March  2017, 6(1): 77-91. doi: 10.3934/eect.2017005

Identification problems of retarded differential systems in Hilbert spaces

Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea

* Corresponding author: Jin-Mun Jeong

Received  October 2015 Revised  August 2016 Published  December 2016

This paper deals with the identification problem for the $L^1$-valued retarded functional differential equation. The unknowns are parameters and operators appearing in the given systems. In order to identify the parameters, we introduce the solution semigroup and the structural operators in the initial data space, and provide the representations of spectral projections and the completeness of generalized eigenspaces. The sufficient condition for the identification problem is given as the so called rank condition in terms of the initial values and eigenvectors of adjoint operator.

Citation: Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005
References:
[1]

S. Agmon, On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure. Appl. Math., 15 (1962), 119-147.  doi: 10.1002/cpa.3160150203.  Google Scholar

[2]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 119-147.  doi: 10.1007/BF02384306.  Google Scholar

[3]

D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach space valued functions, Proc. Conf. Harmonic Analysis, University of Chicago, 1 (1981), 270-286.   Google Scholar

[4]

P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Springer-verlag Belin-Heidelberg-NewYork, 1967. doi: 10.1007/978-3-642-64981-3.  Google Scholar

[5]

G. Di BlasioK. Kunisch and E. Sinestrari, $L^2-$regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl., 102 (1984), 38-57.  doi: 10.1016/0022-247X(84)90200-2.  Google Scholar

[6]

G. Di Blasio and A. Lorenzi, Identification problems for integro-differential delay equations, Differential Integral Equations, 16 (2003), 1385-1408.   Google Scholar

[7]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.  doi: 10.1007/BF01163654.  Google Scholar

[8]

J. M. Jeong, Retarded functional differential equations with $L^1$-valued controller, Funkcial. Ekvac., 36 (1993), 71-93.   Google Scholar

[9]

J. M. Jeong, Supplement to the paper ''Retarded functional differential equations with $L^1$-valued controller", Funkcial. Ekvac., 38 (1995), 267-275.   Google Scholar

[10]

J. M. Jeong, Spectral properties of the operator associated with a retarded functional differential equation in Hilbert space, Proc. Japan Acad., 65A (1989), 98-101.  doi: 10.3792/pjaa.65.98.  Google Scholar

[11]

S. Kitamura and S. Nakagiri, Identifiability of spatially varying and constant parameters in distributes systems of parabolic type, SIAM J. Control & Optim., 15 (1977), 785-802.  doi: 10.1137/0315050.  Google Scholar

[12]

C. Kravaris and J. H. Seinfeld, Identifiability of spatially-varying conductivity from point observation as an inverse Sturm-Liouville problem, SIAM J. Control & Optim., 24 (1986), 522-542.  doi: 10.1137/0324030.  Google Scholar

[13]

S. Lenhart and C. C. Travis, Stability of functional partial differential equations, J. Differential Equation, 58 (1985), 212-227.  doi: 10.1016/0022-0396(85)90013-0.  Google Scholar

[14]

A. Manitius, Completeness and F-completeness of eigenfunctions associated with retarded functional differential equation, J. Differential Equations, 35 (1980), 1-29.  doi: 10.1016/0022-0396(80)90045-5.  Google Scholar

[15]

S. Nakagiri, Identifiability of linear systems in Hilbert spaces, SIAM J. Control & Optim., 21 (1983), 501-530.  doi: 10.1137/0321031.  Google Scholar

[16]

S. Nakagiri and M. Yamamoto, Identifiability of linear retarded systems in Banach space, Funkcial. Ekvac., 31 (1988), 315-329.   Google Scholar

[17]

S. Nakagiri, Controllability and identifiability for linear time-delay systems in Hilbert space. Control theory of distributed parameter systems and applications, Lecture Notes in Control and Inform. Sci. , 159, Springer, Berlin, (1991), 116-125. doi: 10.1007/BFb0004443.  Google Scholar

