Generalized differential equations in the space of regulated functions (boundary value problems and controllability)

Milan Tvrdý

Mathematica Bohemica (1991)

  • Volume: 116, Issue: 3, page 225-244
  • ISSN: 0862-7959

Abstract

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Boundary value problems for generalized linear differential equations and the corresponding controllability problems are dealt with. The adjoint problems are introduced in such a way that the usual duality theorems are valid. As a special case the interface boundary value problems are included. In contrast to the earlier papers by the author the right-hand side of the generalized differential equations as well as the solutions of this equation can be in general regulated functions (not necessarily of bounded variation). Similar problems in the space of regulated functions were treated e.g. by Ch. S. Hönig, L. Fichmann and L. Barbanti, who made use of the interior (Dushnik) integral. In this paper the integral is the Perron-Stieltjes (Kurzweil) integral.

How to cite

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Tvrdý, Milan. "Generalized differential equations in the space of regulated functions (boundary value problems and controllability)." Mathematica Bohemica 116.3 (1991): 225-244. <http://eudml.org/doc/29328>.

@article{Tvrdý1991,
abstract = {Boundary value problems for generalized linear differential equations and the corresponding controllability problems are dealt with. The adjoint problems are introduced in such a way that the usual duality theorems are valid. As a special case the interface boundary value problems are included. In contrast to the earlier papers by the author the right-hand side of the generalized differential equations as well as the solutions of this equation can be in general regulated functions (not necessarily of bounded variation). Similar problems in the space of regulated functions were treated e.g. by Ch. S. Hönig, L. Fichmann and L. Barbanti, who made use of the interior (Dushnik) integral. In this paper the integral is the Perron-Stieltjes (Kurzweil) integral.},
author = {Tvrdý, Milan},
journal = {Mathematica Bohemica},
keywords = {generalized differential equations; adjoint operators; existence and uniqueness theorems; controllability; regulated function; boundary value problem; adjoint problem; interface problem; Peron-Stieltjes integral; Kurzweil integral; generalized differential equations; adjoint operators; existence and uniqueness theorems; controllability},
language = {eng},
number = {3},
pages = {225-244},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized differential equations in the space of regulated functions (boundary value problems and controllability)},
url = {http://eudml.org/doc/29328},
volume = {116},
year = {1991},
}

TY - JOUR
AU - Tvrdý, Milan
TI - Generalized differential equations in the space of regulated functions (boundary value problems and controllability)
JO - Mathematica Bohemica
PY - 1991
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 116
IS - 3
SP - 225
EP - 244
AB - Boundary value problems for generalized linear differential equations and the corresponding controllability problems are dealt with. The adjoint problems are introduced in such a way that the usual duality theorems are valid. As a special case the interface boundary value problems are included. In contrast to the earlier papers by the author the right-hand side of the generalized differential equations as well as the solutions of this equation can be in general regulated functions (not necessarily of bounded variation). Similar problems in the space of regulated functions were treated e.g. by Ch. S. Hönig, L. Fichmann and L. Barbanti, who made use of the interior (Dushnik) integral. In this paper the integral is the Perron-Stieltjes (Kurzweil) integral.
LA - eng
KW - generalized differential equations; adjoint operators; existence and uniqueness theorems; controllability; regulated function; boundary value problem; adjoint problem; interface problem; Peron-Stieltjes integral; Kurzweil integral; generalized differential equations; adjoint operators; existence and uniqueness theorems; controllability
UR - http://eudml.org/doc/29328
ER -

