Abstract
A parabolic equation defined on a bounded domain is considered, with input acting on theboundary expressed as a specifiedfeedback of the solution. Both Dirichlet and mixed (in particular, Neumann) boundary conditions are treated. Algebraic conditions based on the finitely many unstable eigenvalues are given, ensuring the existence ofboundary vectors, for which all the solutions to theboundary feedback parabolic equation decay exponentially to zero ast→+∞ in (essentially) the strongest possible space norm. A semigroup approach is employed.
Similar content being viewed by others
References
P. J. Antsaklis and W. A. Wolovich, Arbitrary pole placement using linear output feedback compensation,Int. Journ. Control 25, n. 6, (1977), 915–925
A. V. Balakrishnan, Filtering and control problems for partial differential equations,Proceedings of the 2nd Kingston Conference on Differential Games and Control Theory, University of Rhode Island, Marcel Dekker Inc., New York, 1976.
A. V. Balakrishnan, Boundary control of parabolic equations:L-Q-R theory,Proceedings of the Conference on Theory of Nonlinear Equations, September 1977, published by Akademie-Verlag, Berlin, 1978.
R. F. Curtain and A. Pritchard,Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences (N.P., Springer Verlag 1978 (Chapter 8, Section 4)
E. J. Davison and S. H. Wang, On pole assignment in linear multivariable systems using output feedback,IEEE AC-20, August 1975, 516–518.
A. Friedman,Partial differential equations, reprinted by Robert E. Krieger Publishing Company, Huntington, New York, 1976.
D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,Proc. Japan Acad. 43 (1967), 82–86.
T. Kato,Perturbation theory of linear operators, Springer-Verlag, New York-Berlin, 1966.
H. Kimura, Pole assignment by gain output feedback,IEEE AC, August 1975, 20, 509–516.
V. Lakshmikantham and S. Leela,Differential and integral inequalities, Vol. I, Academic Press, New York, 1969.
I. Lasiecka, Boundary control of parabolic systems: regularity of optimal solutions,Appl. Math. and Opt., 4, 301–327 (1978).
I. Lasiecka, State constrained control problems for parabolic systems: regularity of optimal solutions,App. Math. and Opt., 6, 1–29 (1980).
J. L. Lions and E. Magenes,Non-homogeneous boundary value problems and applications, vol. I; II, 1972, III, 1973, Springer-Verlag, New York-Heidelberg-Berlin.
J. L. Lions, Espaces d'interpolation et domaines de puissances fractionnaires d'operateurs,J. Math. Soc. Japan 14, No. 2, 1962, 233–241.
J. L. Lions and E. Magenes, Problem sux limites non homogenes,IV Ann. Scuola Normale Superiore Pisa 15 (1961), 311–325.
I. Lasiecka, Unified theory for abstract parabolic boundary problems: a semigroup approach,Appl. Math. and Opt., to appear.
J. Necas,Les methodes directes en theorie des equations elliptiques, Mosson et Cie, Editeurs, 1967.
T. Nambu, Feedback stabilization of distributed parameter systems of parabolic type,J. Diff. Eq. 33, 2, Aug. 79.
A. Pazy,Semigroups of operators and applications to partial differential equations, Lecture Notes #10, University of Maryland, College Park, Maryland.
E. J. P. G. Schmidt and N. Weck, On the boundary behaviour of solutions to elliptic and parabolic equations,SIAM J. Control and Optim. 16, n. 4, (1978), 593–598.
R. Triggiani, On the stabilizability problem in Banach space,J. Math. Anal. Applic. 52 (1975) 383–403. Addendum 56(1976).
R. Triggiani, Well-posedness and regularity of boundary feedback parabolic systems, to appear.
R. Triggiani, On Nambu's boundary stabilizability problem for diffusion processes,J. Differential Equations 33, No. 2, 189–200 (1979)
D. C. Washburn, A semigroup theoretic approach to modeling of boundary input problems,Proceedings of IFIP Working Conference, University of Rome, Italy, Lecture Notes in Control and Information Sciences, No. 1, 446–458, Springer-Verlag, 1977.
D. C. Washburn, A bound on the boundary input map for parabolic equations with application to time optimal control,SIAM J. Control and Optim. Based on a Ph.D. thesis at UCLA, 1974.
M. Wonham, On pole assignment in multi-input controllable linear systems,IEEE Trans-AC 12 (6), 1967, 660–665.
J. Zabczyk, On stabilization of boundary control systems, Report 785, Centre de Recherches mathematiques, Universite de Montreal, March 1978.
J. Zabczyk, Stabilization of boundary control systems,Proceedings of International Symposium on Systems Optimization and Analysis, held at I.R.I.A., December 1978.
R. Triggiani, A cosine operator approach to modelling boundary input hyperbolic systems,Proceedings 8th IFIP, Springer-Verlag Lecture Notes on Control N 6, 380–390 (1978).
Author information
Authors and Affiliations
Additional information
Communicated by W. Fleming
Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-77-3338.
A preliminary version of this paper has appeared in the Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, held at Delft University, The Netherlands, July 3–6, 1979; pp. 428–433.
Rights and permissions
About this article
Cite this article
Triggiani, R. Boundary feedback stabilizability of parabolic equations. Appl Math Optim 6, 201–220 (1980). https://doi.org/10.1007/BF01442895
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01442895