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The Largest Class of Hereditary Systems Defining a C0 Semigroup on the Product Space

Published online by Cambridge University Press:  20 November 2018

M. C. Delfour
M. C. Delfour*
Affiliation:
Université de Montréal, Montréal, Québec
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The object of this paper is to characterize the largest class of autonomous linear hereditary differential systems which generates a strongly continuous semigroup of class C0 on the product space Mp = Rn × Lp(-h, 0), 1 ≦ p < ∞, 0 < h ≦ + ∞ (R is the field of real numbers and Lp(– h, 0) is the space of equivalence classes of Lebesgue measurable maps x:[ – h, 0] ⌒ R → Rn which are p-integrable in [ –h, 0] ⌒R.) Our results extend and complete those of [4] and [15], [16] for linear hereditary differential equations possessing “finite memory” (h < + ∞ ) and those of [14], [5] and [6] in the “infinite memory case (h = + ∞ )”.

Consider the autonomous linear hereditary differential equation

(1.1)

where x(t) ∊ Rn, x:[–h, 0] ⌒ R → Rn is defined as xt(θ) = x(t + θ), C(–h, 0) is the space of bounded continuous functions [–h, 0] ⌒ R → Rn and L:C(–h, 0) → Rn is a continuous linear map.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 4 , April 1980 , pp. 969 - 978
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Adams, R. A., Sobolev spaces (Academic Press, New York, 1975).Google Scholar
2. Barbu, V. and Grossman, S. T., Asymptotic behavior of linear integrodifferential systems, Trans. Amer. Math. Soc. 173 (1972), 277–288.Google Scholar
3. Bielecki, A., Une remarque sur la méthode de Banach-Cacciopoli-Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Acad. Polon. Sci. 4 (1956), 261–264.Google Scholar
4. Borisovic, J. G. and Turbabin, A. S., On the Cauchy problem for linear non-homogeneous differential equations with retarded argument, Solviet Math. Doklady 10 (1969), 401–405.Google Scholar
5. Burns, J . A. and Herdman, T. L., Adjoint semigroup theory for a Volterra integrodifferential system, Bulletin of the American Mathematical Society 81 (1975), 1099–1102.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. Delfour, M. C. and Manitius, A., The structural operator F and its role in the theory of retarded systems I, J. Math. Anal. Appl. 73 (1980), 466–490.Google Scholar
8. Delfour, M. C. and Manitius, A., The structural operator F and its role in the theory of retarded systems II, J. Math. Anal. Appl. (1979), to appear.Google Scholar
9. Delfour, M. C. and Mitter, S. K., Controllability, observability and optimal feedback control of affine hereditary differential systems, SIAM J. Control 10 (1972), 298–328.Google Scholar
10. Hale, J. K., Linear functional-differential equations with constant coefficients, Contributions to Differential Equations 2 (1963), 291–319.Google Scholar
11. Hale, J. K., Functional differential equations (Springer-Verlag, New York, 1971).CrossRefGoogle Scholar
12. Hale, J. K., Theory of functional differential equations (Springer-Verlag, New York, 1977).CrossRefGoogle Scholar
13. Hille, E. and Phillips, R. S., Functional analysis and semi-groups (AMS, Providence, R.I., 1957).Google Scholar
14. Miller, R. K., Linear Volterra integrodifferential equations as semi-groups, Funkcial, Ekvac. 17 (1974), 39–55.Google Scholar
15. Vinter, R. B., On a problem of Zabczyk concerning semigroups generated by operators with non-local boundary conditions, Publication 77/8 (1977), Department of Computing and Control, Imperial College of Science and Technology, London (England).Google Scholar
16. Vinter, R. B., Semigroups on product spaces with applications to initial value problems with non-local boundary conditions, in “Control of distributed parameter systems”, 91–98, (Pergamon Press, Oxford, England, 1978).Google Scholar