[18]

S. Nakagiri, Structural properties of functional differential equations in Banach spaces, Osaka J. Math., 25 (1988), 353-398.   Google Scholar

[19]

S. Nakagiri and H. Tanabe, Structural operators and eigenmanifold decomposition for functional differential equations in Hilbert space, J. Math. Anal. Appl., 204 (1996), 554-581.  doi: 10.1006/jmaa.1996.0454.  Google Scholar

[20]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problems, SIAM J. Control & Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.  Google Scholar

[21]

R. Seeley, Norms and domains of the complex power $A_B^Z$, Amer. J. Math., 93 (1971), 299-309.  doi: 10.2307/2373377.  Google Scholar

[22]

R. Seeley, Interpolation in $L^p$ with boundary conditions, Studia Math., 44 (1972), 47-60.   Google Scholar

[23]

T. Suzuki, Uniquenee and nonuniqueness in an inverse problem for theparabolic problem, J. Differential Equations, 47 (1983), 294-316.  doi: 10.1016/0022-0396(83)90038-4.  Google Scholar

[24]

H. Tanabe, Functional Analytic Methods for Partial Differential Equations Marcel Dekker. Inc. /New York, Basel, Hong Kong, 1997.  Google Scholar

[25]

H. Triebel, Interpolation Theory, Functional Spaces, Differential Operators North-Holland, 1978. doi: DIO.  Google Scholar

[26]

M. Yamamoto and S. Nakagiri, Identiability of operators for evolution equations in Banach spaces with an application to transport equations, J. Math. Anal. Appl., 186 (1994), 161-181.  doi: 10.1006/jmaa.1994.1292.  Google Scholar

[27]

K. Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

show all references

References:
[1]

S. Agmon, On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure. Appl. Math., 15 (1962), 119-147.  doi: 10.1002/cpa.3160150203.  Google Scholar

[2]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 119-147.  doi: 10.1007/BF02384306.  Google Scholar

[3]

D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach space valued functions, Proc. Conf. Harmonic Analysis, University of Chicago, 1 (1981), 270-286.   Google Scholar

[4]

P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Springer-verlag Belin-Heidelberg-NewYork, 1967. doi: 10.1007/978-3-642-64981-3.  Google Scholar

[5]

G. Di BlasioK. Kunisch and E. Sinestrari, $L^2-$regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl., 102 (1984), 38-57.  doi: 10.1016/0022-247X(84)90200-2.  Google Scholar

[6]

G. Di Blasio and A. Lorenzi, Identification problems for integro-differential delay equations, Differential Integral Equations, 16 (2003), 1385-1408.   Google Scholar

[7]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.  doi: 10.1007/BF01163654.  Google Scholar

[8]

J. M. Jeong, Retarded functional differential equations with $L^1$-valued controller, Funkcial. Ekvac., 36 (1993), 71-93.   Google Scholar

[9]

J. M. Jeong, Supplement to the paper ''Retarded functional differential equations with $L^1$-valued controller", Funkcial. Ekvac., 38 (1995), 267-275.   Google Scholar

[10]

J. M. Jeong, Spectral properties of the operator associated with a retarded functional differential equation in Hilbert space, Proc. Japan Acad., 65A (1989), 98-101.  doi: 10.3792/pjaa.65.98.  Google Scholar

[11]

S. Kitamura and S. Nakagiri, Identifiability of spatially varying and constant parameters in distributes systems of parabolic type, SIAM J. Control & Optim., 15 (1977), 785-802.  doi: 10.1137/0315050.  Google Scholar

[12]

C. Kravaris and J. H. Seinfeld, Identifiability of spatially-varying conductivity from point observation as an inverse Sturm-Liouville problem, SIAM J. Control & Optim., 24 (1986), 522-542.  doi: 10.1137/0324030.  Google Scholar

[13]

