# American Institute of Mathematical Sciences

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 Blasio, K. 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

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##### 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 Blasio, K. 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
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