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ЛИТЕРАТУРА
1. Жуковский Е.С. Непрерывная зависимость от параметров решений уравнений Вольтерра // Матем. сб. М., 2006. Т. 197, №10. С. 33-56.
2. Жуковский Е.С. Alves M.J. О единственности решений уравнений Volterra // Вестник Тамбовского университета. Серия: Естественные и технические науки. Тамбов, 2006. Т. 11. Вып. 3. С. 262-267.
3. Вайнико Г.М. Регулярная сходимость операторов и приближенное решение уравнений // Математический анализ (Итоги науки и техники. ВИНИТИ АН СССР). М., 1979. Т. 16. С. 5-53.
Жуковский Евгений Семенович Тамбовский государственный ун-т Россия, Тамбов e-mail: [email protected]
Manuel Joaquim Alves Eduardo Mondlane University Maputo, Mozambique e-mail: [email protected]
Поступила в редакцию 30 апреля 2007 г.
LYAPUNOV REDUCIBILITY AND STABILIZATION OF NONSTATIONARY
SYSTEMS WITH OBSERVER 1
© V. A. Zaitsev
Consider a linear control system with observer
x = A(t)x + B(t)u, y = C(t)x, (t,x,u,y) G M1+ra+m+fc, (1)
with measurable bounded coefficients. Let us construct the estimator x of the state x for system (!);
x = A(t)x + V(t)(y(t) — C(t)x) + B(t)u, x G M™.
Let the control law be u = U(t)x. Then closed-loop 2n-dimensional system is
f A(t) B(t)U (t) \ (x
%) = (t)c(t) A(t) + B(t)u(t) - v(t)c(t)) •
Let X = x — x. If we replace (x, X) by (x, X) in (2), we obtain
X \ = / A(t) + B (t)U (t) —B (t)U (t) \ /x
x) \ 0 A(t) — V (t)C (t) J l^X
(2)
(3)
Theorem 1. Suppose system, (1) is uniformly completely controllable and uniformly completely observable (when u = 0). Then for any numbers X, ^ £ R there exist measurable bounded control functions U(t),V(t) and a bounded piecewise continuous matrix S(t) such that
^The work is supported by RFBR (grant №06-01-00258).
system (3) (and system (2)) with these control functions is reducible by som,e Lyapunov transformation to the system
^x\ = f A(t) + XI S (t) \ (x
xj V 0 A(t) + ¿I) \x) ' (4)
Corollary 1. Suppose system (1) is uniformly completely controllable and uniformly
completely observable (when u = 0). Then the closed-loop system is uniformly stabilizable.
Theorem 2. Suppose system (1) is uniformly completely controllable and uniformly completely observable (when u = 0). Let the coefficients of system (1) be periodic. Then for any numbers X, ¡1 £ R there exist periodic measurable bounded control functions U(t),V(t) and a bounded periodic matrix S(t) such that system (3) (and system (2)) with these control functions is reducible by some periodic Lyapunov transformation to the system (4).
C o r o l l a r y 2. Suppose system (1) is uniformly completely controllable and uniformly
completely observable (when u = 0). Let the coefficients of system (1) be periodic. Then for any v > 0 there exist periodic measurable bounded control functions U(t),V(t) such that system (3) (and system (2)) with these control functions is reducible by some periodic Lyapunov transformation to the system with the constant matrix P whose eigenvalues Xi satisfy condition Re Xi < —v for all i = 1,, 2n.
Zaitsev Vasily Alexandrovich Udmurtia State University Russia, Izhevsk e-mail: [email protected]
Поступила в редакцию 23 апреля 2007 г.
ОБ УСТОЙЧИВОМ ЧИСЛЕННОМ МЕТОДЕ РЕШЕНИЯ ЗАДАЧИ СИНЬОРИНИ С МОДИФИЦИРОВАННЫМ ФУНКЦИОНАЛОМ
ЛАГРАНЖА
© А. Я. Золотухин
Вариационная задача Синьорини имеет вид [1]
J(и) = 2 У \’Vu\2dü — J fudü------------min (1)
п п
на выпуклом замкнутом множестве
G = {и £ W),(ii) : yu ^ ф п. в. на Г}, (2)
где Q С R2 — ограниченная область с достаточно регулярной границей Г, f £ L2(0,) и ф £ L2(Г) — заданные функции, yu £ W1“2^) есть след функции и £ W2(Q) на Г.
