algorithms for searching for hidden oscillations in the aizerman and kalman problems

ISSN 1064�5624, Doklady Mathematics, 2011, Vol. 84, No. 1, pp. 475–481. © Pleiades Publishing, Ltd., 2011.Original Russian Text © G.A. Leonov, N.V. Kuznetsov, 2011, published in Doklady Akademii Nauk, 2011, Vol. 439, No. 2, pp. 167–173.

475

Consider the following system with one scalar non�linearity:

(1)

where P is a constant n × n matrix, q and r are constantn�vectors, * denotes transposition, ψ(σ) is a piecewisecontinuous scalar function, and ψ(0) = 0. The solu�tions of system (1) are understood in the sense of Fil�ippov [1]. We assume the complete controllability ofthe pair (P, q) and the complete observability of thepair (P, r).

In 1949, Aizerman [2] posed the following prob�lem: Suppose that all linear systems of the form (1)with

(2)

are asymptotically stable. Is system (1) with any non�linearity ψ(σ) satisfying the condition

absolutely stable (i.e., the trivial solution of system (1)is asymptotically stable and any solution tends to zeroas t → +∞)?

In 1957, Kalman [3] modified Aizerman’s problemas follows: Suppose that ψ(σ) is a piecewise differen�tiable function (i.e., a function having finitely manypoints of discontinuity of the first kind on any intervaland differentiable on the intervals of continuity) and

at all points of differentiability. Is system (1) absolutelystable provided that all linear systems (1) with ψ(σ) =μσ for μ ∈ (μ1, μ2) are asymptotically stable?

The current state�of�the�art in the Aizerman andKalman problems is reviewed in [7].

This paper develops the ideas of [4–7]. We presenta further development of the justification of the har�

dxdt�� Px qψ r*x( ), x+ �

n,∈=

ψ σ( ) μσ, μ μ1 μ2,( ),∈=

μ1ψ σ( )

σ�� μ2, σ∀ 0,≠< <

μ1 ψ' σ( ) μ2< <

monic balance method (describing function method)and the construction of counterexamples to Kalman’sproblem.

1. AN ANALYTICAL CRITERION FOR THE EXISTENCE OF HIDDEN

OSCILLATIONS

Let us prove an analytical criterion for the existenceof periodic solutions, which makes it possible to detectthe existence of hidden oscillations, a basin of attrac�tion of which does not contain neighborhoods of equi�libria.

Consider system (1) of the special form

(3)

where y1 and y2 are scalar quantities, y3 is an (n – 2)�vector, b3 and c3 are (n – 2)�vectors, b1 and b2 are realnumbers, A3 is an (n – 2) × (n – 2) matrix all of whoseeigenvalues have negative real parts, and the nonlin�earity ϕ0(σ) has the form

(4)

Here, ϕ1(σ) and ϕ2(σ) are piecewise differentiablefunctions (the νi are discontinuity points) satisfyingthe conditions

(5)

where μ > 0 and L are some numbers. In what follows,without loss of generality, we assume that

(6)

These conditions hold for, e.g., the nonlinearity

y· 1 ω0y2– b1ϕ0 y1 c3*y3+( ),+=

y· 2 ω0y1 b2ϕ0 y1 c3*y3+( ),+=

y· 3 A3y3 b3ϕ0 y1 c3*y3+( ),+=

ϕ0 σ( )ϕ1 σ( ) σ∀ ε≤

ε3ϕ2 σ( ) σ∀ ε.>⎩⎨⎧

=

ϕ1 σ( ) μ σ ,≤

b2 c3*b3 b1+( )ϕ1 σ( ) b1ω0σ+( )ϕ1 σ( ) σd

ε–

ε

∫

= Lε3 O ε4( ),+

ϕ2 σ( ) 0 σ∀ ε– ε,[ ].∈=

Algorithms for Searching for Hidden Oscillationsin the Aizerman and Kalman Problems

Corresponding Member of the RAS G. A. Leonova and N. V. Kuznetsova, b

Received March 31, 2011

DOI: 10.1134/S1064562411040120

a Mathematics and Mechanics Faculty, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504 Russiab University of Jyväskylä, Finlande�mail: [email protected], [email protected]

MATHEMATICS

476

DOKLADY MATHEMATICS Vol. 84 No. 1 2011

LEONOV, KUZNETSOV

(7)

where M is a positive number.Consider the following set in the phase space of the

nonlinear system (3):

(8)

where the a1, 2 are positive numbers and the numberD is found by using the following lemma.

