For more details: https://www.atmschools.org/school/2023/NCMW/ctpde
| 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
