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August  2013, 18(6): 1611-1631. doi: 10.3934/dcdsb.2013.18.1611

Parameter estimation by quasilinearization in differential equations with state-dependent delays

1. 

Department of Mathematics and Computing, University of Pannonia, H-8201 Veszprém, P.O.Box 158

Received  January 2012 Revised  February 2012 Published  March 2013

In this paper we study a parameter estimation method in functional differential equations with state-dependent delays using a quasilinearization technique. We define the method, prove its convergence under certain conditions, and test its applicability in numerical examples. We estimate infinite dimensional parameters such as coefficient functions, delay functions and initial functions in state-dependent delay equations. The method uses the derivative of the solution with respect to the parameters. The proof of the convergence is based on the Lipschitz continuity of the derivative with respect to the parameters.
Citation: Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611
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Funkcialaj Ekvacioj, 31 (1988), 315-329.  Google Scholar

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show all references

References:
[1]

SIAM J. Control and Opt., 19 (1981), 791-828. doi: 10.1137/0319051.  Google Scholar

[2]

SIAM J. Control and Opt., 21 (1983), 895-915. doi: 10.1137/0321054.  Google Scholar

[3]

J. Math. Anal. Appl., 42 (1973), 91-109.  Google Scholar

[4]

SIAM. J. Math. Anal. Appl., 13 (1982), 607-620. doi: 10.1137/0513039.  Google Scholar

[5]

in "Proc. 27th IEEE Conf. on Decision and Control," Austin, TX, (1988), 806-809. Google Scholar

[6]

Quart. Appl. Math., 51 (1993), 1-22.  Google Scholar

[7]

Appl. Math. Optim., 21 (1990), 45-52. doi: 10.1007/BF01445156.  Google Scholar

[8]

Appl. Math. Comp., 7 (1980), 281-311. doi: 10.1016/0096-3003(80)90023-5.  Google Scholar

[9]

Front. Math. China, 5 (2010), 221-286. doi: 10.1007/s11464-010-0005-9.  Google Scholar

[10]

Contrib. Differential Equations, 1 (1961), 317-336.  Google Scholar

[11]

Acta Sci. Math. (Szeged), 66 (2000), 71-84.  Google Scholar

[12]

Appl. Math. Letters, 8 (1995), 19-24. doi: 10.1016/0893-9659(95)00079-6.  Google Scholar

[13]

J. Diff. Eqns., 92 (1991), 14-26. doi: 10.1016/0022-0396(91)90061-D.  Google Scholar

[14]

Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[15]

Func. Diff. Eqns., 4 (1997), 65-79.  Google Scholar

[16]

Nonlinear Anal., 47 (2001), 4557-4566. doi: 10.1016/S0362-546X(01)00569-7.  Google Scholar

[17]

J. Dynam. Differential Equations, 23 (2011), 843-884. doi: 10.1007/s10884-011-9218-1.  Google Scholar

[18]

Ann. Mat. Pura Appl., 192 (2011), 17-47. doi: 10.1007/s10231-011-0210-5.  Google Scholar

[19]

arXiv:1201.0269v1, 2011. Google Scholar

[20]

in "Proceedings of ASME Fifteenth Biennial Conference on Mechanical Vibration and Noise," DE-Vol. 84-3, Vol. 3, Part C, Boston, Massachusetts, (1995), 1061-1066. Google Scholar

[21]

in "Proceedings of the 3rd IEEE Mediterranean Symposium on New Directions in Control and Automation," Cyprus, (1995), 291-298. Google Scholar

[22]

Appl. Numer. Math., 24 (1997), 393-409. doi: 10.1016/S0168-9274(97)00035-4.  Google Scholar

[23]

Appl. Math. and Comp., 89 (1998), 147-160. doi: 10.1016/S0096-3003(97)81654-2.  Google Scholar

[24]

Nonlin. Anal., 39 (2000), 305-325. doi: 10.1016/S0362-546X(98)00169-2.  Google Scholar

[25]

in "Handbook of Differential Equations: Ordinary Differential Equations. Vol. III" (eds. A. Canada, P. Drek and A. Fonda), Elsevier, North-Holland, (2006), 435-545. doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

[26]

J. Diff. Eqns., 135 (1997), 192-237. doi: 10.1006/jdeq.1996.3238.  Google Scholar

[27]

J. Nonlinear Analysis: Theory, Methods and Applications, 29 (1997), 1303-1318. doi: 10.1016/S0362-546X(96)00100-9.  Google Scholar

[28]

J. Optim. Th. Appl., 113 (2002), 227-250. doi: 10.1023/A:1014874707485.  Google Scholar

[29]

Differential and Integral Equations, 11 (1998), 589-603.  Google Scholar

[30]

Int. J. Adaptive Control Signal Processing, 15 (2001), 655-678. Google Scholar

[31]

in "Séminaires IRIA, Analyse et Contrôle de Systèmes," IRIA, France, (1975), 149-198. Google Scholar

[32]

SIAM J. Appl. Math., 50 (1990), 972-1000. doi: 10.1137/0150060.  Google Scholar

[33]

Funkcialaj Ekvacioj, 31 (1988), 315-329.  Google Scholar

[34]

Int. J. Qual. Theory Differential Equations Appl., 1 (2007), 88-114. Google Scholar

[35]

Functional Differential Equations, 17 (2010), 253-293.  Google Scholar

[36]

J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.  Google Scholar

[37]

(Russian) Sovrem. Mat. Fundam. Napravl., 1 (2003), 40-55; translation in J. Math. Sci. (N. Y.), 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.  Google Scholar

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