Model inversion of boundary controlled parabolic partial differential equations using summability methodsFootnote

Revised and expanded version of a paper presented at the 5th Vienna International Conference on Mathematical Modelling (MATHMOD), Vienna, Austria, 2006.

References

• Jensen , K. and Ray , W. 1982 . The bifurcation behavior of tubular reactors . Chem. Eng. Sci. , 37 : 199 – 222 .
   Web of Science ®Google Scholar
• Winkin , J. , Dochain , D. and Ligarius , P. 2000 . Dynamical analysis of distributed parameter tubular reactors . Automatica , 36 : 349 – 361 .
   Web of Science ®Google Scholar
• Curtain , R. and Zwart , H. 1995 . An Introduction to Infinite‐Dimensional Linear Systems Theory, Texts in Applied Mathematics 21 , New York : Springer-Verlag .
   Google Scholar
• Butkovskiy , A. 1969 . Distributed Control Systems , New York : Elsevier .
   Google Scholar
• Lions , J. 1971 . Optimal Control of Systems Governed by Partial Differential Equations , Springer-Verlag .
   Google Scholar
• Ray , W. 1981 . Advanced Process Control , New York : McGraw-Hill .
   Google Scholar
• Fursikov , A. 1999 . Optimal Control of Distributed Systems. Theory and Applications AMS (American Mathematical Society), Providence
   Google Scholar
• Fliess , M. , Lévine , J. , Martin , P. and Rouchon , P. 1996 . Flatness and defect of non-linear systems: Introductory theory and examples . Int. J. Control , 61 : 1327 – 1361 .
   Web of Science ®Google Scholar
• Rothfuß , R. , Rudolph , J. and Zeitz , M. 1996 . Flatness-based control of a non-linear chemical reactor model . Automatica , 32 : 1433 – 1439 .
   Web of Science ®Google Scholar
• Sira-Ramirez , H. and Agrawal , S. 2004 . Differentially Flat Systems , New York, Basel : Marcel Dekker Inc. .
   Google Scholar
• Fliess , M. , Mounier , H. , Rouchon , P. and Rudolph , J. 1997 . Systèmes linéaires sur les opérateurs de Mikusiński et commande d'une poutre flexible . ESAIM Proc , 2 : 183 – 193 .
   Google Scholar
• Laroche , B. , Martin , P. and Rouchon , P. 2000 . Motion planning for the heat equation . Int. J. Robust Non-linear Control , 10 : 629 – 643 .
   Web of Science ®Google Scholar
• Lynch , A. and Rudolph , J. 2002 . Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems . Int. J. Control , 75 : 1219 – 1230 .
   Web of Science ®Google Scholar
• Dunbar , W. , Petit , N. , Rouchon , P. and Martin , P. 2003 . Motion planning for a non-linear Stefan problem . ESAIM-COCV , 9 : 275 – 296 .
   Web of Science ®Google Scholar
• Rudolph , J. 2003 . Beiträge zur flachheitsbasierten Folgeregelung linearer und nichtlinearer Systeme endlicher und unendlicher Dimension , Aachen : Shaker-Verlag, Berichte aus der Steuerungs- und Regelungstechnik .
   Google Scholar
• Meurer , T. 2005 . Feedforward and Feedback Tracking Control of Diffusion-Convection-Reaction Systems using Summability Methods , VDI Verlag : Fortschr.-Ber. VDI Reihe 8 Nr. 1081 Düsseldorf .
   Google Scholar
• Hardy , G. 1964 . Divergent Series , Vol. 3 , Oxford : Clarendon Press .
   Google Scholar
• Balser , W. 2000 . Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations , New York : Springer-Verlag .
   Google Scholar
• Delattre , C. , Dochain , D. and Winkin , J. 2002 . Observability analysis of a non-linear tubular bioreactor . Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems (MTNS) . 2002 , Notre Dame, USA.
   Google Scholar
• Atkinson , B. 1974 . Biochemical Reactors , London : Pion Limited .
   Google Scholar
• Meurer , T. and Zeitz , M. 2005 . Feedforward and feedback tracking control of non-linear diffusion-convection-reaction systems using summability methods . Ind. Eng. Chem. Res. , 44 : 2532 – 2548 .
   Web of Science ®Google Scholar
• Hua , C. and Rodino , L. General theory of partial differential equations and micrological analysis . Proceedings of the Workshop on General Theory of PDEs and Micrological Analysis . Edited by: Min-You , Q. and Rodino , L. pp. 6 – 81 . (Int. Center for Theoretical Physics, Trieste (I): Longman) .
   Google Scholar
• Rudin , W. 1999 . Reelle und Komplexe Analysis , Oldenbourg : Wissenschaftsverlag GmbH .
   Google Scholar
• Wagner , M. , Meurer , T. and Zeitz , M. K-summable power series as a design tool for feedforward control of diffusion-convection-reaction systems . Proceedings of the 6th IFAC Symposium “Non-linear Control Systems” (NOLCOS 2004) . pp. 149 – 154 . Stuttgart (D) .
   Google Scholar
• Borel , E. 1928 . Leçons sur les séries divergentes , Paris : Gauthier‐Villar .
   Google Scholar
• Ramis , J. P. 1980 . “ Les seriés k-sommables et leurs applicationsin ” . In Analysis, Micrological Calculus and Relativistic Quantum Theory, Vol. 126 of Lecture Notes in Physics , 178 – 199 . Springer-Verlag .
   Google Scholar
• Malgrange , B. 1995 . Sommation des séries divergentes . Expositiones Mathematicae , 13 : 163 – 222 .
   Google Scholar
• Balser , W. 1994 . From Divergent Power Series to Analytic Functions , Springer-Verlag .
   Google Scholar
• Lutz , D. , Miyake , M. and Schäfke , R. 1999 . On the Borel Summability of Divergent Solutions of the Heat Equation . Nagoya Math. J. , 154 : 1 – 29 .
   Web of Science ®Google Scholar
• Balser , W. and Miyake , M. 1999 . Summability of formal solutions of certain partial differential equations . Acta Sci. Math (Szeged) , 65 : 543 – 551 .
   Google Scholar
• Balser , W. 2001 . Summability of formal power series solutions of ordinary and partial differential equations . Functional Diff. Eq. , 8 : 11 – 24 .
   Google Scholar
• Ramis , J. P. 1993 . Séries Divergentes et Théorie Asymptotiques, Panoramas et synthèses Vol. 121 , Paris : Society of Mathematics France .
   Google Scholar
• Balser , W. and Braun , R. 2000 . Power series methods and multisummability . Math. Nachr. , 212 : 37 – 50 .
   Web of Science ®Google Scholar
• Brezinski , C. 1977 . Accéleration de la convergence en analyse numérique, Lecture Notes in Mathematics , Berlin : Springer-Verlag .
   Google Scholar
• Weniger , E. 1989 . Non-linear sequence transformations for the acceleration of convergence and the summation of divergent series . Compt. Phys. Rep. , 10 : 189 – 371 .
   Google Scholar