Lemma 1. Solutions of system (3) with initial datafrom Ω can be represented as

(9)

moreover, there exist numbers D1 ≥ D > 0 such that if

for sufficiently small ε > 0, then

(10)

and

(11)

Consider the Poincaré map F of the set Ω for thetrajectories of system (3), for which

(12)

where T is a positive number such that

and the relations

do not hold.Consider the describing function

(13)

Theorem 1. If the inequalities

(14)

hold, then, for sufficiently small ε > 0, the Poincarémap (12) of the set Ω is a self�map, i.e., FΩ ⊂ Ω.

This theorem and Brouwer’s fixed point theoremimply the following result.

Theorem 2. Suppose that there exists a number a0 > 0for which

ϕ0 σ( )μσ σ∀ ε≤

Mε3 σ( ) σ∀sgn ε,>⎩⎨⎧

=

Ω y1 c3*y3+{ 0, y2 a1– a2–,[ ],∈= =

y3 Dε2 },≤

y1 t( ) ω0t( )y2 0( ) O ε2( ),+sin–=

y2 t( ) ω0t( )y2 0( ) O ε2( ),+cos=

y3 t( ) A3t( )y3 0( ) On 2– ε2( )+exp On 2– ε2( ),= =

t 0 T,[ ],∈

y3 0( ) Dε2≤

y3 T( ) Dε2≤

y3 t( ) D1ε2 t∀ 0 T,[ ].∈≤

Fy1 0( )

y2 0( )

y3 0( )

y1 T( )

y2 T( )

y3 T( )

,=

y1 T( ) c3*y3 T( )+ 0, y2 T( ) 0<=

y1 t( ) c3*y3 t( )+ 0, y2 t( ) 0 t∀ 0 T,( )∈<=

Φ a( ) ϕ2 a ω0t( )sin( ) ω0t( )sin t.d

0

2π/ω0

∫=

b1Φ a1( ) 2

ω02a1

2��L, b1Φ a2( )– 2

ω02a2

2��L,–<>

(15)

Then, for sufficiently small ε > 0, system (3) has aperiodic solution of the form (9) with initial data

(16)

and period

Corollary 1. For the nonlinearity (7),

and (16) implies

(17)

Sketch of the proof of Theorems 1 and 2. Takinginto account the form of the nonlinearity ϕ0 and therepresentation of solutions (9), we obtain

(18)

Therefore, | (τ)| > κ > 0 for |σ(τ)| ≤ ε. Thus, (18) and(9) imply the existence of numbers

(19)

such that (see Fig. 1)

(20)

This and the first relation in (18) imply the followinglemma.

Lemma 2. The following relations hold:

b1Φ a0( ) 2

ω02a0

2��L, a0 νi,≠–=

b1dΦ a( )

da��

a a0=

4

ω02a0

3��L.>

y1 0( ) O ε2( ), y2 0( ) a0– O ε( ),+= =

y3 0( ) On 2– ε2( )=

T 2πω0

�� O ε2( ).+=

L 23�� b2 c3*b3 b1+( )μ b1ω0+( )μ,=

Φ a0( ) M 4ω0

��=

y1 0( ) O ε2( ),=

y2 0( ) = μ3ω0b1M�� b2 c3*b3 b1+( )μ b1ω0+( )–– O ε( ),+

y3 0( ) On 3– ε2( ).=

σ t( ) y1 t( ) c3*y3 t( )+ ω0t( )y2 0( ) O ε2( ),+sin–= =

σ· t( ) y· 1 t( ) c3*y· 3 t( )+ ω0 ω0t( )y2 0( ) O ε( ).+cos–= =

σ·

0 τ0 τ1 τ2 τ3 τ4 τ5< < < < < T,= =

τ1: t∀ 0 τ1,( ), σ t( ) 0 ε,( ), σ τ1( )∈ ∈ ε;=

τ2: t∀ τ1 τ2,( ), σ t( ) ε> , σ τ2( )∈ ε;=

τ3: t∀ τ2 τ3,( ), σ t( ) ε– ε,( ), σ τ3( )∈ ∈ ε;–=

τ4: t∀ τ3 τ4,( ), σ t( ) ε– , σ τ4( )<∈ ε;–=

τ5 T= : t∀ τ4 T,( ), σ t( ) ε– 0,( )∈ , σ T( )∈ 0.=

τ1ε

ω0 y2 0( )�� O ε2( ),+=

τ2 τ1– πω0

�� 2εω0 y2 0( )��– O ε2( ),+=


ALGORITHMS FOR SEARCHING FOR HIDDEN OSCILLATIONS 477

(21)

