimal control problem the objective is to minimize a criterion depending on the observation of the state and on the control variable. This workshop will be a sequel or follow up of the last NCM Workshop on Control of ODE (https://www.atmschools.org/school/2022/NCMW/ctde) in IISER Kolkata, in Dec 2022.

 
In the first week we will recall the notion of controllability and stabilizability for the finite dimensional linear systems, various criteria and techniques to check controllability or stabilizability of a system which was covered in the previous workshop. Then we will discuss some topics from time dependent Sobolev spaces and Semigroup theory, (Energy methods) that are required to introduce the notion of weak solutions/ transposition solutions of evolution equations, namely transport, wave, heat and KDV equations. Parallelly we introduce the notion of controllability for the infinite dimensional linear systems. We will discuss controllability, stabilizability and optimal control problem of some classical linear partial differential equations, Hyperbolic or Parabolic or mixed in nature namely transport equations, wave equations, heat equations and KDV equation. In the first week, we will focus on the transport equation and wave equation and in the second week will be devoted to control problems related to heat equations and KDV equations.
 
In the process several Direct methods and Indirect duality methods (Based on observability) to prove controllability and stabilizability such as Spectral methods (The moment methods, Ingham inequalities and non- harmonic Fourier series), method of multiplier, method of Lebeau-Roobbiano, method of Fursikov-Imanuvilov (Carleman estimates), fundamental solution methods, transmutation method, flatness approach, back-stepping design will be discussed.At the end of the second week, we will briefly discuss Control of nonlinear versions of above PDEs using Fixed point method, Source term method, Return method, Quasi-static deformation method and Power series expansion method (as explained in the book and recent papers of Jean Michel Coron). These methods have wide ranging applications in current research on control of Non-linear Partial Differential Equations, which is an important research area of the current interest nationally as well as internationally

Dates: 

Monday, December 4, 2023 - 19:00 to Saturday, December 16, 2023 - 19:00

Venue: 

Venue Address: 

Indian Institute of Science Education and Research Thiruvananthapuram (IISER TVM), Maruthamala PO,
Vithura, Thiruvananthapuram – 695551. Kerala, India

Venue State: 

Venue City: 

PIN: 

211019

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
IIT Kanpur

 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.
Datko’s Theorem, Gramian, Urquiza’s method, Ricatti based feedback, Backstepping.
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
Relation between Optimal control, Stabilization, Approximate and Null controllability.
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:

  1. 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.
  2. Jean-Michel Coron, Control and nonlinearity. Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.
  3. 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.
  4. Vilmos Komornik, Paola Loreti, Fourier series in control theory. Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.
  5. Vilmos Komornik;. Exact controllability and stabilization. The multiplier method. RAM:Research in Applied Mathematics.Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.
  6. M. Krstic, A. Smyshlyaev,.: Boundary control of PDEs: A course on back-stepping designs,vol. 16, SIAM, Philadelphia, 2008
  7. 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.
  8. Jean-Pierre RAYMOND, Optimal Control of Partial Differential Equations, Ficus Lecture Notes https://www.math.univ-toulouse.fr/~raymond/book-ficus.pdf
  9. Fredi Tröltzsch, Optimal control of partial differential equations. Theory, methods and applications.
  10. Marius Tucsnak and George Weiss. Observation and control for operator semigroups.Birkhauser Verlag, Basel, 2009.
  11. Jerzy Zabczyk, Mathematical control theory. An introduction. Modern Birkhäuser Classics. ,Inc., Boston, MA, 1995.
  12. 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/
Discussion

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

Selected Applicants: 

NOTE: Selected outstation applicants should confirm their participation through email to  sheetal@iisertvm.ac.in (with CC to shirshendu@iiserkol.ac.in) by 1st November, 2023 .  If a selected outstation candidate fails to confirm his/her participation by 1st November, 2023 then WAIT LISTED Candidates will be given the opportunity and they will be informed through email. 

