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

Tvrdý, Milan

Mathematica Bohemica, Tome 116 (1991) no. 3, pp. 225-244

Abstract (VO)

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.

Détail

DOI : 10.21136/MB.1991.126174

Classification : 26A45, 34A34, 34B10, 34B15, 34H05, 93B05
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

Bibliographie

[Au] Aumann G.: Reelle Funktionen. Springer-Verlag, Berlin-Heidelberg-New York, 1969. | MR | Zbl

[Ba] Barbanti L.: Linear Volterra-Stieltjes integral equations and control. Lecture Notes in Mathematics 1017, Springer-Verlag (1983), 67-72. | MR | Zbl

[Br] Bryan R. N: A nonhomogeneous linear differential system with interface conditions and their adjoints. Proc. A.M.S. 22 (1969), 270-276.

[Co] Conti R.: On ordinary differential equations with interface conditions. Journ. Diff. Eq. 4 (1968), 4-11. | DOI | MR | Zbl

[Fi] Fichmann L.: Volterra-Stieltjes Integral Equations and Equations of the Neutral Type. (in Portuguese), Thesis. University of Sao Paulo (1984).

[Fra] Fraňková D.: Regulated functions. Math. Boh. (Časopis pěst. mat.) 116 (1991), 20-59. | MR

[Ha] Halanay A.: Optimal control of periodic solutions. Rev. Roum. Math. Pures et Appl. 14 (1974), 3-16. | MR | Zbl

[Hi] Hildebrandt T. H.: Introduction to the Theory of Integration. Academic Press, New York-London, 1963. | MR | Zbl

[Hö1] 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. | MR

[Hö2] 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. | MR

[Ka] Kaltenborn H. S.: Linear functional operations on functions having discontinuities of the first kind. Bulletin A.M.S. (1934), 702-708. | MR | Zbl

[Ku1] Kurzweil J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (82) (1957), 418-449. | MR | Zbl

[Ku2] Kurzweil J.: Nichtabsolute konvergente Integrale. BSB B. G. Teubner Verlagsgesselschaft, Leipzig, 1980. | MR

[La] Lando Yu. K.: Controllable integro-differential operators. (in Russian). Diff. Uravn. 9 (1973), 2227-2230. | MR

[Ma] Marchiò C: (M, N, F)-controllabilità completa. Questioni di controllabilità. Istituto U. Dini, Firenze (1973/2), 14-26.

[Rol] Rolewicz S.: Functional Analysis and Control Analysis (Lineaг Systems). PWN-Polish Scientific Publishers and D. Reidel, Warszawa and Dordrecht, 1987.

[Rud] Rudin W.: Functional Analysis. МcGraw-Hill, New York, 1973. | MR | Zbl

[Rus] Russell D. L.: Mathematics of Finite-Dimensional Control Systems (Theory and Design). (LNPAM 43). M. Dekker, New York and Basel, 1979. | MR | Zbl

[Sa] Saks S.: Theory of the Integral. Monografie Matematyczne, Warszawa-Lwów, 1937. | Zbl

[Sch1] Schwabik Š.: Generalized Differential Equations (Fundamental Results). Rozpravy ČSAV, řada MPV, 95 (6), Academia, Praha, 1985. | MR | Zbl

[Sch2] Schwabik Š.: On the relation between Young's and Kurzweil's concept of Stieltjes integral. Časopis pěst. mat. 98 (1973), 237-251. | MR | Zbl

[SchЗ] Schwabik Š.: Differential equations with interface conditions. Časopis pěst. mat. 105 (1980), 391-408. | MR | Zbl

[ST] Schwabik Š., Tvrdý M.: Boundary value problems for generalized linear differential equations. Czech. Math. J. 29 (104) (1974), 451-477. | MR

[STV] Schwabik Š., Tvrdý M., Vejvoda O.: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia and D. Reidel, Praha and Dordrecht, 1979. | MR

[T1] Tvrdý M.: Boundary value problems for linear generalized differential equations and their adjoints. Czech. Math. J. 23 (98) (1973), 183-217. | MR

[T2] Tvrdý M.: Boundary value problems for generalized linear integrodifferential equations with left-continuous solutions. Časopis pěst. mat. 99 (1974), 147-157. | MR

[TЗ] Tvrdý M.: Regulated functions and the Perron-Stieltjes integral. Časopis pěst. mat. 114 (1989), 187-209. | MR

[Wa] Ward A. J.: The Perron-Stieltjes integral. Math. Zeitschr. 41 (1936), 578-604. | DOI | MR | Zbl

[We] Wexler D.: On boundary value problems for an ordinary linear differential system. Аnn. Mat. pura ed appl. 80 (1968), 12З-1З4. | MR

[Ze] Zettl A.: Аdjoint and self-adjoint boundary value problems with interface conditions. Јourn. Аppl. Mat. 16 (1968), 851-859. | MR