Lemma 3. The following estimate is valid:

(22)

Proof. The continuity of σ(t) and the boundednessof the function ϕ2(σ) on finite intervals imply

(23)

Suppose that ±y2(0) ≠ νi. If –sin(ω0τ)y2(0) = νi, then

ω0τ ≠ + πk. Therefore, for sufficiently small ε,

according to (18), we have | (τ)| > κ > 0. Thus, for alltime intervals [tj, tj + 1], outside the neighborhoods (τ –mε, τ + mε) of moments of time τ corresponding to

for sufficiently small ε, we have

(24)

Let ±y2(0) = νi. Then, for all time intervals [tj, tj + 1]outside the neighborhoods (τ – mε, τ + mε) of

moments of time ω0τ = + πk, for sufficiently small ε,

we have

Choosing m so that

we obtain

τ3 τ2– 2εω0 y2 0( )�� O ε2( ),+=

τ4 τ3– πω0

�� 2εω0 y2 0( )��– O ε2( ),+=

T τ4– εω0 y2 0( )�� O ε2( ).+=

ϕ2 σ t( )( ) td

0

2π/ω0

∫

= ϕ2 ω0t( )y2 0( )sin–( ) td

0

2π/ω0

∫ O ε( ).+

ϕ2 σ t( )( ) td

τ mε–

τ mε+

∫ O ε( ).=

π2��

σ·

ω0τ( )y2 0( )sin– νi,=

ω0t( )y2 0( )sin– νi,≠

σ t( ) ω0t( )y2 0( ) O ε2( ) νi≠+sin–=

t∀ tj tj 1+,[ ].∈

π2��

σ t( ) ω0t( )y2 0( )sin– D1ε2+≤

= νi 1 12�� ω0mε( )2– O ε3( )+⎝ ⎠

⎛ ⎞ D1ε2+

t∀ tj tj 1+,[ ].∈

νi12�� ω0m( )2 D1,>

(25)

Relations (24) and (25), together with the bound�edness of the derivative of ϕ2(σ) on the intervals ofcontinuity, imply

(26)

The estimates (23) and (26) obtained above are uni�form on [0, 2π]. This implies the assertion of thelemma.

Lemma 4. For sufficiently small ε > 0,

(27)

Proof. Consider the function

where y1(t) and y2(t) are solutions with initial datafrom Ω. The derivative V(t) = V(y1(t), y2(t)) by virtue ofsystem (3) has the form

(28)

According to (9), we have the estimate V(T) – V(0) =

(T) – (0) + O(ε4). Let us estimate the increment

(29)

ω0t( )y2 0( )sin– νi,≠

σ t( ) y2 0( )< νi t∀ tj tj 1+,[ ].∈=

ϕ2 σ t( )( ) ϕ2 ω0t( )y2 0( )sin–( ) O ε2( ),+=

t tj tj 1+,[ ].∈

y22 T( ) y2

2 0( )–

= 2 y2 0( ) 2

ω02 y2 0( ) 2

��L b1Φ y2 0( )( )+⎝ ⎠⎛ ⎞ ε3 O ε4( ).+

V y1 y2,( ) y12 y2

2,+=

V· y1 t( ) y2 t( ),( ) 2 y1 t( )b1 y2 t( )b2+( )ϕ0 σ t( )( ).=

y22 y2

2

V T( ) V 0( )– V· t( ) td

0

T

∫=

= 2 y1 t( )b1 y2 t( )b2+( )ϕ0 σ t( )( ) t,d

0

T

∫

Fig. 1. Projection of the first approximation to the solutiononto the plane (y1, y2) and the nonlinearity ϕ0(σ) in (7).

τ3 τ2

ϕ0(σ)y2

y1

σ

ε

−ε

0

τ4 τ1

y2(0)

0T

0

478


LEONOV, KUZNETSOV

by representing the integral over the interval [0, T] as asum of integrals over the intervals [τi, τi + 1].

