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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 and Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611
References:
[1]

H. T. Banks, J. A. Burns and E. M. Cliff, Parameter estimation and identification for systems with delays, SIAM J. Control and Opt., 19 (1981), 791-828. doi: 10.1137/0319051.

[2]

H. T. Banks and P. K. Daniel Lamm, Estimation of delays and other parameters in nonlinear functional differential equations, SIAM J. Control and Opt., 21 (1983), 895-915. doi: 10.1137/0321054.

[3]

H. T. Banks and G. M. Groome, Jr., Convergence theorems for parameter estimation by quasilinearization, J. Math. Anal. Appl., 42 (1973), 91-109.

[4]

D. W. Brewer, The differentiability with respect to a parameter of the solution of a linear abstract Cauchy problem, SIAM. J. Math. Anal. Appl., 13 (1982), 607-620. doi: 10.1137/0513039.

[5]

D. W. Brewer, Quasi-Newton methods for parameter estimation in functional differential equations, in "Proc. 27th IEEE Conf. on Decision and Control," Austin, TX, (1988), 806-809.

[6]

D. W. Brewer, J. A. Burns and E. M. Cliff, Parameter identification for an abstract Cauchy problem by quasilinearization, Quart. Appl. Math., 51 (1993), 1-22.

[7]

M. Brokate and F. Colonius, Linearizing equations with state-dependent delays, Appl. Math. Optim., 21 (1990), 45-52. doi: 10.1007/BF01445156.

[8]

J. A. Burns and P. D. Hirsch, A difference equation approach to parameter estimation for differential-delay equations, Appl. Math. Comp., 7 (1980), 281-311. doi: 10.1016/0096-3003(80)90023-5.

[9]

Y. Chen, Q. Hu and J. Wu, Second-order differentiability with respect to parameters for differential equations with adaptive delays, Front. Math. China, 5 (2010), 221-286. doi: 10.1007/s11464-010-0005-9.

[10]

R. D. Driver, Existence theory for a delay-differential system, Contrib. Differential Equations, 1 (1961), 317-336.

[11]

I. Gyõri and F. Hartung, On the exponential stability of a state-dependent delay equation, Acta Sci. Math. (Szeged), 66 (2000), 71-84.

[12]

I. Gyõri, F. Hartung and J. Turi, On numerical approximations for a class of differential equations with time- and state-dependent delays, Appl. Math. Letters, 8 (1995), 19-24. doi: 10.1016/0893-9659(95)00079-6.

[13]

J. K. Hale and L. A. C. Ladeira, Differentiability with respect to delays, J. Diff. Eqns., 92 (1991), 14-26. doi: 10.1016/0022-0396(91)90061-D.

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[15]

F. Hartung, On differentiability of solutions with respect to parameters in a class of functional differential equations, Func. Diff. Eqns., 4 (1997), 65-79.

[16]

F. Hartung, Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: a numerical study, Nonlinear Anal., 47 (2001), 4557-4566. doi: 10.1016/S0362-546X(01)00569-7.

[17]

F. Hartung, Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, J. Dynam. Differential Equations, 23 (2011), 843-884. doi: 10.1007/s10884-011-9218-1.

[18]

F. Hartung, On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays, Ann. Mat. Pura Appl., 192 (2011), 17-47. doi: 10.1007/s10231-011-0210-5.

[19]

F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays, arXiv:1201.0269v1, 2011.

[20]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems, 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.

[21]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems: A numerical study, in "Proceedings of the 3rd IEEE Mediterranean Symposium on New Directions in Control and Automation," Cyprus, (1995), 291-298.

[22]

F. Hartung, T. L. Herdman and J. Turi, On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays, Appl. Numer. Math., 24 (1997), 393-409. doi: 10.1016/S0168-9274(97)00035-4.

[23]

F. Hartung, T. L. Herdman and J. Turi, Parameter identification in classes of hereditary systems of neutral type, Appl. Math. and Comp., 89 (1998), 147-160. doi: 10.1016/S0096-3003(97)81654-2.

[24]

F. Hartung, T. L. Herdman and J. Turi, Parameter identification in neutral functional differential equations with state-dependent delays, Nonlin. Anal., 39 (2000), 305-325. doi: 10.1016/S0362-546X(98)00169-2.

[25]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, 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.