Sr. SID Full Name Gender Affiliation Position In
College / University		 Univ/ Inst
M.Sc. /M.A.		 Year Of
Passing
M.Sc./M.A		 Ph.D.
Deg.
Date
Outstation participants
1 49684 Mr Ajith Kumar T Male Niser Bhubaneswar Phd University Of Madras 2019  
2 49915 Ms. Stuti Das Female Indian Institute Of Technology Phd Jadavpur University 2021  
3 50171 Mr Panneer Selvam A Male National Institute Of Technology Puducherry Ugc-Srf
(Research Scholar)		 Bharathidasan University 2016  
4 50365 Mr. Ashish Ashish Male Iit Roorkee Phd Iit Delhi 2022  
5 50411 Mr. Shiba Biswas Male Department Of Mathematics, Indian Institute Of Technology Kharagpur Phd University Of North Bengal 2019  
6 50435 Mr. Mohmedmunavvar Mubarak Bapu Male Iit Kanpur Phd Scholar Spu 2018  
7 50517 Mr. Vishal Neeraje Male International Center For Theoretical Sciences Phd Indian Statistical Institute, Bangalore 2021  
8 50531 Ms. Sinchita Lahiri Female Tulane University Phd Student Tata Institute Of Fundamental Research, Center For Applicable Mathematics 2019  
9 50629 Mr Rajkumar M Male Vellore Institute Of Technology Ph.D The American College 2022  
10 50685 Mr Sabarish Kumar V Male Bharathiar University M.Phil Research Scholar M.Sc Mathematics With Ca 2021  
11 50815 Ms. Nitu Lakhmara Female Indian Institute Of Technology Kharagpur Phd Indian Institute Of Technology Kharagpur 2019  
12 50848 Mr. Bhanu Prakash Male Sri Sathya Sai Institute Of Higher Learning Phd Student Sri Sathya Sai Institute Of Higher Learning 2020  
13 50870 Mr Sagar Gautam Male Iit Roorkee Phd University Of Lucknow 2019  
14 51006 Mr. Bholanath Kumbhakar Male Indian Institute Of Technology Roorkee Phd The University Of Burdwan 2018  
15 51008 Ms. Nidhi Shukla Female Iit Roorkee Phd Institute Of Science, Banaras Hindu University 2019  
16 51182 Mr. Vishnukumar K S Male National Institute Of Technology Puducherry Phd Thiagarajar College 2021  
17 51315 Mr. Debanjit Mondal Male Iiser Kolkata Integrated Phd Iiser Kolkata 2022  
18 51319 Mr. Arghya Kundu Male Iit Kharagpur Research Scholar

Ballygunge Science College (Calcutta University)

