American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions
  • DCDS-B Home
  • This Issue
  • Next Article
    Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions
August  2013, 18(6): 1611-1631. doi: 10.3934/dcdsb.2013.18.1611

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

Ferenc Hartung 1,

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.
Keywords: quasilinearization, Parameter estimation, delay differential equation, state-dependent delay, differentiability with respect to parameters..
Mathematics Subject Classification: Primary: 34K05; Secondary: 93B3.
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
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.  doi: 10.1137/0319051.  Google Scholar

[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.  doi: 10.1137/0321054.  Google Scholar

[3]

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

[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.  doi: 10.1137/0513039.  Google Scholar

[5]

D. W. Brewer, Quasi-Newton methods for parameter estimation in functional differential equations,, in, (1988), 806.   Google Scholar

[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.   Google Scholar

[7]

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

[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.  doi: 10.1016/0096-3003(80)90023-5.  Google Scholar

[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.  doi: 10.1007/s11464-010-0005-9.  Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1016/0893-9659(95)00079-6.  Google Scholar

[13]

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

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[15]

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

[16]

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

[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.  doi: 10.1007/s10884-011-9218-1.  Google Scholar

[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.  doi: 10.1007/s10231-011-0210-5.  Google Scholar

[19]

F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays,, , (2011).   Google Scholar

[20]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems,, in, (1995), 84.   Google Scholar

[21]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems: A numerical study,, in, (1995), 291.   Google Scholar

[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.  doi: 10.1016/S0168-9274(97)00035-4.  Google Scholar

[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.  doi: 10.1016/S0096-3003(97)81654-2.  Google Scholar

[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.  doi: 10.1016/S0362-546X(98)00169-2.  Google Scholar

[25]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006), 435.  doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.3238.  Google Scholar

[27]

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

[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.  doi: 10.1023/A:1014874707485.  Google Scholar

[29]

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

[30]

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

[31]

A. Manitius, On the optimal control of systems with a delay depending on state, control, and time,, in, (1975), 149.   Google Scholar

[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.  doi: 10.1137/0150060.  Google Scholar

[33]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar

[37]

H.-O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay,, (Russian) Sovrem. Mat. Fundam. Napravl., 1 (2003), 40.  doi: 10.1023/B:JOTH.0000047253.23098.12.  Google Scholar

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.  doi: 10.1137/0319051.  Google Scholar

[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.  doi: 10.1137/0321054.  Google Scholar

[3]

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

[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.  doi: 10.1137/0513039.  Google Scholar

[5]

D. W. Brewer, Quasi-Newton methods for parameter estimation in functional differential equations,, in, (1988), 806.   Google Scholar

[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.   Google Scholar

[7]

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

[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.  doi: 10.1016/0096-3003(80)90023-5.  Google Scholar

[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.  doi: 10.1007/s11464-010-0005-9.  Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1016/0893-9659(95)00079-6.  Google Scholar

[13]

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

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[15]

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

[16]

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

[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.  doi: 10.1007/s10884-011-9218-1.  Google Scholar

[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.  doi: 10.1007/s10231-011-0210-5.  Google Scholar

[19]

F. Hartung, On second-order differentiability with respect to parameters for differential equations with state-dependent delays,, , (2011).   Google Scholar

[20]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems,, in, (1995), 84.   Google Scholar

[21]

F. Hartung, T. L. Herdman and J. Turi, Identifications of parameters in hereditary systems: A numerical study,, in, (1995), 291.   Google Scholar

[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.  doi: 10.1016/S0168-9274(97)00035-4.  Google Scholar

[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.  doi: 10.1016/S0096-3003(97)81654-2.  Google Scholar

[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.  doi: 10.1016/S0362-546X(98)00169-2.  Google Scholar

[25]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006), 435.  doi: 10.1016/S1874-5725(06)80009-X.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.3238.  Google Scholar

[27]

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

[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.  doi: 10.1023/A:1014874707485.  Google Scholar

[29]

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

[30]

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

[31]

A. Manitius, On the optimal control of systems with a delay depending on state, control, and time,, in, (1975), 149.   Google Scholar

[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.  doi: 10.1137/0150060.  Google Scholar

[33]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jde.2003.07.001.  Google Scholar

[37]

H.-O. Walther, Smoothness properties of semiflows for differential equations with state dependent delay,, (Russian) Sovrem. Mat. Fundam. Napravl., 1 (2003), 40.  doi: 10.1023/B:JOTH.0000047253.23098.12.  Google Scholar

[1]

A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701
[2]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[3]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993
[4]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[5]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167
[6]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23
[7]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[8]

Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : ⅰ-ⅰ. doi: 10.3934/dcdss.20201i
[9]

Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002
[10]

Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded state-dependent delay equation. Conference Publications, 2001, 2001 (Special) : 56-65. doi: 10.3934/proc.2001.2001.56
[11]

Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365
[12]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[13]

Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607
[14]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038
[15]

F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 85-104. doi: 10.3934/dcdss.2020005
[16]

Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633
[17]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319
[18]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 47-66. doi: 10.3934/dcdss.2020003
[19]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039
[20]

Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences