×

A stability theorem for a class of distributed parameter control systems. (English) Zbl 1152.49033

From the introduction: This paper presents an optimal control problem governed by a hyperbolic equation. The control may be act in the cost functional and in the right side of this equation. The difference approximations problem for the considered problem is obtained. A stability estimate of the difference approximations problem is established.

MSC:

49M25 Discrete approximations in optimal control
49K20 Optimality conditions for problems involving partial differential equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] A.G. Butkovsky, Optimal control of distributed parameter systems , Automat. Remote Control 22 (1962), 1156-1170.
[2] A.G. Butkovsky and A.Ya. Lerner, On a certain control problem of distributed parameter systems , Automat. Remote Control 21 (1960), 472-480.
[3] ——–, Optimal control theory for systems with distributed parameters , Soviet Phys. Dokl. 5 (1961), 936-950.
[4] S.P. Chaudhuri, Distributed optimal control in a nuclear reactor , Internat. J. Control 16 (1972), 927-937.
[5] ——–, The derivation of maximum principle for a distributed parameter system , Proc. Third Southeastern Sympos. on Systems Theory, Vol. 1, Georgia Institute of Technology, 1971.
[6] S.H. Farag and M.H. Farag, Necessary optimality conditions in control problems for hyperbolic equations , J. Egyptian Math. Soc. 8 (2000), 1-10. · Zbl 0973.49018
[7] J.H. Holliday and C. Storey, Numerical solution of certain nonlinear distributed parameter optimal control problems , Internat. J. Control, 18 (1973), 817-825. · Zbl 0276.49022
[8] O.A. Ladyzhenskaya, Boundary value problems of mathematical physics , Nauka, Moscow, Russia, 1973. · Zbl 0285.76010
[9] J.-L Lions, Optimal control by systems described by partial differential equations , Mir, Moscow, SSSR, 1972.
[10] V. Rehbock, S. Wang and K.L. Teo, A computational method for the solution of optimal control problems subject to a linear second order hyperbolic PDE, J. Aust. Math. Soc. 40 (1998), 266-287. · Zbl 0917.65060
[11] A.P. Sage and S.P. Chaudhuri, Gradient and quasilinearization computational techniques for distributed parameter systems , Internat. J. Control 6 (1967), 81-89.
[12] A.A. Samarskii, Introduction to the numerical methods , Moscow, Russia, 1982.
[13] M.M. Sobh, Difference method of solution of optimal control for hyperbolic equation , in VI Republican Confer. High Schools, Baku, (1983), 183-185.
[14] R.K. Tagiev, Difference method of problems with controls in coefficients of hyperbolic equation , Cpornek Numerical Methods 12 (1982), 109-119.
[15] A.N. Tikhonov and N.Ya. Arsenin, Methods for the solution of incorrectly posed problems , Nauka, Moscow, Russia, 1974.
[16] P.K.C. Wang, Advances in control systems , Vol. 1 (C.T. Leondes, ed.), Academic Press, New York, 1964.
[17] Y. Yavin, Optimal control of the vibrations of a nonlinear string , Internat. J. Control 16 (1972), 559-562. · Zbl 0238.49032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.