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SOME CONTROL PROBLEMS FOR LINEAR ABSTRACT FUNCTIONAL-DIFFERENTIAL SYSTEMS 1
© V. P. Maksimov
The linear abstract functional differential equation [1] is the equation
Lx = f, (1)
where L : D ^ B is a linear bounded operator, D and B are Banach spaces such that D is isomorphic to the direct product B x Rn. Let us denote by J = {A, Y} : B x Rn ^ D an isomorphism and let J-1 = [5, rj. Suppose that the so called principal boundary value problem Lx = f,rx = a is uniquely solvable for any f £ B and a £ Rn. In such a situation, the solution of the problem has the form x = Gf + Xa, where G is the Green operator, X the fundamental vector. Consider the control problem
Lx = Fu + f, (2)
rx = a, lx = 3, (3)
with a linear bounded operator F : H ^ B, a linear bounded vector functional l = [li,...,lmj : D ^ Rm (it defines the aim of controlling), and the control u from a Hilbert space, H. The equality XiU = liGFu defines the linear bounded functional Xi : H ^ R. Let us denote by the same symbol Xi the element of H that generates the functional Xi, that is, XiU = (Xi,u), where ( , ) stands for the inner product in H. Necessary and sufficient conditions for the solvability of the control problem (2),(3) are presented in the talk. Those are formulated in terms of the matrix r = {(Xi,Xj)}i,j=i,.,m.
Problem (2), (3) covers a wide class of control problems for ordinary differential systems, differential delay systems, some singular and impulsive systems. In the talk, some examples of the above problems are considered, and questions of developing contemporary computer-assisted techniques for the study of the control problems are discussed. The key idea of the computer-aided approach to the study of the problem (2), (3) with use of Computer Algebra Systems is based on the fact that the property of controllability of system (2) retains under small disturbance of the parameters. Thus effective tests of the controllability can be formulated in terms of the elements of a matrix To that approximates r with a precision high enough. This requires a high precision of the approximation to X,G,F,l within classes of so called «computable» functions and operators [1]. The questions of such approximations are discussed as well. Some applied control problems being reducible to the form (2), (3) are presented.
1The work is supported by RFBR (grant №06-01-00744).
REFERENCES
1. Azbelev N.V., Maksimov V.P., and Rakhmatullina L.F. Introduction to the Theory of Functional Differential Equations: Methods and Applications // New York-Cairo: Hindawi Publishing, 2007.
Maksimov Vladimir Petrovich Perm State University Russia, Perm
e-mail: [email protected]
Поступила в редакцию 5 мая 2007 г.
ИНВАРИАНТЫ АФФИННОЙ ГРУППЫ ПРЯМОЙ В ПРОСТРАНСТВЕ МНОГОЧЛЕНОВ 1
© Н. А. Малашонок
Рассматриваются орбиты и инварианты аффинной группы прямой О = {х ^ ах + в}, действующей сопряжениями в пространстве многочленов / (х).
Пусть Уп — пространство многочленов / (х) степени ^ п над полем М :
/ (х) = а0 + а1х + ... + апхп, (1)
где ао,а1 ,...,ап € М и переменная х пробегает М. Пространство Уп имеет размерность п + 1. Обозначим Уп+, V—, У,° подмножества Уп, состоящие из многочленов / (х) с ап > 0, ап < 0, ап = 0, соответственно. Подмножество У,° может быть естественным образом отождествлено с Уп_1. Таким образом, мы имеем фильтрацию
М = Уо С У1 С У2 С ... С Уп С .... (2)
Пусть О — группа аффинных преобразований ф прямой М :
х ^ ф(х) = ах + в, (3)
где а, в € М, а > 0. Она действует в пространстве Уп многочленов /(х) следующим образом:
Т(ф)/ = <р_1 о / о <р,
или
1в (Т(ф)/)(х) = -/(ах + в) - -. аа
хРабота выполнена при поддержке РФФИ (проект №05-01-00074а), научной программы «Развитие Научного Потенциала Высшей Школы» РНП. 2.1.1.351 и темплана № 1.2.02.
