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6
Для T = 0.01 и T = 0.001 оба метода не дали точных результатов, что согласуется с тем, что в подобных задачах существует некоторое минимальное время, за которое возможно реализовать управление [5].
Из результатов расчетов видно, что предложенный в данной работе улучшенный метод почти всегда достигает заданной точности за меньшее время и на меньшем отрезке [0, S], чем оригинальный метод. Данное улучшение заметно лишь для больших T и для довольно больших At, что согласуется с тем, что введенные поправки пропорциональны At, и при малых At они будут тоже очень малы. К тому же для некоторых больших значений At оригинальный метод вовсе не сходится, а улучшенный метод достигает требуемой точности.
Выводы
Были выведены новые формулы для расчета оптимальных управлений в квантовой системе в виде поправок к формулам из оригинального метода D-MORPH [1-5]. На численном эксперименте из области квантовых вычислений было продемонстрировано, что новые формулы действительно дают ускорение вычислений по сравнению с оригинальным методом, т. е. позволяют достичь заданной точности реализации желаемого оператора эволюции за меньшее число шагов метода ode45 и за меньшее время, даже при включении только одного дополнительного
поправочного коэффициента, пропорционального At. Таким образом, включение в метод дополнительной информации о гамильтониане системы (коммутаторы) положительно сказалось на точности результатов и на скорости их получения, что является преимуществом при проведении расчетов на компьютерах со средней производительностью.
Литература
1. Moore, K.W. Search complexity and resource scaling for the quantum optimal control of unitary transformations / R. Chakrabarti, G. Riviello, H. Rabitz // Phys. Rev. A. 2011. Vol. 83(1).
2. Moore, K.W. Exploring constrained quantum control landscapes / H. Rabitz // The Journal of Chemical Physics. 2012. Vol. 137(13).
3. Moore Tibbetts, K. Exploring the trade-off between fidelity and time optimal control of quantum unitary transformations / C. Brif, M.D. Grace, A. Donovan, et. al. // Phys. Rev. A. 2012. Vol. 86(6).
4. Riviello, G. Searching for quantum optimal controls in the presence of singular critical points / C. Brif, R. Long, R. Wu, K. Moore Tibbetts, T. Ho, H. Rabitz // Phys. Rev. A. 2014. Vol. 90(1).
5. Riviello, G. Searching for quantum optimal controls under sever constraints / K. Moore Tibbetts, C. Brif, R. Long, et. al. // Phys. Rev. A. 2015. Vol. 91(4).
DOI: 10.18454/IRJ.2016.48.139 Перов А.И.1, Каверина В.К.2
1 Доктор физико-математических наук, Воронежский государственный университет 2Кандидат физико-математических наук, Воронежский государственный архитектурно-строительный университет Работа выполнена при поддержке гранта РФФИ№16-01-00197 НЕЛИНЕЙНЫЕ ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ ВЫСШИХ ПОРЯДКОВ
Аннотация
В статье изучается вопрос существования периодических решений у возмущенного дифференциального уравнения П -го порядка. Полученные теоремы навеяны проблемой В.И. Зубова и связаны с понятиями асимптотической устойчивости в целом и устойчивости по Дирихле. Доказательства этих теорем носят топологический характер и опираются на понятие степени отображения. Рассматривается периодически возмущенная автономная система обыкновенных дифференциальных уравнений, указываются достаточные условия, при которых система имеет периодическое решение.
Ключевые слова: нелинейные скалярные дифференциальные уравнения высшего порядка, периодические решения (свободные и вынужденные колебания), топологическая степень отображения, метод направляющих функций.
Perov A.I.1, Kaverina V.K.2
1PhD in Physics and Mathematics,
Voronezh State University,
2PhD in Physics and Mathematics, Voronezh State University of Architecture and Civil Engineering NON-LINEAR DIFFERENTIAL EQUATIONS OF HIGHER ORDER
Abstract
In this article we study the problem of the existence ofperiodic solutions of the perturbed differential equations of higher order. These theorems were inspired by the work of V.I. Zubov and associated with such notions as asymptotically stability in the large and stability in the Dirichlet's sense in the large. Proofs of these theorems have a topological sense. We consider periodically perturbed autonomous system of the ordinary differential equations and denote the sufficient conditions of the existence of periodic solution.
Keywords: nonlinear scalar differential equations of higher order, periodic solutions (free and forced oscillations), topological degree of mapping, method of guiding functions.
