Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T08:28:40.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Necessary conditions for constrained distributed parameter systems with deviating argument

Published online by Cambridge University Press:  17 February 2009

Mohammad A. Kazemi
Mohammad A. Kazemi
Affiliation:
Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina28223.
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.

In this paper we consider an optimal control problem governed by a system of nonlinear hyperbolic partial differential equations with deviating argument, Darboux-type boundary conditions and terminal state inequality constraints. The control variables are assumed to be measurable and the state variables are assumed to belong to a Sobolev space. We derive an integral representation of the increments of the functionals involved, and using separation theorems of functional analysis, obtain necessary conditions for optimality in the form of a Pontryagin maximum principle. The approach presented here applies equally well to other nonlinear constrained distributed parameters with deviating argument.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 4 , April 1996 , pp. 495 - 511
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Ahmed, N. U. and Teo, K. L., Optimal control of distributed parameter systems (Elsevier North Holland, New York, 1981).Google Scholar
[2]Casas, E., “Boundary control of semilinear elliptic equations with pointwise state constraints”, SIAM J. Control Optim. 31 (1993) 993–1006.CrossRefGoogle Scholar
[3]Giorecki, M. et al. , Analysis and synthesis of time delay systems (John Wiley & Sons, New York, 1991).Google Scholar
[4]Kazemi, M. A., “Optimal control of systems governed by partial differential equations with integral inequality constraints”, Nonlinear Analysis, Theory, Methods, Appl. 8 (1984) 1409–1425.CrossRefGoogle Scholar
[5]Kazemi, M. A., “Optimal control of distributed hyperbolic systems with time delay”, Far East J. Math. Sci. 1 (1993) 53–68.Google Scholar
[6]Kowalewski, A., “Optimal control of distributed parabolic systems involving time lags”, IMA J. Math Control. Infor. 7 (1991) 375–393.CrossRefGoogle Scholar
[7]Lions, J. L., Optimal control of systems governed by partial differential equations (Springer-Verlag, Berlin, 1971).CrossRefGoogle Scholar
[8]Plotnikov, V. I. and Sumin, V. I., “Optimization of objects with distributed parameters described by Goursah-Darboux systems”, Comput. Maths. Math. Phys. 12 (1972) 67–77.Google Scholar
[9]Sadek, L. S., “Optimal control of time-delay systems with distributed parameters”, J. Optim. Theory Appl. 67 (1990) 567–585.CrossRefGoogle Scholar
[10]Salukvadze, M. E. and Teuteunava, M. T., “An optimal control problem for systems described by partial differential equations of hyperbolic type with delay”, J. Optim. Theory Appl. 52 (1987) 311–322.CrossRefGoogle Scholar
[11]Suryanarayana, M. B., “Necessary conditions for optimization problems with hyperbolic partial differential equations”, SIAMJ. Control 2 (1973) 130–147.CrossRefGoogle Scholar
[12]Teo, K. L., “Optimal control of systems governed by time delayed second order linear parabolic partial differential equations with a first boundary condition”, J. Optim. Theory Appl. 29 (1979) 437–481.CrossRefGoogle Scholar
[13]Vasil'ev, O. V. and Srochko, V. A., “Optimization of a class of controlled processes with distributed parameters”, Sibirian Math. J. 19 (1978) 328–331.CrossRefGoogle Scholar
[14]Wang, P. K. C., “Optimal control of parabolic systems with boundary conditions having time delays”, SIAMJ. Control optim. 13 (1975) 274–293.CrossRefGoogle Scholar
[15]Wheeden, R. L. and Zygmund, A., Measure and integral (Marcel Dekker, New York, 1977).CrossRefGoogle Scholar
[16]Wong, K. H., “Optimal control computation for parabolic systems with boundary conditions in-volving time delays”, J. Optim. Theory Appl. 53 (1987) 475–507.CrossRefGoogle Scholar
[17]Wu, Z. S. and Teo, K. L., Computational methods for optimizing distributed systems (Academic Press, 1984).Google Scholar
[18]Wu, Z. S. and Teo, K. L., “A conditional gradient method for an optimal control problem involving a class of nonlinear second order hyperbolic partial differential equations”, J. Math. Anal. Appl. 91 (1983) 376–393.CrossRefGoogle Scholar
[19]Wu, Z. S. and Teo, K. L., “A convex optimal control problem involving a class of linear hyperbolic systems”, J. Optim. Theory App. 39 (1983) 341–560.Google Scholar