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Volume 7, Issue 4
On the Optimal Control Problem of Laser Surface Hardening

N. Gupta, N. Nataraj & A. K. Pani

Int. J. Numer. Anal. Mod., 7 (2010), pp. 667-680.

Published online: 2010-07

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  • Abstract

We discuss an optimal control problem of laser surface hardening of steel which is governed by a dynamical system consisting of a semilinear parabolic equation and an ordinary differential equation with a non differentiable right hand side function $f_+$. To avoid the numerical and analytic difficulties posed by $f_+$, it is regularized using a monotone Heaviside function and the regularized problem has been studied in literature. In this article, we establish the convergence of solution of the regularized problem to that of the original problem. The estimates, in terms of the regularized parameter, justify the existence of solution of the original problem. Finally, a numerical experiment is presented to illustrate the effect of regularization parameter on the state and control errors.

  • Keywords

Laser surface hardening of steel, semilinear parabolic equation, ODE with non-differentiable forcing function, regularized Heaviside function, regularised problem, convergence with respect to regularization parameter, numerical experiments.

  • AMS Subject Headings

65N35, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-7-667, author = {Gupta , N.Nataraj , N. and Pani , A. K.}, title = {On the Optimal Control Problem of Laser Surface Hardening}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2010}, volume = {7}, number = {4}, pages = {667--680}, abstract = {

We discuss an optimal control problem of laser surface hardening of steel which is governed by a dynamical system consisting of a semilinear parabolic equation and an ordinary differential equation with a non differentiable right hand side function $f_+$. To avoid the numerical and analytic difficulties posed by $f_+$, it is regularized using a monotone Heaviside function and the regularized problem has been studied in literature. In this article, we establish the convergence of solution of the regularized problem to that of the original problem. The estimates, in terms of the regularized parameter, justify the existence of solution of the original problem. Finally, a numerical experiment is presented to illustrate the effect of regularization parameter on the state and control errors.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/745.html} }
TY - JOUR T1 - On the Optimal Control Problem of Laser Surface Hardening AU - Gupta , N. AU - Nataraj , N. AU - Pani , A. K. JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 667 EP - 680 PY - 2010 DA - 2010/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/745.html KW - Laser surface hardening of steel, semilinear parabolic equation, ODE with non-differentiable forcing function, regularized Heaviside function, regularised problem, convergence with respect to regularization parameter, numerical experiments. AB -

We discuss an optimal control problem of laser surface hardening of steel which is governed by a dynamical system consisting of a semilinear parabolic equation and an ordinary differential equation with a non differentiable right hand side function $f_+$. To avoid the numerical and analytic difficulties posed by $f_+$, it is regularized using a monotone Heaviside function and the regularized problem has been studied in literature. In this article, we establish the convergence of solution of the regularized problem to that of the original problem. The estimates, in terms of the regularized parameter, justify the existence of solution of the original problem. Finally, a numerical experiment is presented to illustrate the effect of regularization parameter on the state and control errors.

Gupta , N.Nataraj , N. and Pani , A. K.. (2010). On the Optimal Control Problem of Laser Surface Hardening. International Journal of Numerical Analysis and Modeling. 7 (4). 667-680. doi:
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