Abstract
Generalizations of well-known conditions for controllability of linear abstract autonomous systems defined on Banach spaces to the case where there is a closed non invertible operator at the derivative are established. Presence of this operator implies that suitable controllers must be chosen. If the operators entering in the equation satisfy certain hypotheses, approximate controllability by use of this class of controllers is expressed only in terms of the coefficients of the system. In particular, approximate controllability in finite time is then equivalent to approximate controllability, according to the usual definition, of a corresponding non degenerate system. This is the case, for example, when the concerned spaces are finite dimensional. Some applications to partial differential equations are given.
Similar content being viewed by others
References
R. F. Curtain and J. Pritchard,Functional Analysis in Modern Applied Mathematics, ed., Academic Press, 1977.
H. O. Fattorini, Some remarks on complete controllability,SIAM J. Control 4, 686–694, 1966.
A. Favini, Laplace transform method for a class of degenerate evolution problems,Rend. Mat., Rome, to appear.
A. Favini, Abstract potential operators and spectral methods for a class of degenerate evolution problems, to appear.
A. Friedman,Partial Differential Equations, ed. Holt-Rinehart-Winston, 1969.
M. L. J. Hautus, Controllability and observability conditions of linear autonomous systems, Nederl. Akad. Wetensch., Proc., Ser. A 72, 443–448, 1969.
L. Pandolfi, Controllability and stabilization for linear systems of algebraic and differential equations,J. Optim. Theory Appl., to appear.
A. G. Rutkas, Cauchy's problem for the equationAx′(t)+Bx(t)=f(t), (russian), Diff. Uravn. 11, n. 11, 1996–2010, 1975.
R. Triggiani, Controllability and observability in Banach space with bounded operators,Siam J. Control 13, 462–491, 1975.
Author information
Authors and Affiliations
Additional information
Communicated by A. V. Balakrishnan
This paper was written under the auspices of the Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni of the Consiglio Nazionale delle Ricerche
Rights and permissions
About this article
Cite this article
Favini, A. Controllability conditions of linear degenerate evolution systems. Appl Math Optim 6, 153–167 (1980). https://doi.org/10.1007/BF01442890
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01442890