Let n be some positive integer. We consider the ordinary nonlinear differential scalar equations of order n
r(K)_/Yrj. v(w—1) ^
~ J )
where f (x, x2,..., xn ):R x...x R (n - pa3R has the following properties: 10. f (0,0,...,0) = 0, 20. f (x,0,...,0) * 0, if x * 0, 30 f (x , x2 , ..., xn ) is a continuous function,
40. f (X
, x2,..., xn) is a locally Lipschitz function
n
|f(xp X2,..., xn ) - f(yx, y2,..., yn )| <^lAXj - yi , (2)
/=1
X, y £ K, where K is any bounded set in the phase space Rn, l = l (K), i = 1,..., n. One equation of order 11 (1) is equivalent to the system of 11 first-order equations
X = x2, x2 =x3, ...,xn_x =xn, xn = f(xl,x2,...,xn), (3)
or
x = F(x), (4)
where x £ Rn, the mapping F(x): RM ^ Rn be defined in a natural way. Condition 10 and 20 mean, that F(0) = 0 and F(x) ^ 0 for x ^ 0, i.e. the origin is a equilibrium state and this equilibrium state is single. Condition 30 tells us that the mapping F(x): Rn ^ Rn is continuous, and condition 40 means that it is a locally Lipschitz mapping.
As the vector field F(x): Rn ^ Rn is continuous, we can tell about ind(F, 0) Kronecker-Poincare index of singular
points 0 [5, p. 89]. If Q is any nonempty bounded open set in Rn, the singular points 0 does not lie in its boundary eQ
(i.e. 0 £ eQ), and so F(x) ^ 0 if x £ eQ, then degree of mapping is defined deg(F, eQ) = deg(F, Q) . Let us note
that degree of mapping can take on only the values of -1, 0, +1 in the discussed case. Together with equation (1) we consider perturbed equation
X^=f(x,jc,...,x^)+h(t), (5)
where h(t): R ^ R is a continuous CO -periodic function
h(t + c) = h(t),
where CO> 0 - some positive number.
This equation is equivalent to periodic perturbed system
\ =X2> = "*3> •••'•^M- 1 = Xin = fiX\ 5 ''''' Xn ) + ),
or
x = i7(x)+h(/), where h(t) is a continuous CD -periodic vector function
h(/ + c) = h(/) ,
h(t) = col (0,..., 0, h(t)): R ^ Rn.
Let be satisfied two more conditions:
50. Any solution x(t) of the equation (1) is defined for all t :0 < t <+(&. (It is referred to the system (4).)
60. Differential equation (1) is asymptotically stable in the large, i.e. its zero solution x(t) = 0 is stable and for any other solution x(t)
x(t),x(t),...,xr*(t)->0 if t->00.
(It is referred to the system (4).)
T h e o r e m 1. Let us consider differential equation (1) under conditions 10 - 60. Then periodic perturbed differential equation (5) with any continuous CO-periodic function h(t) has at least one CO-periodic solution x(t)
x(t + CO) = x(t),
(forced oscillation), if this condition is satisfied
f (R,0,...,0) = R.
This theorem was inspired by the work of V.I. Zubov [1], [3, c. 220].
Let follow to study differential equation (1). Let be satisfied two more conditions:
70. Any solution x(t) of the equation (1) is defined for all t, < t < .
80. Any solution x(t) is bounded with its derivatives up to n — 1 order
|x(t)| < Ci, |x(t)| < С2,..., x(n—^(t)
< с ,, — ад< t . (6)
и—1' v '
Under condition 70 system (3) (or (4)) define a dynamical system in R n [7, p. 272].
Condition 80 means that equation (1) is stable in the Dirichlet's sense in the large, i.e. its zero solution is stable in the Dirichlet's sense and condition (6) is satisfied for any other solution.
If the bounded solution of the equation (1) is periodic x(t + <) = x(t), then it is a free oscillation. Since Lipschitz
condition (2) is satisfied near the periodic solution, by York theorem [6, p. 20], [9] period <7 > 0 is bounded below some number <70, depending on Lipschitz constants /, l2,..., ln .
90. Suppose that periods of all natural oscillations of the system (3) (see also (4)) are bounded below
0 <<<<.
T h e o r e m 2. Let us consider differential equation (1) under conditions 10 - 40 and 70 - 90. Then periodic perturbed differential equation (5) with any continuous CO-periodic function h(t) has at least one CO-periodic solution x(t)
x(t + C) = x(t) ,
(forced oscillation), if the period CO is small enough
0 < c < <.
and
f (R, 0,...,0) = R.
