Abstract
In this paper we consider space semi-discretization of some integro-differential equations using the harmonic analysis method. We study the problem of boundary observability, i. e., the problem of whether the initial data of solutions can be estimated uniformly in terms of the boundary observation as the net-spacing \(h\rightarrow 0\). When \(h\rightarrow 0\) these finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We shall consider the piecewise Hermite cubic orthogonal spline collocation semi-discretization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Loreti, P., Sforza, D.: Reachability problems for a class of integro-differential equations. J. Differ. Equ. 248, 1711–1755 (2010)
Pandolfi, L.: Boundary controllability and source reconstruction in a viscoelastic string under external traction. J. Math. Anal. Appl. 407, 464–479 (2013)
Pandolfi, L.: Traction, deformation and velocity of deformation in a viscoelastic string. Evol. Equ. Control Theory 2, 471–493 (2013)
Avdonin, S.A., Belinskiy, B.P.: On controllability of a non-homogeneous elastic string with memory. J. Math. Anal. Appl. 398, 254–269 (2013)
Bialecki, B., Fairweather, G., Bennett, K.R.: Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations. SIAM J. Numer. Anal. 29(1), 156–173 (1992)
Dyksen, W.R.: Tensor product generalized ADI methods for separable elliptic problems. SIAM J. Numer. Anal. 24(1), 59–76 (1987)
Infante, J.A., Zuazua, E.: Boundary observability for the space semi-discretizations of the 1-D wave equation. Math. Model. Numer. Anal. (M2AN) 33(2), 407–438 (1999)
Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005)
Boulmezaoud, T.Z., Urquiza, J.M.: On the eigenvalues of the spectral second order differentiation operator and application to the boundary observability of the wave equation. J. Sci. Comput. 31(3), 307–345 (2007)
Ingham, A.E.: Some trigonometrical inequalities with applications to the theory of series. Math. Zeitschrift 41, 367–379 (1936)
Negreanu, M., Zuazua, E.: Uniform observability of the wave equation via a discrete Ingham inequality. In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, pp. 3158–3163
Negreanu, M., Zuazua, E.: Discrete Ingham inequalities and applications. SIAM J. Numer. Anal. 44(1), 412–448 (2006)
Yan, Yi, Fairweather, G.: Orthogonal spline collocation methods for some partial integro-differential equations. SIAM J. Numer. Anal. 20(3), 755–768 (1992)
Fairweather, G.: Spline collocation methods for a class of hyperbolic partial integro-differential equations. SIAM J. Numer. Anal. 31(2), 444–460 (1994)
Pani, A.K., Fairweather, G., Fernandes, R.I.: Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term. SIAM J. Numer. Anal. 46(1), 344–364 (2008)
Pani, A.K., Fairweather, G., Fernandes, R.I.: ADI orthogonal spline collocation methods for parabolic partial integro-differential equations. IMA J. Numer. Anal. 30, 248–276 (2010)
Bialecki, B., Fairweather, G.: Orthogonal spline collocation methods for partial differential equations. J. Comput. Appl. Math. 128, 55–82 (2001)
Loreti, P., Pandolfi, L., Sforza, D.: Boundary controllability and observability of a viscoelastic string. SIAM J. Control Optim. 50(2), 820–844 (2012)
Fairweather, G., Yang, X., Xu, D., Zhang, H.: An ADI Crank-Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation. J. Sci. Comput. 65, 1217–1239 (2015)
Fairweather, G., Zhang, H., Yang, X., Xu, D.: A backward Euler orthogonal spline collocation method for the time-fractional Fokker-Planck equation. Numer. Methods Partial Differ. Equ. 31, 1534–1550 (2015)
Lions, J.-L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 2. Recherches en Mathématiques Appliquées, Masson, Paris, 1988
Marica, A., Zuazua, E.: Symmetric Discontinuous Galerkin Approximations of 1-D waves. Fourier Analysis, Propagation, Observability and Applications. With a Foreword by Roland Glowinski. Springer Briefs in Mathematics. Springer, New York (2014)
Aurora, Marica, Zuazua, Enrique: On the quadratic finite element approximation of 1-D waves: propagation, observation, control, and numerical implementation, The Courant-Friedrichs-Lewy (CFL) condition, 75–99. Birkhuser/Springer, New York (2013)
Haraux, A.: Séries lacunaires et contrôle semiinterne des vibrations d’une plaque rectangulaire. J. Math. Pures Appl. 68, 457–465 (1989)
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China, Contract Grant Numbers 11271123, 10971062, the Innovation and Open Research Project for College of Hunan Province (Contract Grant 12K028), and the Construct Program of the Key Discipline in Hunan Province, Performance Computing and Stochastic Information Processing (Ministry of Education of China).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, D. Boundary Observability of Semi-Discrete Second-Order Integro-Differential Equations Derived from Piecewise Hermite Cubic Orthogonal Spline Collocation Method. Appl Math Optim 77, 73–97 (2018). https://doi.org/10.1007/s00245-016-9367-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-016-9367-z
Keywords
- Orthogonal spline collocation methods
- Second-order integro-differential equations
- Observability
- Filtering