[26]

F. Hartung and J. Turi, On differentiability of solutions with respect to parameters in state-dependent delay equations, J. Diff. Eqns., 135 (1997), 192-237. doi: 10.1006/jdeq.1996.3238.

[27]

F. Hartung and J. Turi, Identification of parameters in delay equations with state-dependent delays, J. Nonlinear Analysis: Theory, Methods and Applications, 29 (1997), 1303-1318. doi: 10.1016/S0362-546X(96)00100-9.

[28]

T. L. Herdman, P. Morin and R. D. Spies, Parameter identification for nonlinear abstract Cauchy problems using quasilinearization, J. Optim. Th. Appl., 113 (2002), 227-250. doi: 10.1023/A:1014874707485.

[29]

V.-M. Hokkanen and G. Moroşanu, Differentiability with respect to delay, Differential and Integral Equations, 11 (1998), 589-603.

[30]

S. M. Verduyn Lunel, Parameter identifiability of differential delay equations, Int. J. Adaptive Control Signal Processing, 15 (2001), 655-678.

[31]

A. Manitius, On the optimal control of systems with a delay depending on state, control, and time, in "Séminaires IRIA, Analyse et Contrôle de Systèmes," IRIA, France, (1975), 149-198.

[32]

K. A. Murphy, Estimation of time- and state-dependent delays and other parameters in functional-differential equations, SIAM J. Appl. Math., 50 (1990), 972-1000. doi: 10.1137/0150060.

[33]

S. Nakagiri and M. Yamamoto, Identifiability of linear retarded systems in Banach spaces, Funkcialaj Ekvacioj, 31 (1988), 315-329.

[34]

B. Slezák, On the parameter-dependence of the solutions of functional differential equations with unbounded state-dependent delay I. The upper-semicontinuity of the resolvent function, Int. J. Qual. Theory Differential Equations Appl., 1 (2007), 88-114.

[35]

B. Slezák, On the smooth parameter-dependence of the solutions of abstract functional differential equations with state-dependent delay, Functional Differential Equations, 17 (2010), 253-293.

[36]

H.-O. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with state dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[37]

H.-O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay, (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.

show all references

References:
[1]

H. T. Banks, J. A. Burns and E. M. Cliff, Parameter estimation and identification for systems with delays, SIAM J. Control and Opt., 19 (1981), 791-828. doi: 10.1137/0319051.

[2]

H. T. Banks and P. K. Daniel Lamm, Estimation of delays and other parameters in nonlinear functional differential equations, SIAM J. Control and Opt., 21 (1983), 895-915. doi: 10.1137/0321054.

[3]

H. T. Banks and G. M. Groome, Jr., Convergence theorems for parameter estimation by quasilinearization, J. Math. Anal. Appl., 42 (1973), 91-109.

[4]

D. W. Brewer, The differentiability with respect to a parameter of the solution of a linear abstract Cauchy problem, SIAM. J. Math. Anal. Appl., 13 (1982), 607-620. doi: 10.1137/0513039.

[5]

D. W. Brewer, Quasi-Newton methods for parameter estimation in functional differential equations, in "Proc. 27th IEEE Conf. on Decision and Control," Austin, TX, (1988), 806-809.

[6]

D. W. Brewer, J. A. Burns and E. M. Cliff, Parameter identification for an abstract Cauchy problem by quasilinearization, Quart. Appl. Math., 51 (1993), 1-22.

[7]

M. Brokate and F. Colonius, Linearizing equations with state-dependent delays, Appl. Math. Optim., 21 (1990), 45-52. doi: 10.1007/BF01445156.

[8]

J. A. Burns and P. D. Hirsch, A difference equation approach to parameter estimation for differential-delay equations, Appl. Math. Comp., 7 (1980), 281-311. doi: 10.1016/0096-3003(80)90023-5.

[9]

Y. Chen, Q. Hu and J. Wu, Second-order differentiability with respect to parameters for differential equations with adaptive delays, Front. Math. China, 5 (2010), 221-286. doi: 10.1007/s11464-010-0005-9.

[10]

R. D. Driver, Existence theory for a delay-differential system, Contrib. Differential Equations, 1 (1961), 317-336.

[11]

I. Gyõri and F. Hartung, On the exponential stability of a state-dependent delay equation, Acta Sci. Math. (Szeged), 66 (2000), 71-84.

