Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T08:39:55.686Z Has data issue: false hasContentIssue false

Optimal control for quasilinear retarded parabolic systems

Published online by Cambridge University Press:  17 February 2009

Liping Pan
Affiliation:
Laboratory of Mathematics for Nonlinear Sciences and Department of Mathematics, Fudan University, Shanghai 200433, China.
Jiongmin Yong
Affiliation:
Laboratory of Mathematics for Nonlinear Sciences and Department of Mathematics, Fudan University, Shanghai 200433, China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study an optimal control problem for a quasilinear parabolic equation which has delays in the highest order spatial derivative terms. The cost functional is Lagrange type and some terminal state constraints are presented. A Pontryagin-type maximum principle is derived.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Ahmed, N. U. and Teo, K. L., “On the optimal controls of a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions”, Ann. Mat. Pura. Appl. 122 (1979) 6182.Google Scholar
[2]Ardito, A. and Ricciardi, P., “Existence and regularity for linear delay partial differential equations”, Nonlinear Anal. 4 (1980) 411414.CrossRefGoogle Scholar
[3]Banks, H. T., “Necessary conditions for control problems with variable time lags”, SIAM J. Control 6 (1968) 947.CrossRefGoogle Scholar
[4]Banks, H. T., “Variational problems involving functional differential equations”, SIAM J. Control 7 (1969) 117.CrossRefGoogle Scholar
[5]Beckurts, K. H. and Wirtz, K., Neutron physics (Springer, Berlin, 1964).CrossRefGoogle Scholar
[6]Di Blasio, G., “The linear-quadratic optimal control problem for delay equations”, Rend. Accad. Naz. Lincei 71 (1981) 156161.Google Scholar
[7]Di Blasio, G., Kunisch, K. and Sinestrari, E., “L 2-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives”, J. Math. Anal. Appl. 102 (1984) 3857.CrossRefGoogle Scholar
[8]Fattorini, H. O. and Frankowska, H., “Necessary conditions for infinite-dimensional control problems”, Math. Control Signals Sys. 4 (1991) 4167.CrossRefGoogle Scholar
[9]Gurtin, M. E. and Pipkin, A. C., “A general theory of heat conduction with finite wave speeds”, Arch. Rat. Mech. Anal. 31 (1968) 113126.CrossRefGoogle Scholar
[10]Li, X. and Yao, Y., “Maximum principle of distributed parameter systems with time lags”, in Distributed parameter systems, Lecture Notes in Control and Inform. Sci. 75, (Springer, 1985) 410427.CrossRefGoogle Scholar
[11]Li, X. and Yong, J., “Necessary conditions of optimal control for distributed parameter systems”, SIAM J. Control Optim. 29 (1991) 895908.CrossRefGoogle Scholar
[12]Li, X. and Yong, J., Optimal control theory for infinite dimensional systems (Birkhäuser, Boston, 1995).CrossRefGoogle Scholar
[13]Nababan, S. and Noussair, E. S., “On the existence of optimal controls of systems governed by quasilinear parabolic partial delay-differential equations”, Internal. T. Systems Sci. 17 (1986) 12451259.Google Scholar
[14]Nababan, S. and Teo, K. L., “Necessary conditions for optimal controls for systems governed by parabolic partial delay-differential equations in divergence form with first boundary conditions”, J. Optim. Theory Appl. 36 (1982) 565613.CrossRefGoogle Scholar
[15]Nakagiri, S., “Optimal control of linear retarded systems in Banach spaces”, J. Math. Anal. Appl. 120 (1986) 169210.CrossRefGoogle Scholar
[16]Nunziato, J. W., “On heat conduction in materials with memory”, Quart. Appl. Math. 29 (1971) 187204.CrossRefGoogle Scholar
[17]Showalter, R. E. and Walkington, N. J., “A hyperbolic Stefan problem”, Quart. Appl. Math. 45 (1987) 769781.CrossRefGoogle Scholar
[18]Tanabe, H., Functional analytic methods for partial differential equations (Marcel Dekker, New York, 1997).Google Scholar
[19]Wang, P. K. C., “Optimal control of parabolic system with boundary conditions involving time delays”, SIAM J. Control 13 (1975) 274293.CrossRefGoogle Scholar
[20]Yong, J. and Pan, L., “Quasi-linear parabolic partial differential equations with delays in the highestorder spatial derivatives”, J. Austral. Math. Soc., Ser. A 54 (1993) 174203.CrossRefGoogle Scholar
[21]You, Y. and Lee, E. B., “Quadratic optimization for infinite-dimensional linear differential difference type system”, SIAM J. Control Optim. 28 (1990) 265293.Google Scholar