Parameter identifiability under approximation

Kunisch, K.; White, L. W.

doi:10.1090/qam/860900

Authors: K. Kunisch and L. W. White
Journal: Quart. Appl. Math. 44 (1986), 475-486
MSC: Primary 49D15; Secondary 93B30, 93C05
DOI: https://doi.org/10.1090/qam/860900
MathSciNet review: 860900
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Abstract: The problem of injectivity of the parameter-to-state map is discussed for Galerkin approximations of a linear parabolic equation. A necessary and sufficient condition is derived and illustrated by means of simple examples. Finally, output least squares identifiability of the Galerkin approximations is discussed.

A. Bamberger, G. Chavent, and P. Lailly, About the stability of the inverse problem in $1$-D wave equations—applications to the interpretation of seismic profiles, Appl. Math. Optim. 5 (1979), no. 1, 1–47. MR 526426, DOI https://doi.org/10.1007/BF01442542

H. T. Banks and P. L. Daniel, Estimation of variable coefficients in parabolic systems, LCDS Report 82-22, Brown University, Sept. 1982; IEEE Trans. Autom. Control. 30, 386–398 (1985)

H. T. Banks and K. Kunisch, An approximation theory for nonlinear partial differential equations with applications to identification and control, SIAM J. Control Optim. 20 (1982), no. 6, 815–849. MR 675572, DOI https://doi.org/10.1137/0320059
G. Chavent, Local stability of the output least square parameter estimation technique, Mat. Apl. Comput. 2 (1983), no. 1, 3–22 (English, with Portuguese summary). MR 699367
I. M. Gel′fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253–304. MR 0073805
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
S. Kitamura and S. Nakagiri, Identifiability of spatially-varying and constant parameters in distributed systems of parabolic type, SIAM J. Control Optim. 15 (1977), no. 5, 785–802. MR 459856, DOI https://doi.org/10.1137/0315050
K. Kunisch and L. White, The parameter estimation problem for parabolic equations and discontinuous observation operators, SIAM J. Control Optim. 23 (1985), no. 6, 900–927. MR 809541, DOI https://doi.org/10.1137/0323052
Alan Pierce, Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim. 17 (1979), no. 4, 494–499. MR 534419, DOI https://doi.org/10.1137/0317035
Takashi Suzuki, Uniqueness and nonuniqueness in an inverse problem for the parabolic equation, J. Differential Equations 47 (1983), no. 2, 296–316. MR 688107, DOI https://doi.org/10.1016/0022-0396%2883%2990038-4

L. W. White, Identification of a friction parameter in a first order linear hyperbolic equation, Proc. 22nd IEEE Conf. Decision and Control, San Antonio, 56–59 (1983)

Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 564653