[12]

I. Gyõri, F. Hartung and J. Turi, On numerical approximations for a class of differential equations with time- and state-dependent delays, Appl. Math. Letters, 8 (1995), 19-24. doi: 10.1016/0893-9659(95)00079-6.

[13]

J. K. Hale and L. A. C. Ladeira, Differentiability with respect to delays, J. Diff. Eqns., 92 (1991), 14-26. doi: 10.1016/0022-0396(91)90061-D.

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[15]

F. Hartung, On differentiability of solutions with respect to parameters in a class of functional differential equations, Func. Diff. Eqns., 4 (1997), 65-79.

[16]

F. Hartung, Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: a numerical study, Nonlinear Anal., 47 (2001), 4557-4566. doi: 10.1016/S0362-546X(01)00569-7.

[17]

F. Hartung, Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays, J. Dynam. Differential Equations, 23 (2011), 843-884. doi: 10.1007/s10884-011-9218-1.

[18]

F. Hartung, On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays, Ann. Mat. Pura Appl., 192 (2011), 17-47. doi: 10.1007/s10231-011-0210-5.

[19]

F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays, arXiv:1201.0269v1, 2011.

[20]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems, 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.

[21]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems: A numerical study, in "Proceedings of the 3rd IEEE Mediterranean Symposium on New Directions in Control and Automation," Cyprus, (1995), 291-298.

[22]

F. Hartung, T. L. Herdman and J. Turi, On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays, Appl. Numer. Math., 24 (1997), 393-409. doi: 10.1016/S0168-9274(97)00035-4.

[23]

F. Hartung, T. L. Herdman and J. Turi, Parameter identification in classes of hereditary systems of neutral type, Appl. Math. and Comp., 89 (1998), 147-160. doi: 10.1016/S0096-3003(97)81654-2.

[24]

F. Hartung, T. L. Herdman and J. Turi, Parameter identification in neutral functional differential equations with state-dependent delays, Nonlin. Anal., 39 (2000), 305-325. doi: 10.1016/S0362-546X(98)00169-2.

[25]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, 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.

[26]

F. Hartung and J. Turi, On differentiability of solutions with respect to parameters in state-dependent delay equations, J. Diff. Eqns., 135 (1997), 192-237. doi: 10.1006/jdeq.1996.3238.

[27]

F. Hartung and J. Turi, Identification of parameters in delay equations with state-dependent delays, J. Nonlinear Analysis: Theory, Methods and Applications, 29 (1997), 1303-1318. doi: 10.1016/S0362-546X(96)00100-9.

[28]

T. L. Herdman, P. Morin and R. D. Spies, Parameter identification for nonlinear abstract Cauchy problems using quasilinearization, J. Optim. Th. Appl., 113 (2002), 227-250. doi: 10.1023/A:1014874707485.

[29]

V.-M. Hokkanen and G. Moroşanu, Differentiability with respect to delay, Differential and Integral Equations, 11 (1998), 589-603.

[30]

S. M. Verduyn Lunel, Parameter identifiability of differential delay equations, Int. J. Adaptive Control Signal Processing, 15 (2001), 655-678.

[31]

A. Manitius, On the optimal control of systems with a delay depending on state, control, and time, in "Séminaires IRIA, Analyse et Contrôle de Systèmes," IRIA, France, (1975), 149-198.

[32]

K. A. Murphy, Estimation of time- and state-dependent delays and other parameters in functional-differential equations, SIAM J. Appl. Math., 50 (1990), 972-1000. doi: 10.1137/0150060.

[33]

S. Nakagiri and M. Yamamoto, Identifiability of linear retarded systems in Banach spaces, Funkcialaj Ekvacioj, 31 (1988), 315-329.

[34]

B. Slezák, On the parameter-dependence of the solutions of functional differential equations with unbounded state-dependent delay I. The upper-semicontinuity of the resolvent function, Int. J. Qual. Theory Differential Equations Appl., 1 (2007), 88-114.

[35]

B. Slezák, On the smooth parameter-dependence of the solutions of abstract functional differential equations with state-dependent delay, Functional Differential Equations, 17 (2010), 253-293.

[36]

H.-O. Walther, The solution manifold and $C^1$-smoothness of solution operators for differential equations with state dependent delay, J. Differential Equations, 195 (2003), 46-65. doi: 10.1016/j.jde.2003.07.001.

[37]

H.-O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay, (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.

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