1. According to (20), for t ∈ [τ1, τ2] ∪ [τ3, τ4], thedefinition of the nonlinearity (4) and (29) imply

Here, taking into account (6) and the form (9) of solu�tions, we have

Applying (22) and taking into account (13), we obtain

2. Let us obtain estimates for t ∈ [0, τ1] ∪ [τ2, τ3] ∪[τ4, T]. According to (9), for y1(t) and y2(t), we have

(30)

the derivative (t) by virtue of systems (3) is estimated as

(31)

Therefore, in view of estimates (5) and (21), we have

and we can consider the function t(σ) inverse to σ(t)on the intervals of sign constancy of (t). Substituting

V· t( ) td

τ1 τ2,[ ] τ3 τ4,[ ]∪

∫

= 2 y1 t( )b1 y2 t( )b2+( )ε3ϕ2 σ t( )( ) t.d

τ1 τ2,[ ] τ3 τ4,[ ]∪

∫

V· t( ) td

τ1 τ2,[ ] τ3 τ4,[ ]∪

∫

= 2 y1 t( )b1 y2 t( )b2+( )ε3ϕ2 σ t( )( ) td

0

2π/ω0

∫

= 2 ω0t( )y2 0( )b1sin–(

0

2π/ω0

∫

+ ω0t( )y2 0( )b2 )ε3cos

× ϕ2 ω0t( )y2 0( )sin– O ε2( )+( )dt O ε4( ).+

V· t( ) td

τ1 τ2,[ ] τ3 τ4,[ ]∪

∫ 2y2 0( )b1ε3 ω0t( )sin

0

2π/ω0

∫–=

× ϕ2 ω0t( )y2 0( )sin–( )dt O ε4( )+

= 2 y2 0( ) b1ε3Φ y2 0( )( ) O ε4( ).+

y1 t( ) σ t( ) O ε2( ),+=

y2 t( ) ω0t( )y2 0( ) O ε2( )+cos=

= y2 0( ) O ε2( ), t 0 τ1,[ ] τ4 T,[ ]∪∈+

y2 0( )– O ε2( ), t τ2 τ3,[ ],∈+⎩⎨⎧

σ·

σ· t( ) =

ω0y2 0( )– b1ϕ1 σ t( )( ) c3*b3ϕ1 σ t( )( ) O ε3( ),+ + +

t 0 τ1,[ ] τ4 T,[ ]∪∈

ω0y2 0( ) b1ϕ1 σ t( )( ) c3*b3ϕ1 σ t( )( ) O ε2( ),+ + +

t τ2 τ3,[ ].∈⎩⎪⎪⎨⎪⎪⎧

σ· t( ) ω0 ω0t( )cos– O ε( ) 0≠+=

σ·

(30) into the expression (28) for (t) and using (31),we obtain

The summation of the resulting expressions yields

It follows from Lemmas 1 and 4 that inequali�ties (14) imply the inclusion FΩ ⊂ Ω; according toBrouwer’s theorem, the map F has a fixed point and,hence, system (3) with initial data from Ω has a peri�odic solution.

2. AN ANALYTICAL�NUMERICAL ALGORITHM FOR LOCALIZING

HIDDEN OSCILLATIONS

To detect hidden oscillation of system (1), we firstchoose the harmonic linearization coefficient k so thatthe matrix of the linear system

(32)

V·

V· t( ) td

0

τ1

∫V· t σ( )( )σ· t σ( )( )�� σd

0

ε

∫=

= 2σb1 y2 0( )b2+( )ϕ1 σ( )

ω0y2 0( )– b1ϕ1 σ( ) c3*b3ϕ1 σ( )+ +�� σd

0

ε

∫ O ε4( ),+

V· t( ) td

τ4

T

∫V· t σ( )( )σ· t σ( )( )�� σd

ε–

0

∫=

= 2σb1 y2 0( )b2+( )ϕ1 σ( )

ω0y2 0( )– b1ϕ1 σ( ) c3*b3ϕ1 σ( )+ +�� σd

ε–

0

∫ O ε4( ),+

V· t( ) td

τ2

τ3

∫V· t σ( )( )σ· t σ( )( )�� σd

ε–

ε

∫=

= 2–σb1 y2 0( )b2–( )ϕ1 σ( )

