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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
##### 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
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