S. Lenhart and C. C. Travis, Stability of functional partial differential equations, J. Differential Equation, 58 (1985), 212-227.  doi: 10.1016/0022-0396(85)90013-0.  Google Scholar

[14]

A. Manitius, Completeness and F-completeness of eigenfunctions associated with retarded functional differential equation, J. Differential Equations, 35 (1980), 1-29.  doi: 10.1016/0022-0396(80)90045-5.  Google Scholar

[15]

S. Nakagiri, Identifiability of linear systems in Hilbert spaces, SIAM J. Control & Optim., 21 (1983), 501-530.  doi: 10.1137/0321031.  Google Scholar

[16]

S. Nakagiri and M. Yamamoto, Identifiability of linear retarded systems in Banach space, Funkcial. Ekvac., 31 (1988), 315-329.   Google Scholar

[17]

S. Nakagiri, Controllability and identifiability for linear time-delay systems in Hilbert space. Control theory of distributed parameter systems and applications, Lecture Notes in Control and Inform. Sci. , 159, Springer, Berlin, (1991), 116-125. doi: 10.1007/BFb0004443.  Google Scholar

[18]

S. Nakagiri, Structural properties of functional differential equations in Banach spaces, Osaka J. Math., 25 (1988), 353-398.   Google Scholar

[19]

S. Nakagiri and H. Tanabe, Structural operators and eigenmanifold decomposition for functional differential equations in Hilbert space, J. Math. Anal. Appl., 204 (1996), 554-581.  doi: 10.1006/jmaa.1996.0454.  Google Scholar

[20]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic problems, SIAM J. Control & Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.  Google Scholar

[21]

R. Seeley, Norms and domains of the complex power $A_B^Z$, Amer. J. Math., 93 (1971), 299-309.  doi: 10.2307/2373377.  Google Scholar

[22]

R. Seeley, Interpolation in $L^p$ with boundary conditions, Studia Math., 44 (1972), 47-60.   Google Scholar

[23]

T. Suzuki, Uniquenee and nonuniqueness in an inverse problem for theparabolic problem, J. Differential Equations, 47 (1983), 294-316.  doi: 10.1016/0022-0396(83)90038-4.  Google Scholar

[24]

H. Tanabe, Functional Analytic Methods for Partial Differential Equations Marcel Dekker. Inc. /New York, Basel, Hong Kong, 1997.  Google Scholar

[25]

H. Triebel, Interpolation Theory, Functional Spaces, Differential Operators North-Holland, 1978. doi: DIO.  Google Scholar

[26]

M. Yamamoto and S. Nakagiri, Identiability of operators for evolution equations in Banach spaces with an application to transport equations, J. Math. Anal. Appl., 186 (1994), 161-181.  doi: 10.1006/jmaa.1994.1292.  Google Scholar

[27]

K. Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[1]

Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125

[2]

A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213

[3]

B. Cantó, C. Coll, E. Sánchez. The problem of global identifiability for systems with tridiagonal matrices. Conference Publications, 2011, 2011 (Special) : 250-257. doi: 10.3934/proc.2011.2011.250

[4]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147

[5]

Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27

[6]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205

[7]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

[8]

Sebastián Buedo-Fernández. Global attraction in a system of delay differential equations via compact and convex sets. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3171-3181. doi: 10.3934/dcdsb.2020056

[9]

Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020104

[10]

Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008

[11]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283

[12]

Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157

[13]

Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51

[14]

Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929

[15]

Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems & Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23

[16]

Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369

[17]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[18]

Yinzheng Sun, Qin Wang, Kyungwoo Song. Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4899-4920. doi: 10.3934/cpaa.2020217

[19]

Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias, Mai Zurwatul Ahlam Mohd Jaffar. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 89-94. doi: 10.3934/naco.2017005

[20]

Zuowei Cai, Jianhua Huang, Lihong Huang. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3591-3614. doi: 10.3934/dcdsb.2017181

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (43)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]