ω0y2 0( ) b1ϕ1 σ( ) c3*b3ϕ1 σ( )+ +�� σd

ε–

ε

∫ O ε4( ).+

V· t( ) td

0 τ1,[ ] τ2 τ3,[ ] τ4 T,[ ]∪ ∪

∫

= 2b1σ b2y2 0( )+

ω0y2 0( )– c3*b3ϕ1 σ( ) b1ϕ1 σ( )+ +��⎝

⎛

ε–

ε

∫

–b1σ b2y2 0( )–

ω0y2 0( ) c3*b3ϕ1 σ( ) b1ϕ1 σ( )+ +��⎠

⎞ ϕ1 σ( )dσ

+ O ε4( ) 4b2 c3*b3 b1+( )ϕ1 σ( ) b1ω0σ+( )

ω02 y2 0( )

��ϕ1 σ( ) σd

ε–

ε

∫=

+ O ε4( ) 4

ω02 y2 0( )

��Lε3 O ε4( ).+=

dzdt�� P0z, P0 P kqr*+= =



has a pair of purely imaginary eigenvalues ±iω0 (ω0 > 0)and the remaining eigenvalues have negative real parts.We assume that such k exists. Let us rewrite system (1)in the form

(33)

which can be reduced to form (3) by a nonsingulartransformation x = Sy.

To find periodic solutions of system (33), we intro�duce a finite sequence of functions ϕ0(σ), ϕ1(σ), …,ϕm(σ) such that the graphs of neighboring functionsϕ j(σ) and ϕ j + 1(σ) are close in a certain sense, thefunction ϕ0(σ) is of the form (4), and ϕm(σ) = ϕ(σ).Then, at the first step of the algorithm for the system

(34)

using the form of the function ϕ0(σ) and Theorem 2,we can find a nearly harmonic stable periodic solutionx0(t). Either all points of this stable periodic solutionbelong to the domain of attraction of a stable periodicsolution x1(t) of the system

(35)

with j = 1, or, under the passage from system (34) tosystem (35) with j = 1, the bifurcation of stability lossand disappearance of a periodic solution occurs. In thefirst case, we can determine x1(t) numerically by con�sidering the trajectory of system (35) with j = 1 fromthe initial point x0(0). The computational procedurestarts at the point x0(0), performs a transition process,reaches the periodic solution x1(t), and evaluates it.For this purpose, the interval [0, T] on computationsare performed must be sufficiently large.

After x1(t) is computed, we proceed to the nextsystem of the form (35) with j = 2 and compute a peri�odic solution x2(t) by considering the trajectory of sys�tem (35) with j = 2 from the initial point x1(T), whichapproaches the periodic trajectory x2 (t) with increas�ing t.

Continuing this procedure and successively calcu�lating xj(t) by considering trajectories of system (35)with initial data xj – 1(T), we finally either compute aperiodic solution of system (35) with j = m (i.e., of theinitial system (33)) or observe the bifurcation of stabil�ity loss and disappearance of a periodic solution atsome step.

Note that the algorithm described above makes itpossible to detect not only hidden periodic oscillationsbut also hidden strange attractors, a basin of attractionof which does not contain neighborhoods of equilibriaeither [8, 9] (unlike classical attractors, such as, e.,g.,those in Lorenz, Rossler, and some other systems, inwhich the linearized systems in neighborhoods ofequilibria have eigenvalues with positive real part andtrajectories going from these equilibria are attracted bythe attractor).

dxdt�� P0x qϕ r*x( ), ϕ σ( )+ ψ σ( ) kσ,–= =

dxdt�� P0x qϕ0 r*x( ),+=

dxdt�� P0x qϕj r*x( )+=

3. HIDDEN OSCILLATIONS IN COUNTEREXAMPLES TO AIZERMAN’S

AND KALMAN’S PROBLEMS

To implement the algorithm described above, wetake a finite sequence of functions

(36)

satisfying Aizerman’s condition for the sector (0, μ2).Here, μ and M are some positive numbers and μ < μ2.We choose m so that the graphs of the functions ϕ j andϕ j + 1 are close to each other.

At the first step of the algorithm ( j = 1), for a suffi�ciently large time interval [0, T], apply (17) to calcu�late a solution x1(t) with initial data

(37)

If the solution obtained in the course of computationstends to a periodic solution, then, according to thealgorithm, we compute a solution of the system with ε2with initial data x1(T).

