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Exact and approximate controllability for distributed parameter systems

Published online by Cambridge University Press:  07 November 2008

R. Glowinski  and
J.L. Lions
R. Glowinski
Affiliation:
University of Houston, Houston, Texas, USAUniversité Pierre et Marie Curie, Paris, France C.E.R.F.A.C.S., Toulouse, France
J.L. Lions
Affiliation:
Collège de FranceRue d'Ulm, 75005 Paris, France
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Extract

We consider a system whose state is given by the solution y to a Partial Differential Equation (PDE) of evolution, and which contains control functions, denoted by v.

Type
Research Article
Information
Acta Numerica , Volume 3 , January 1994 , pp. 269 - 378
Copyright
Copyright © Cambridge University Press 1994

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References

REFERENCES

Abergel, F. and Temam, R. (1990), ‘On some control problems in fluid mechanics’, Theoret. Comput. Fluid Dyn. 1, 303–326.CrossRefGoogle Scholar
Buschnell, D.M. and Hefner, J.N. (1990), Viscous Drag Reduction in Boundary Layers, American Institute of Aeronautics and Astronautics (Washington, DC).CrossRefGoogle Scholar
Carthel, C., Glowinski, R. and Lions, J.L. (1994), ‘On exact and approximate boundary controllabilities for the heat equation: a numerical approach’, J. Opt. Theoret. Appl. 82, 3.CrossRefGoogle Scholar
Ciarlet, P.G. (1978), The Finite Element Method for Elliptic Problems, North-Holland (Amsterdam).Google Scholar
Ciarlet, P.G. (1989), Introduction to Numerical Linear Algebra and Optimization, Cambridge University Press (Cambridge).CrossRefGoogle Scholar
Ciarlet, P.G. (1990a), ‘A new class of variational problems arising in the modeling of elastic multi-structures’, Numer. Math. 57, 547–560.CrossRefGoogle Scholar
Ciarlet, P.G. (1990b), Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson (Paris).Google Scholar
Ciarlet, G., Le Dret, H. and Nzengwa, R. (1989), ‘Junctions between three-dimensional and two-dimensional linearly elastic structures’, J. Math. Pures et Appl. 68, 261–295.Google Scholar
Daniel, J. (1970), The Approximate Minimization of Functionals, Prentice Hall (En-glewood Cliffs, NJ).Google Scholar
Dean, E.J., Glowinski, R. and Li, C.H. (1989), ‘Supercomputer solution of partial differential equation problems in computational fluid dynamics and in control’, Comput. Phys. Commun. 53, 401–439.CrossRefGoogle Scholar
Dennis, J. and Schnabel, R.B. (1983), Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall (Englewood Cliffs, NJ).Google Scholar
Diaz, J.I. (1991), ‘Sobre la controlabilidad aproximada de problemas no lineales disipativos’, Jornadas Hispano-Francesas Sobre Control de Sistemas Distribuidos, Universidad de Malaga 41–48.Google Scholar
Downer, J.D., Park, K.C. and Chiou, J.C. (1992), ‘Dynamics of flexible beams for multibody systems: a computational procedure’, Comput. Meth. Appl. Mech. Engrg 96, 373–408.CrossRefGoogle Scholar
Ekeland, I. and Temam, R. (1974), Analyse Convexe et Problèmes Variationnels, Dunod (Paris).Google Scholar
Engquist, B., Gustafsson, B. and Vreeburg, J. (1978), ‘Numerical solution of a PDE system describing a catalytic converter’, J. Comput. Phys. 27, 295–314.CrossRefGoogle Scholar
Fabre, C., Puel, J.P. and Zuazua, E. (1993), ‘Contrôlabilité approchée de l'équation de la chaleur linéaire avec des contrôles de norme L∞ minimale’, C. R. Acad. Sci. I 316, 679–684.Google Scholar
Friedman, A. (1988), ‘Modeling catalytic converter performance’, in Mathematics in Industrial Problems, Part 4 (Friedman, A., ed.) Springer (New York) Ch. 7, 70–77.CrossRefGoogle Scholar
Friend, C.M. (1993), ‘Catalysis and surfaces’, Scientific American (04), 74–79.CrossRefGoogle Scholar
Fujita, H. and Suzuki, T. (1991), ‘Evolution problems’, in Handbook of Numerical Analysis vol. II (Ciarlet, P.G. and Lions, J.L., eds.) North-Holland (Amsterdam), 789–928.Google Scholar
Gabay, D. (1982), ‘Application de la méthode des multiplicateurs aux inéquations variationnelles’, in Méethodes de Lagrangien Augmenté (Fortin, M. and Glowinski, R., eds.) Dunod-Bordas, (Paris).Google Scholar
Gabay, D. (1983) ‘Application of the method of multipliers to variational inequalities’, in Augmented Lagrangian Methods (Fortin, M. and Glowinski, R., eds.) North-Holland (Amsterdam).Google Scholar
Glowinski, R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer (New York).CrossRefGoogle Scholar
