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On Semigroups in Rn X Lp Corresponding to Differential Equations with Delays

Published online by Cambridge University Press:  20 November 2018

C. Bernier  and
A. Manitius
C. Bernier
Affiliation:
Université de Montréal, Montréal, Quebec
A. Manitius
Affiliation:
Université de Montréal, Montréal, Quebec
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In this paper we study some properties of the semigroups associated with the linear retarded functional differential equations (FDE) in the setting of Banach spaces Rn X Lp( — h, 0, Rn), 1 < p < ∞. Earlier investigations of these equations via semigroups defined on the customary space C([ — h, 0], Rn) played an important role in problems of stability, oscillations, bifurcation, asymptotic behavior etc. [15].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 5 , 01 October 1978 , pp. 897 - 914
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Banks, H. T. and Burns, J. A., Eigenmanifold decomposition for retarded functional differential equations in Hilbert space, Report TR-1, Department of Mathematics, Virginia Polytechnic Institute, Blacksburg, Virginia, U.S.A., Nov. 1974.Google Scholar
2. Banks, H. T. and Burns, J. A., Hereditary control problems: numerical methods based on averaging approximations, SIAM J. on Control and Optimization 16 (1978), 169–208.Google Scholar
3. Banks, H. T. and Manitius, A., Projection series for retarded functional differential equations with applications to optimal control problems, J. Diff. Equations 18 (1975), 296–332.Google Scholar
4. Bellman, R. and Cooke, K. L., Differential-difference equations (Academic Press, New York, 1963).Google Scholar
5. Borisovic, J. G. and Turbabin, A. S., On the Cauchy problem for linear nonhomogeneous differential equations with retarded argument, Soviet Math. Doklady 10 (1969), 401–405.Google Scholar
6. Burns, J. A. and Herdman, T. L., Adjoint semigroup theory for a class of functional differential equations, SIAM J. Math. Anal. 7 (1976), 729–745.Google Scholar
7. Butzer, P. L. and Berens, H., Semi-groups of operators and approximation (Springer Verlag, New York, 1967).Google Scholar
8. Coleman, B. D. and Mizel, V. J., Norms and semigroups in the theory of fading memory, Arch. Rational Mech. Anal. 23 (1966), 87–123.Google Scholar
9. Courtain, R. and A . J . Pritchard, The infinite dimensional Riccati equation for systems defined by evolution operators, SIAM J. Control and Optimization 14 (1976), 951–983.Google Scholar
10. Delfour, M. C., State theory of linear hereditary differential systems, J. Math. Anal. Appl. 60 (1977), 8–35.Google Scholar
11. Delfour, M. C., Lee, E. B. and Manitius, A., F-reduction of the operator Riccati equation for hereditary differential systems, Automatica 14 (1978), to appear.Google Scholar
12. Delfour, M. C. and Manitius, A., Control systems with delays: areas of applications and present status of the linear theory, Proceedings of the International Symposium on New Trends in Systems Analysis, IRIA, France, Dec. 13–17, 1976, to be published by Springer Verlag.Google Scholar
13. Delfour, M. C. and Mitter, S. K., Hereditary differential systems with constant delays: II, a class of affine systems and the adjoint problem, J. Differential Equations 18 (197“)), 18–28.Google Scholar
14. Dunford, N. and Schwartz, J. T., Linear operators, Part I: General theory (Interscience, New York, 1958).Google Scholar
15. Hale, J. K., Theory of functional differential equations (Springer Verlag, New York, 1977).Google Scholar
16. Hartman, S. and Mikusinski, J., The theory of Lebesgue measure and integration (Pergamon Press, Inc., New York, 1961).Google Scholar
17. Henry, D., The adjoint of a linear functional differential equation and boundary value problems, J. Diff. Equations 9 (1971), 55–66.Google Scholar
18. Hewitt, E. and Stromberg, K., Real and abstract analysis (Springer Verlag, New York, 1965).Google Scholar
19. Krasovskii, N. N., The approximation of a problem of analytic design of controls in a system with time-lag, J. Appl. Math. Mech. 28 (1964), 876–885.Google Scholar
20. Manitius, A., On the controllability conditions for systems with distributed delays in state and control, Arch. Autom. Telem. 17 (1972), 363–377.Google Scholar
21. Manitius, A., Controllability, observability and stabilizability of retarded systems, Proceedings of the 1976 IEEE Conference on Decision and Control (IEEE Publications, New York, 1976), 752–758.Google Scholar
22. Manitius, A., Completeness and F-completeness of eigenfunction s associated with retarded functional differential equations, Report CRM-755, Centre de Recherches Mathématiques, Université de Montréal, Montréal, Canada, 1977.Google Scholar
23. Manitius, A. and Triggiani, R., Function space controllability of linear retarded systems: a derivation from abstract operator conditions, in SIAM J. Control and Optimization 16 (1978), 599–645.Google Scholar
24. Natanson, I. P., Theory of functions of a real variable (Ungar, New York, 1955).Google Scholar
25. Pazy, A., Semigroups of linear operators and application to partial differential equations, Univ. of Maryland Lecture Notes, Vol. 10, College Park, Md., 1974.Google Scholar
26. Vinter, R. B., On the evolution of the state linear differential delay equations in M2: properties of the generator, Electronic Systems Lab., Technical Report ESL-R-541, M.I.T., Cambridge, 1974, J. Inst. Math. Appl., to appear.Google Scholar
27. Zabczyk, J., On decomposition of generators, in SIAM J. Control and Optimization 16 (1978), 523–534.Google Scholar