 2018  
19 51325 Mr. Pardeep Kumar Male Indian Institute Of Technology Roorkee, Roorkee, Uttarakhand, India - 247667 Phd Student South Asian University, New Delhi 2015  
20 51363 Ms. Maria Robert Female National Institute Of Technology Calicut Phd Student Cochin University Of Science And Technology 2019  
21 51455 Mr. Brijendra Kumar Chaurasiya Male Vit Ap University Phd Nit Jamshedpur (M.Sc.) 2019  
22 51475 Mr Balakrishna Chhatria Male Iit Kharagpur Ph.D. Student Iiser Kolkata 2019  
23 51504 Ms. Aditi Chattaraj Female Indian Institute Of Technology Hyderabad Phd Bankura University 2019  
24 51508 Mr. Sunil Sunil Male Iit Guwahati Phd Jaipur National University, Jaipur 2017  
25 51529 Mr. Pranav Kumar Male Iiser Bhopal Int-Ph.D. M.Sc 2022 01/08/2022
26 51548 Mr. Rishabh Sharma Male Indian Institute Of Technology Patna Ph.D. Indian Institute Of Technology, Patna 2018  
27 51556 Mr Muthuraj T Male Indira Gandhi Centre For Atomic Research Senior Research Fellow Phd Bharathiar University 2018  
28 51595 Mr. Sudipta Chattopadhyay Male Iit Bombay Phd Iiest Shibpur M.Tech 2018  
29 51600 Mr Anubrato Bhattacharyya Male Presidency University Ph.D. Tifr-Cam 2020  
30 51627 Mr. Aman Batra Male Indian Institute Of Technology Bombay Phd Indian Institute Of Technology Kharagpur 2019  
31 51640 Dr. Sathishkumar G Male Srm Institute Of Science And Technology, Ramapuram Campus Assistant Professor Periyar University Department 2015 20/01/2020
32 51645 Ms. Aroonima Sahoo Female National Institute Of Technology Rourkela Phd Utkal University, Vani Vihar 2018  
33 51648 Mr. Soham Chatterjee Male Indian Institute Of Technology Bombay Phd Indian Institute Of Technology Kanpur 2017  
34 51656 Mr. Siyad Rahman. M Male Iiser Thiruvananthapuram Msc Student Iiser Thiruvananthapuram Appeared  
35 52126 Mr. Vivek Sahu Male Iit Kanpur Phd Iit Guwahati 2020  
36 52133 Ms. Srishti Siddharth Female Indian Insititute Of Technology Bombay Phd      
37 52172 Mr Deepak K.D Male Isi Bangalore Phd Chennai Mathematical Institute 2016  
38 52191 Ms. Anamika Purohit Female Indian Institute Of Technology Gandhinagar Phd The Maharaja Sayajirao University Of Baroda 2019  
39 51432 Ms. Arshi Vaishnavi Female Indian Institute Of Technology (Bhu) Phd (Research Scholar) Banaras Hindu University 2021  
40 51658 Mr. Kushal Roy Male Iit Guwahati Ph.D Bankura University 2021  
Local participants
1 51027 Ms. Kalpana Rawat Female Indian Institute of Space Science and Technology, Thiruvananthapuram PhD Kumaun University Nainital 2020  
2 51195 Mr Ankur Upadhyay Male IISER Thiruvananthapuram PhD MNIT JAIPUR 2022  
3 51380 Ms. Sarbani Pramanik Female IISER TVM PhD Student IISER TVM    
4 51394 Ms Manika Bag Female IISER-TVM PhD IISER-TVM 2021 01/08/2022
5 51399 Mr. Nishant Ranwan Male IISER TVM PhD Indian Institute of Technology, Patna 2021  
6 51401 Mr. Ritabrata Jana Male IISER Thiruvananthapuram PhD Banaras Hindu University 2019  
7 51415 Ms Greeshma K Female IISER Thiruvananthapuram PhD Central University of Kerala 2020  
8 51583 Mrs Dyuti Dwibedi Female IISER TVM INTEGRATED PHD STUDENT IISER TVM Appeared  
9 52261 Ms. Aleena Thomas Female Indian Institute of Space Science and Technology Research Scholar University of Kerala 2020  
10 52174 Mr Dev Prakash Jha Male Indian Institute of Space Science and Technology PhD IIT Jodhpur 2020  
Waiting list
1 51430 Mr. Nitin Kumar Maurya Male Indian Institute of Technology(BHU),Varanasi Research Scholar National Institute of Technology, Warangal 2022  
2 51345 Dr. Malar Kandasamy Female Erode Arts and Science College Associate Professor Gobi Arts and Science College 1996 22/06/2018
3 51591 Mr. Pankaj Roy Male IIT Guwahati PhD IIT Kanpur 2022  
4 51236 Mr. Shishu Pal Singh Male Rajiv Gandhi Institute of petroleum technology PhD Delhi University 2020  
5 51650 Mr. Pratyay Mondal Male Indian Institute of Technology Guwahati PhD M.Sc from Banaras Hindu University 2022  
6 51460 Ms. Kausika Chellamuthu Female Bharathiar University Student Bharathiar University 2012 09/06/2023
7 51321 Ms. Jyoti Jindal Female Indian Institute of Technology, Roorkee PhD Punjabi University, Patiala 2020  
8 51465 Ms Asmita Rai Female Indian Institute of Technology (BHU) Ph.D Udai Pratap Autonomous College 2019  
9 50556 Mr Harshit Bajpai Male IIT Roorkee PhD IIT Roorkee 2022  
10 51328 Mr. Akhilesh Verma Male IIT ROORKEE PhD BANARAS HINDU UNIVERSITY VARANASI U P 2019  
11 50505 Dr. Rajender Kumar Male Deshbandhu College (University of Delhi) Asso. Prof. CCS University 2007 15/01/2020
12 51561 Mr Harikaran B Male Vellore Institute of Technology, Chennai Research Scholar Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore.(Affiliated to Bharathiar University) 2022  
13 52266 Mr. Nikhil Chanauria Male IIT Mandi Ph.D. IIT Delhi 2020  
14 49680 Ms. Ramya K Female National Institute of Technology, Puducherry (NITPY) Ph.D Ethiraj College for Women 2020  
15 50533 Mr. Thangeshwaran M Male PERIYAR UNIVERSITY CENTRE FOR POST GRADUATE AND RESEARCH STUDIES GUEST LECTURER PERIYAR UNIVERSITY 2013  
16 50598 Mr Gaurav Gaurav Male Indian Institute of Technology (B.H.U) PhD Banaras Hindu University 2014  
17 50595 Mr Baskar A Male Vellore Institute of Technology PhD Guru Nanak College 2018  
18 50924 Mr. Abhishek Tiwari Male NIT Jamshedpur PhD University of Allahabad 2021  
19 50961 Mr Ankit Pandey Male South Asian Umniversity Ph.D. Sout Asian University 2022  
20 50537 Mr Sumit Kumar Male NIT Jamshedpur PHD CENTRAL UNIVERSITY OF SOUTH BIHAR 2020  
21 51523 Ms. Gunjan Bhandari Female NIT KURUKSHETRA P.hd Doon University 2021  
22 51033 Ms. Sowmica S Female Bharathiar University PhD Bharathiar University 2020  
23 52228 Ms Shraddha Salwahan Female Indian Institute of Technology Mandi PhD Panjab University 2019  
24 51055 Mr Sombir Dhaniya Male Central University of Punjab PhD Kurukshetra University Kurukshetra 2019  
25 51271 Ms. Angelin Shena Female Vellore Institute of Technology Ph.D The American College, Madurai 2022  
26 50438 Mr Vijayakumar Muthusamy Male Anna University Regional Campus Coimbatore PhD Gobi Arts and Science College / Bharathiar University, M.Sc 2016  

 

 

How to Reach: 

TBA

School Short Name: 

ctpde

Last Date Application: 

Monday, August 15, 2022

School Type: 

NCMW

Separate faculty form: 

0