Glowinski, R. (1992a), ‘Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation’, J. Comput. Phys. 103, 189–221.CrossRefGoogle Scholar
Glowinski, R. (1992b), ‘Boundary controllability problems for the wave and heat equations’, in Boundary Control and Boundary Variation (Zolesio, J.P., ed.) Lecture Notes in Control and Information Sciences, Vol. 178, Springer (Berlin) 221–237.CrossRefGoogle Scholar
Glowinski, R. and Le Tallec, P. (1989), Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM (Philadelphia, PA).CrossRefGoogle Scholar
Glowinski, R. and Li, C.H. (1990), ‘On the numerical implementation of the Hilbert uniqueness method for the exact boundary controllability of the wave equation’, C. R. Acad. Sci. I 311, 135–142.Google Scholar
Glowinski, R., Li, C.H. and Lions, J.L. (1990), ‘A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: description of the numerical methods’, Japan. J. Appl. Math. 7, 1–76.CrossRefGoogle Scholar
Glowinski, R., Lions, J.L. and Tremolières, R. (1976), Analyse Numérique des Inéquations Variationnelles, Dunod (Paris).Google Scholar
Glowinski, R., Lions, J.L. and Tremolierès, R. (1981), Numerical Analysis of Variational Inequalities, North-Holland (Amsterdam).Google Scholar
Golub, G.H. and Van Loan, C. (1989), Matrix Computations, Johns Hopkins Press (Baltimore, MD).Google Scholar
Ladyzenskaya, O.A., Solonnikov, V.A. and Ural'ceva, N.N. (1968), Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society (Providence, RI).CrossRefGoogle Scholar
Lagnese, J.E. and Leugering, G.. To appear.Google Scholar
Laursen, T.A. and Simo, J.C. (1993), ‘A continuum-based finite element formulation for the implicit solution of multi-body, large deformation frictional contact problems’, Int. J. Numer. Meth. Engrg 36, 3451–3486.CrossRefGoogle Scholar
Lions, J.L. (1961), Équations Différentielles Op´erationnelles et Problèmes aux Limites, Springer (Heidelberg).Google Scholar
Lions, J.L. (1968), Contrôle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod (Paris).Google Scholar
Lions, J.L. (1988a), ‘Exact controllability, stabilization and perturbation for distributed systems’, SIAM Rev. 30, 1–68.CrossRefGoogle Scholar
Lions, J.L. (1988b), Controlabilité Exacte, Perturbation et Stabilisation des Systèmes Distribués, vols. 1 and 2, Masson (Paris).Google Scholar
Lions, J.L. (1990), El Planeta Tierra, Instituto de España, Espasa Calpe, S.A., Madrid.Google Scholar
Lions, J.L. (1993), ‘Quelques remarques sur la controlabilité en liaison avec des questions d'environnement’, in Les Grands Systémes des Sciences et de la Technologie (Horwitz, J. and Lions, J.L., eds.), Masson (Paris) 240–264.Google Scholar
Lions, J.L. and Magenes, E. (1968), Problèmes aux Limites Non Homogènes, Vol. 1. Dunod (Paris).Google Scholar
Lions, P.L. and Mercier, B. (1979), ‘Splitting algorithms for the sum of two nonlinear operators’, SIAM J. Numer. Anal. 16, 964–979.CrossRefGoogle Scholar
Mizohata, S. (1958), ‘Unicité’ du prolongement des solutions pour quelques opérateurs différentiels paraboliques’, Mem. Coll. Sci. Univ. Kyoto A 31, 219–239.Google Scholar
Nocedal, J. (1992), ‘Theory of algorithms for unconstrained optimization’, Acta Numerica 1992, Cambridge University Press, (Cambridge) 199–242.Google Scholar
Park, K.C., Chiou, J.C. and Downer, J.D. (1990), ‘Explicit-implicit staggered procedures for multibody dynamics analysis’, J. Guidance Control Dyn. 13, 562–570.CrossRefGoogle Scholar
Peaceman, D. and Rachford, H. (1955), ‘The numerical solution of parabolic and elliptic differential equations’, J. Soc. Ind. Appl. Math. 3, 28–41.CrossRefGoogle Scholar
Raviart, P.A. and Thomas, J.M. (1988), Introduction à, l'Analyse Numérique des Équations aux Dérivés Partielles, Masson (Paris).Google Scholar
Russel, D.L. (1978), ‘Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions’, SIAM Rev. 20, 639–739.CrossRefGoogle Scholar
Sanchez Hubert, J. and Sanchez Palencia, E. (1989), Vibrations and Coupling of Continuous Systems (Asymptotic Methods), Springer (Berlin).CrossRefGoogle Scholar
Saut, J.C. and Scheurer, B. (1987) ‘Unique continuation for some evolution equations’, J. Diff. Eqns 66, 118–139.CrossRefGoogle Scholar
Sellin, R.H. and Moses, T. (1989), Drag Reduction in Fluid Flows, Ellis Horwood (Chichester).Google Scholar
Thomee, V. (1990), ‘Finite difference methods for linear parabolic equations’, in Handbook of Numerical Analysis, vol. I (Ciarlet, P.G. and Lions, J.L., eds.) North-Holland (Amsterdam) 5–196.Google Scholar