Suppose that, using this algorithm, we haveobtained a periodic solution xm(t) of system (35) withcontinuous nondecreasing function ϕm(σ) (“satura�tion”). In this case, to construct a counterexample toKalman’s problem, we organize the following compu�tational procedure [10, 11] for the sequence of nonlin�earities

(38)

where i = 0, 1, …, h, ψ0(σ) = ϕm(σ), and N is a positiveparameter such that hN < μ2. At each step (i = 1, 2, …, h),we obtain a strictly increasing nonlinearity, and thecalculated periodic solution gives a counterexample toKalman’s problem.

Examples given below illustrate the algorithmdescribed above.

Consider the system

ϕj σ( )μσ σ∀ εj≤

M σ( )εj3 σ∀ εj,>sgn⎩

⎨⎧

=

εjj

m�� μ

M��, j 1 2 … m,, , ,= =

S 0 μ3ω0b1M�� b2 c3*b3 b1+( )μ b1ω0+( )–– 0, ,⎝ ⎠

⎛ ⎞ *.

ψi σ( )

= μσ σ∀ εm≤

i σ σ( )εmsgn–( )N σ( )μεm σ∀sgn εm,>+⎩⎨⎧

x· 1 x2,=

x· 2 x4,–=

x· 3 x1 2x4– ϕ x4( ),–=

x· 4 x1 x3 x4– ϕ x4( ),–+=

480


LEONOV, KUZNETSOV

(39)

Determining the matrix S and calculating y(0), weobtain, according to (37), the initial data x(0) = Sy(0) =

ϕ σ( )

=

5σ, σ∀ 15��≤

σ( ) 125�� σ σ( )1

5��sgn–⎝ ⎠

⎛ ⎞ σ∀ 15��.>+sgn

⎩⎪⎪⎨⎪⎪⎧

(0, 0.2309, 0.2309, 0) for the first step of the algorithmat j = 1 for system (39) with nonlinearity ϕ1(σ). Suc�cessively calculating solutions and increasing j, weobtain a periodic solution at the last step of the algo�rithm for the initial nonlinearity ϕ(σ) (see Fig. 2).

We continue the recursive construction of periodicsolutions by replacing the nonlinearity ϕ(σ) by thestrictly increasing function ψi(σ) defined by (38) withμ = 1 and N = 0.01. The periodic solution obtained ati = 3 after the transitional computational process ispresented in Fig. 3.

Now, we apply the algorithm to the system

(40)

Determining the matrix S and calculating y(0), weobtain, according to (37), the initial data x(0) = Sy(0) =(0, 0.1722, 0.1722, 0) for the first step of the algorithm.

We compute the periodic solution at the last step ofthe algorithm for ϕ(σ). Then, we continue the succes�sive construction of periodic solutions for system (40)by replacing the nonlinearity ϕ(σ) by the strictlyincreasing function ψi(σ) defined by (38) with μ = 1

x· 1 x2,=

x· 2 x4,–=

x· 3 x1 2x4– 9131900

��ϕ x4( ),–=

x· 4 x1 x3 x4– 1837180

��ϕ x4( ),–+=

ϕ σ( )σ σ∀ 900

9185��≤

σ( ) 9009185�� σ∀sgn 900

9185��.>⎩

⎪⎨⎪⎧

=

4

−2 4x3

x4

2

0

−2

−4 20

ε = 0.2

Fig. 2. Projection of the trajectory on the plane (x3, x4).

2

−1 2x3

x4

1

0

−1

−2 10

i = 3


2

−1 2x3

x4

1

0

−1

−2 10

i = 5




and N = 0.01, for i = 1, 2, …, 5. The periodic solutionsobtained for i = 5 after the transitional computationalprocess are presented in Fig. 4.

Note that the second part of algorithm, where thesaturation zone is “lifted” and the counterexample toKalman’s problem is constructed, supplementsnumerical results of [12, 13], in which systems (39)and (40) with nonlinearities sign and sat, respectively,are considered. Thus, we have shown that the algo�rithm proposed in this paper works for the modifiedsystems considered in [12, 13].

ACKNOWLEDGMENTS

This work was supported by the Ministry of Educa�tion and Science of Russian Federation, St. PetersburgState University, and the Academy of Finland.

REFERENCES

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