References

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  1. Aumann G., Reelle Funktionen, Springer-Verlag, Berlin-Heidelberg-New York, 1969. (1969) Zbl0181.05801MR0061652
  2. Barbanti L., Linear Volterra-Stieltjes integral equations and control, Lecture Notes in Mathematics 1017, Springer-Verlag (1983), 67-72. (1983) Zbl0537.49021MR0726569
  3. Bryan R. N, A nonhomogeneous linear differential system with interface conditions and their adjoints, Proc. A.M.S. 22 (1969), 270-276. (1969) 
  4. Conti R., 10.1016/0022-0396(68)90045-4, Journ. Diff. Eq. 4 (1968), 4-11. (1968) Zbl0157.14104MR0218642DOI10.1016/0022-0396(68)90045-4
  5. Fichmann L., Volterra-Stieltjes Integral Equations and Equations of the Neutral Type, (in Portuguese), Thesis. University of Sao Paulo (1984). (1984) 
  6. Fraňková D., Regulated functions, Math. Boh. (Časopis pěst. mat.) 116 (1991), 20-59. (1991) MR1100424
  7. Halanay A., Optimal control of periodic solutions, Rev. Roum. Math. Pures et Appl. 14 (1974), 3-16. (1974) Zbl0276.49012MR0344968
  8. Hildebrandt T. H., Introduction to the Theory of Integration, Academic Press, New York-London, 1963. (1963) Zbl0112.28302MR0154957
  9. Hönig Ch. S., The adjoint equation of a linear Volterra-Stieltjes integral equation with a linear constraint, Lecture Notes in Mathematics 957, Springer-Verlag (1982), 118-125. (1982) MR0679143
  10. Hönig Ch. S., Volterra-Stieltjes integral equations, Functional Differential Equations and Bifurcation, Proceedings of the Sao Carlos Conference 1979 (Lecture Notes in Mathematics 799). Spгinger-Verlag, 173-216. (1979) MR0585488
  11. Kaltenborn H. S., Linear functional operations on functions having discontinuities of the first kind, Bulletin A.M.S. (1934), 702-708. (1934) Zbl0010.16905MR1562957
  12. Kurzweil J., Generalized ordinary differential equations and continuous dependence on a parameter, Czech. Math. J. 7 (82) (1957), 418-449. (1957) Zbl0090.30002MR0111875
  13. Kurzweil J., Nichtabsolute konvergente Integrale, BSB B. G. Teubner Verlagsgesselschaft, Leipzig, 1980. (1980) MR0597703
  14. Lando Yu. K., Controllable integro-differential operators, (in Russian). Diff. Uravn. 9 (1973), 2227-2230. (1973) MR0333772
  15. Marchiò C, (M, N, F)-controllabilità completa, Questioni di controllabilità. Istituto U. Dini, Firenze (1973/2), 14-26. (1973) 
  16. Rolewicz S., Functional Analysis and Control Analysis (Lineaг Systems), PWN-Polish Scientific Publishers and D. Reidel, Warszawa and Dordrecht, 1987. (1987) 
  17. Rudin W., Functional Analysis, МcGraw-Hill, New York, 1973. (1973) Zbl0253.46001MR1157815
  18. Russell D. L., Mathematics of Finite-Dimensional Control Systems (Theory and Design), (LNPAM 43). M. Dekker, New York and Basel, 1979. (1979) Zbl0408.93002MR0531035
  19. Saks S., Theory of the Integral, Monografie Matematyczne, Warszawa-Lwów, 1937. (1937) Zbl0017.30004
  20. Schwabik Š., Generalized Differential Equations (Fundamental Results), Rozpravy ČSAV, řada MPV, 95 (6), Academia, Praha, 1985. (1985) Zbl0594.34002MR0823224
  21. Schwabik Š., On the relation between Young's and Kurzweil's concept of Stieltjes integral, Časopis pěst. mat. 98 (1973), 237-251. (1973) Zbl0266.26006MR0322113
  22. Schwabik Š., Differential equations with interface conditions, Časopis pěst. mat. 105 (1980), 391-408. (1980) Zbl0453.34008MR0597916
  23. Schwabik Š., Tvrdý M., Boundary value problems for generalized linear differential equations, Czech. Math. J. 29 (104) (1974), 451-477. (1974) MR0536070
  24. Schwabik Š., Tvrdý M., Vejvoda O., Differential and Integral Equations: Boundary Value Problems and Adjoints, Academia and D. Reidel, Praha and Dordrecht, 1979. (1979) MR0542283
  25. Tvrdý M., Boundary value problems for linear generalized differential equations and their adjoints, Czech. Math. J. 23 (98) (1973), 183-217. (1973) MR0320417
  26. Tvrdý M., Boundary value problems for generalized linear integrodifferential equations with left-continuous solutions, Časopis pěst. mat. 99 (1974), 147-157. (1974) MR0405041
  27. Tvrdý M., Regulated functions and the Perron-Stieltjes integral, Časopis pěst. mat. 114 (1989), 187-209. (1989) MR1063765
  28. Ward A. J., 10.1007/BF01180442, Math. Zeitschr. 41 (1936), 578-604. (1936) Zbl0014.39702MR1545641DOI10.1007/BF01180442
  29. Wexler D., On boundary value problems for an ordinary linear differential system, Аnn. Mat. pura ed appl. 80 (1968), 12З-1З4. (1968) MR0247242
  30. Zettl A., Аdjoint and self-adjoint boundary value problems with interface conditions, Јourn. Аppl. Mat. 16 (1968), 851-859. (1968) MR0234049

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