The Library

Optimal control of the propagation of a graph in inhomogeneous media

Tools

Deckelnick, Klaus, Elliott, Charles M. and Styles, Vanessa. (2009) Optimal control of the propagation of a graph in inhomogeneous media. SIAM Journal on Control and Optimization, Vol.48 (No.3). pp. 1335-1352. ISSN 0363-0129

Preview

PDF
WRAP_Elliott_Optimal_Control.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (297Kb)

Official URL: http://dx.doi.org/10.1137/080723648

Abstract

We study an optimal control problem for viscosity solutions of a Hamilton–Jacobi equation describing the propagation of a one-dimensional graph with the control being the speed function. The existence of an optimal control is proved together with an approximate controllability result in the $H^{-1}$-norm. We prove convergence of a discrete optimal control problem based on a monotone finite difference scheme and describe some numerical results.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Differential equations, Partial -- Research, Eikonal equation, Hamilton-Jacobi equations, Approximation theory
Journal or Publication Title:	SIAM Journal on Control and Optimization
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0363-0129
Official Date:	15 April 2009
Volume:	Vol.48
Number:	No.3
Page Range:	pp. 1335-1352
Identification Number:	10.1137/080723648
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	# G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), pp. 271–283. # J. M. Berg, A. Yezzi, and A. R. Tannenbaum, Phase transitions, curve evolution and the control of semiconductor manufacturing processes, in Proceedings of the IEEE Conference on Decision and Control, Kobe, Japan, 1996, pp. 3376–3381. # J. M. Berg and N. Zhou, Shape-based optimal estimation and design of curve evolution processes with application to plasma etching, IEEE Trans. Automat. Control, 46 (2001), pp. 1862–1873. # C. Castro, F. Palacios, and E. Zuazua, An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks, Math. Models Methods Appl. Sci., 18 (2008), pp. 369–416. [MathRev] # M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1–42. [ISI] # M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), pp. 487–502. [ISI] # K. Deckelnick and C. M. Elliott, Uniqueness and error analysis for Hamilton–Jacobi equations with discontinuities, Interfaces Free Bound., 6 (2004), pp. 329–349. [MathRev] # K. Deckelnick and C. M. Elliott, Propagation of graphs in two-dimensional inhomogeneous media, Appl. Numer. Math., 56 (2006), pp. 1163–1178. [Inspec] [MathRev] # H. Ishii, Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo. Univ., 28 (1985), pp. 33–77. [MathRev] # O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 24, AMS, Providence, RI, 1968. # S. Leung and J. Qian, An adjoint state method for three-dimensional transmission traveltime tomography using first-arrivals, Commun. Math. Sci., 4 (2006), pp. 249–266. [MathRev] # J. A. Sethian, Level Set Methods, Cambridge University Press, Cambridge, UK, 1996. # S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41 (2002), pp. 740–797. [MathRev]
URI:	http://wrap.warwick.ac.uk/id/eprint/2210