Checking the stability (one way or the other) is the most difficult thing in the theorem 1 and 2.
This checking can be easy if there are Lyapunov functions u(x) :Rn ^ R with defined properties [2,c. 344]:
10. u(0) = 0,
20. u(x) > 0 for x * 0, x e Rn,
30. u(x) is a continuous function,
40. u(x) is a continuously differentiate function,
grad u(x) = col jMp,M^,...,|,
50. grad u(0) = 0,
60. grad u(x) ^ 0 for x * 0, x e Rn.
In theorem 1 we suppose that the condition
(grad u(x), F(x)) < 0 if x * 0 (7)
is satisfied.
In theorem 2 we suppose that the condition
(grad u(x), F(x)) = 0 if x * 0 (8)
is satisfied.
70. To ensure the stability in the large we need to demand that
u(x) if lixi (9)
In case (8) we have a compact dynamic system on the level surfaces
u(x) = c > 0, x e Rn.
n — 1 dimension. In this case the space Rn is even-dimensional : n = 2k (k - a positive integer). Standard application is
x + f(x)x + g(x) = h(t)
Lienart equation to theorem 1 and
x + g(x) = h(t)
- to theorem 2.
Different nonlinear differential equations of order 3 and 4 to which the developed theory can be apply are discussed in the monograph [11].
Now we study the more general case. Let us consider the autonomous system of ODE in vector form
x = G(x), xeR", (10)
where G(x): Rn ^ Rn is a locally Lipschitz mapping for which G(0) = 0 and G(x) * 0 for x * 0. We suppose that zero stationary solution of system (10) x(t) = 0 is asymptotically stable in the large.
The problem is to prove or disprove following statement: in order that non-autonomous system of ODE
x = G(x) + h(i), (11)
with any continuous vector function h(t):R ^ R", h(t + a) = h(t) has a periodic solution x(t + a) = x(t) it is necessary and sufficient that the mapping G is a mapping onto
G(R" )=R". (12)
We prove the sufficiency of these condition if we demand that the condition of coercitivity of mapping G :
||G(x)|| if ||x|| (13)
is satisfied in addition to condition (12).
As zero solution of system (10) is asymptotically stable in the large then by Krasovskii-Barbashin theorem [12, p. 37] there is a continuously differentiate function u(x):R" ^ R which is satisfied properties 10 - 60 and conditions (7), (9)
(where F(x) = G(x)). In this case the topological degree of gradient mapping grad u: R" ^ R" on the boundary of any ball containing zero of the space as an internal point is
deg(grad u(x), S") = 1
[4, p. 111]. From condition (7) it follows that vector fields grad u(x) and -G(x) are homotopy on the boundary of the ball S" and that's why
deg(-G (x), S") = (-1)". (14)
We suppose that any solution x(t,t0,xo) of the system (11) with initial condition x(t0) = x0 is defined for 0 < t <+to .
Let us show that perturbed system (11) has at least one periodic solution x(t + a) = x(t) for any function h(t + a) = h(t) . It is well known that the initial value with t = 0 of the CO -periodic solution is a fixed point of the Poincare mapping p(x) :R" ^R" where p(x) = x(t, t0,x0), i.e. x = p(x) .
Let k = max||h(/)||, 0 < t < a. Because of coercitivity property (13) of mapping G(x) we can indicate such r that
||G(x)|| > k for ||x|| = r .
By Rouché theorem using (14) we get
deg(G(x)+h(t), S") = (-1)", 0 < t <c. (15)
Let us assume q(x) = p(x) — x, q(x): R" ^ R" If q(() = 0 for some ( GdS" then perturbed system (11) has CO -
periodic solution x(t) = x(t,0,() . Let q(() ^ 0 for ( g8S" , i.e. vector field q(x) on 8S" is nonsingular. The main part of the proof is to prove formula
deg(q (x), S") = (—1)". (16)
By Kronecker theorem [13, p.162] it follows that mapping q(x) has zero in S" . Let it will be point then x(t) = x(t, 0, () will be CO -periodic solution of the perturbed system (11) and our statement is proved.
Let us note that method of guiding functions also serve for the proof of the existence of the periodic and bounded solutions of the nonlinear system of differential equations and based on the topological notion of degree of mapping, guiding functions resemble Lyapunov functions by their properties [4], [8], [10].
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