## Diffusive realizations for solutions of some operator equations: The one-dimensional case

HTML articles powered by AMS MathViewer

- by Michel Lenczner, Gérard Montseny and Youssef Yakoubi PDF
- Math. Comp.
**81**(2012), 319-344 Request permission

## Abstract:

In this paper we deal with the derivation of state-realizations of linear operators that are solutions to certain operator linear differential equations in one-dimensional bounded domains. We develop two approaches in the framework of diffusive representations: one with complex diffusive symbols; the other with real diffusive symbols. Then, we illustrate the theories and develop numerical methods for a Lyapunov equation arising from optimal control theory of the heat equation. A practical purpose of this approach is real-time computation on a semi-decentralized architecture with low granularity.## References

- Bassam Bamieh, Fernando Paganini, and Munther A. Dahleh,
*Distributed control of spatially invariant systems*, IEEE Trans. Automat. Control**47**(2002), no. 7, 1091–1107. MR**1911479**, DOI 10.1109/TAC.2002.800646 - P. Bidan, T. Lebey, G. Montseny, C. Neascu, and J. Saint-Michel,
*Transient voltage distribution in inverter fed motor windings: Experimental study and modeling*, IEEE Transactions on Power Electronics**16**(2001), no. 1, 92–100. - P. Bidan, T. Lebey, and C. Neacsu,
*Development of a new off-line test procedure for low voltage rotating machines fed by adjustable speed drives (ASD)*, IEEE Transactions on Dielectrics and Electrical Insulation**10**(2003), no. 1, 168–175. - C. Casenave and E. Montseny,
*Time-local dissipative formulation and stable numerical schemes for a class of integrodifferential wave equations*, SIAM J. Appl. Math.**68**(2008), no. 6, 1763–1782. MR**2424962**, DOI 10.1137/070693710 - Eduardo Cuesta, Christian Lubich, and Cesar Palencia,
*Convolution quadrature time discretization of fractional diffusion-wave equations*, Math. Comp.**75**(2006), no. 254, 673–696. MR**2196986**, DOI 10.1090/S0025-5718-06-01788-1 - Raffaello D’Andrea and Geir E. Dullerud,
*Distributed control design for spatially interconnected systems*, IEEE Trans. Automat. Control**48**(2003), no. 9, 1478–1495. MR**2000106**, DOI 10.1109/TAC.2003.816954 - Ivan P. Gavrilyuk and Vladimir L. Makarov,
*Exponentially convergent parallel discretization methods for the first order evolution equations*, Comput. Methods Appl. Math.**1**(2001), no. 4, 333–355. MR**1892950**, DOI 10.2478/cmam-2001-0022 - I. P. Gavrilyuk and V. L. Makarov,
*Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces*, SIAM J. Numer. Anal.**43**(2005), no. 5, 2144–2171. MR**2192335**, DOI 10.1137/040611045 - G. Goavec-Merou, Y. Yakoubi, R. Couturier, M. Lenczner, and J.-M. Friedt,
*Fpga implementation of diffusive realizations for a distributed control operator*, dMEMS 2010, Besançon, France - June 28-29th 2010. - Th. Hélie and D. Matignon,
*Diffusive representations for the analysis and simulation of flared acoustic pipes with visco-thermal losses*, Math. Models Methods Appl. Sci.**16**(2006), no. 4, 503–536. MR**2218212**, DOI 10.1142/S0218202506001248 - Thomas Hélie, Denis Matignon, and Rémi Mignot,
*Criterion design for optimizing low-cost approximations of infinite-dimensional systems: towards efficient real-time simulation*, Int. J. Tomogr. Stat.**7**(2007), no. F07, 13–18. MR**2393796** - Cédric Langbort and Raffaello D’Andrea,
*Distributed control of spatially reversible interconnected systems with boundary conditions*, SIAM J. Control Optim.**44**(2005), no. 1, 1–28. MR**2176664**, DOI 10.1137/S0363012902415803 - Lionel Laudebat, Pierre Bidan, and Gérard Montseny,
*Modeling and optimal identification of pseudodifferential electrical dynamics by means of diffusive representation I. Modeling*, IEEE Trans. Circuits Syst. I. Regul. Pap.**51**(2004), no. 9, 1801–1813. MR**2090345**, DOI 10.1109/TCSI.2004.834501 - Michel Lenczner and Gérard Montseny,
*Diffusive realization of operator solutions of certain operational partial differential equations*, C. R. Math. Acad. Sci. Paris**341**(2005), no. 12, 737–740 (English, with English and French summaries). MR**2188868**, DOI 10.1016/j.crma.2005.09.036 - M. Lenczner and Y. Yakoubi,
