Skip Navigation


IMA Journal of Mathematical Control and Information Advance Access originally published online on June 8, 2006
IMA Journal of Mathematical Control and Information 2007 24(1):115-136; doi:10.1093/imamci/dnl012
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
24/1/115    most recent
dnl012v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Kumar, A.
Right arrow Articles by Pani, A. K.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

On approximation theorems for controllability of non-linear parabolic problems

Anil Kumar**, Mohan C. Joshi*** and Amiya K. Pani****

Industrial Mathematics Group, Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India

** Email: anil{at}math.iitb.ac.in

*** Email: mcj{at}math.iitb.ac.in

**** Email: akp{at}math.iitb.ac.in


  Abstract

In this paper, we consider the following control system governed by the non-linear parabolic differential equation of the form:

[graphic: see PDF]

where A is a linear operator with dense domain and f(t, y) is a non-linear function. We have proved that under Lipschitz continuity assumption on the non-linear function f(t, y), the set of admissible controls is non-empty. The optimal pair (u*, y*) is then obtained as the limit of the optimal pair sequence {(un*, yn*)}, where un* is a minimizer of the unconstrained problem involving a penalty function arising from the controllability constraint and yn* is the solution of the parabolic non-linear system defined above. Subsequently, we give approximation theorems which guarantee the convergence of the numerical schemes to optimal pair sequence. We also present numerical experiment which shows the applicability of our result.

Keywords: controllability; optimal control; non-linear parabolic system; penalty function; Nemytskii operator; C0-semigroup; Lipschitz continuity; generalized Hammerstein equation..

Received on 28 June 2005. accepted on 20 April 2006.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.