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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 |
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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