*Semi-decentralized approximation of optimal control for partial differential equations in bounded domains*, Comptes Rendus Mécanique**337**(2009), no. 4, 245–250. - David Levadoux and Gérard Montseny,
*Diffusive realization of the impedance operator on circular boundary for 2D wave equation*, Mathematical and numerical aspects of wave propagation—WAVES 2003, Springer, Berlin, 2003, pp. 136–141. MR**2077989** - María López-Fernández, Christian Lubich, Cesar Palencia, and Achim Schädle,
*Fast Runge-Kutta approximation of inhomogeneous parabolic equations*, Numer. Math.**102**(2005), no. 2, 277–291. MR**2206466**, DOI 10.1007/s00211-005-0624-3 - María López-Fernández, Christian Lubich, and Achim Schädle,
*Adaptive, fast, and oblivious convolution in evolution equations with memory*, SIAM J. Sci. Comput.**30**(2008), no. 2, 1015–1037. MR**2385897**, DOI 10.1137/060674168 - M. López-Fernández and C. Palencia,
*On the numerical inversion of the Laplace transform of certain holomorphic mappings*, Appl. Numer. Math.**51**(2004), no. 2-3, 289–303. MR**2091405**, DOI 10.1016/j.apnum.2004.06.015 - Christian Lubich,
*Convolution quadrature revisited*, BIT**44**(2004), no. 3, 503–514. MR**2106013**, DOI 10.1023/B:BITN.0000046813.23911.2d - Christian Lubich and Achim Schädle,
*Fast convolution for nonreflecting boundary conditions*, SIAM J. Sci. Comput.**24**(2002), no. 1, 161–182. MR**1924419**, DOI 10.1137/S1064827501388741 - Denis Matignon and Christophe Prieur,
*Asymptotic stability of linear conservative systems when coupled with diffusive systems*, ESAIM Control Optim. Calc. Var.**11**(2005), no. 3, 487–507. MR**2148855**, DOI 10.1051/cocv:2005016 - William McLean and Vidar Thomée,
*Time discretization of an evolution equation via Laplace transforms*, IMA J. Numer. Anal.**24**(2004), no. 3, 439–463. MR**2068831**, DOI 10.1093/imanum/24.3.439 - G Montseny,
*Simple approach to approximation and dynamical realization of pseudodifferential time operators such as fractional ones*, IEEE Transactions on Circuits and Systems II-Express Briefs**51**(2004), no. 11, 613–618. - G. Montseny,
*Représentation diffusive*, Hermès-Sciences, 2005. - Achim Schädle, María López-Fernández, and Christian Lubich,
*Fast and oblivious convolution quadrature*, SIAM J. Sci. Comput.**28**(2006), no. 2, 421–438. MR**2231714**, DOI 10.1137/050623139 - Laurent Schwartz,
*Méthodes mathématiques pour les sciences physiques*, Enseignement des Sciences, Hermann, Paris, 1961 (French). MR**0143360** - Dongwoo Sheen, Ian H. Sloan, and Vidar Thomée,
*A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature*, IMA J. Numer. Anal.**23**(2003), no. 2, 269–299. MR**1975267**, DOI 10.1093/imanum/23.2.269 - A. Talbot,
*The accurate numerical inversion of Laplace transforms*, J. Inst. Math. Appl.**23**(1979), no. 1, 97–120. MR**526286** - L. N. Trefethen, J. A. C. Weideman, and T. Schmelzer,
*Talbot quadratures and rational approximations*, BIT**46**(2006), no. 3, 653–670. MR**2265580**, DOI 10.1007/s10543-006-0077-9 - C. Wagschal,
*Fonctions holomorphes, Equations différentielles*, Hermann, Paris, 2003. - J. A. C. Weideman and L. N. Trefethen,
*Parabolic and hyperbolic contours for computing the Bromwich integral*, Math. Comp.**76**(2007), no. 259, 1341–1356. MR**2299777**, DOI 10.1090/S0025-5718-07-01945-X - K\B{o}saku Yosida,
*Functional analysis*, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. MR**1336382**, DOI 10.1007/978-3-642-61859-8

## Additional Information

**Michel Lenczner**- Affiliation: Femto-St Institute, Time-Frequency 26, Rue de l’Epitaphe, 25030 Besançon, France –and– UTBM, 90010 Belfort Cedex, France
- Email: michel.lenczner@utbm.fr
**Gérard Montseny**- Affiliation: LAAS-CNRS 7, avenue du Colonel Roche 31077 Toulouse Cedex 4, France
- Email: montseny@laas.fr
**Youssef Yakoubi**- Affiliation: UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, F-75005, Paris Cedex, France
- Email: yyakoubi@ann.jussieu.fr
- Received by editor(s): September 24, 2009
- Received by editor(s) in revised form: September 23, 2010
- Published electronically: July 19, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**81**(2012), 319-344 - MSC (2010): Primary 35-xx, 47A62, 01-08, 47G10
- DOI: https://doi.org/10.1090/S0025-5718-2011-02485-3
- MathSciNet review: 2833497