on the aizerman problem

ISSN 0005-1179, Automation and Remote Control, 2009, Vol. 70, No. 7, pp. 1120–1131. c© Pleiades Publishing, Ltd., 2009.Original Russian Text c© G.A. Leonov, 2009, published in Avtomatika i Telemekhanika, 2009, No. 7, pp. 37–49.

DETERMINATE SYSTEMS

On the Aizerman Problem

G. A. Leonov

St. Petersburg State University, St. Petersburg, RussiaReceived July 17, 2008

Abstract—The method of harmonic linearization in degenerate cases is stated and strictlymathematically justified. This method is applied to the search for periodic oscillations forsystems satisfying the generalized Routh–Hurwitz condition.

PACS numbers: 02.30.Yy, 01.60.+q

DOI: 10.1134/S0005117909070042

1. INTRODUCTION

The Aizerman problem [1] stimulated the development of many mathematical methods for theinvestigation of nonlinear systems of automatic control and the formation of the new theory—thetheory of absolute stability [2]. In this case many hundreds of articles, tens of books and reviews(a rather full bibliography is available in [3]) are devoted to the sufficient criteria of absolutestability, but there is quite a small number of effective necessary criteria different from linear ones(such as the Nyquist criterion) and establishing the existence of periodic oscillations for autonomoussystems satisfying the generalized Routh–Hurwitz condition [4–6].

In this work, the development is continued of the methods of the search for periodic oscillationsfor nonlinear systems satisfying the generalized Routh–Hurwitz conditions and the approach issuggested that combines the above methods with the numerical methods and the applied theory ofbifurcations.

This article can also be considered as an extension of the approach suggested in [7] to the cal-culation of periodic oscillations of autonomous systems in the joint use of the method of harmoniclinearization, numerical methods, and the applied theory of bifurcations. The harmonic lineariza-tion method itself proves to be too rough so as to discover periodic oscillations in nonlinear systemsthat satisfy the generalized Routh–Hurwitz conditions. However, its extension to some degeneratemathematical structures and estimates in the sense of classical investigations of critical cases in thetheory of motion stability [8] made it possible to obtain effective estimates for periodic oscillationsin nonlinear systems satisfying the generalized Routh–Hurwitz conditions.

Thus, we will consider the system

dx

dt= Px+ qϕ(r∗x), (1)

where P is the constant n × n-matrix, q and r are the constant n-dimensional vectors, ∗ is theoperation of transposition.

We will assume that the piecewise continuous function1 ϕ(σ) satisfies the condition

k1σ2 < ϕ(σ)σ < k2σ

2, ∀σ �= 0. (2)

Here k1 and k2 are some numbers.1 We call here the function ϕ(σ) the piecewise continuous one if it has a finite number of discontinuity points in any

finite interval and for any discontinuity point σ there exist finite limits limσ→σ−0

ϕ(σ) and limσ→σ+0

ϕ(σ).

1120

ON THE AIZERMAN PROBLEM 1121

The solution of the system (1) is understood in the sense of A.F. Filippov [9].We will say that the system (1) satisfies the generalized Routh–Hurwitz conditions if at any

k ∈ (k1, k2) the linear system

dx

dt= Px+ kqr∗x (3)

is stable on the whole.M.A. Aizerman proposed the hypothesis for the fact that all systems (1) satisfying the generalized

Routh–Hurwitz condition are stable on the whole.The necessary criteria of absolute stability that are obtained in [4–6] disprove this hypothesis.

However, the problem remains for the development of algorithms of the search for periodic solutionsfor various specific systems of the form (1) that satisfy the generalized Routh–Hurwitz condition.Thus, this article is devoted to the description of these algorithms.

2. CONSTRUCTION OF THE POINCARE MAPPING FOR PIECEWISELINEAR SYSTEMS WITH A SMALL PARAMETER

Without diminishing generality, we will consider that k1 = 0, k2 = k > 0. We will also assumethat the matrix P has two pure imaginary eigenvalues ±iω0 (ω0 > 0), and its remaining eigenvalueshave negative real parts. This property of the matrix P is typical if we consider a finite maximumHurwitz sector.2

We will put into consideration the class of functions ϕ(σ) of the special form

ϕ(σ) = μσ, ∀σ ∈ (−ε, ε),ϕ(σ) = Mε3, ∀σ > ε, (4)

ϕ(σ) = −Mε3, ∀σ < −ε.Here μ and M are some positive numbers and ε is a small positive parameter. In this case, for smallparameters ε and for any solution x(t) of the system (1) the following inequality is fulfilled3 [9]:

|ϕ(r∗x(t))| ≤ με. (5)

We will now compare the solutions x(t) of the system (1) with the solutions y(t) of the system

dy

dt= Py (6)

in a certain fixed interval [0, T ] with the same initial conditions x(0) = y(0).

Lemma 1. Let the following inequality be met :4

|y(t)| ≤ ρ, ∀ t ∈ [0, T ]. (7)

Then for a sufficiently small number ε with respect to T , ρ, |P |, |P |−1, μ the solution x(t) of thesystem (1) with initial data x(0) = y(0) satisfies the estimates

|x(t)| ≤ 2ρ, ∀ t ∈ [0, T ], (8)

|x(t) − y(t)| ≤ εμ|q||P |−1 exp(|P |T ), ∀ t ∈ [0, T ]. (9)

2 We will call here the sector (k1, k2) the maximum Hurwitz sector if for all k ∈ (k1, k2) the system (3) is stable onthe whole and at k = k1 and k = k2 the system (3) is not stable on the whole.

3 The inequality (5) is obvious beyond the discontinuity points σ = ±ε. For the solutions in the sense of Filippovthis inequality also takes place for those t where r∗x(t) = ±ε [9].

4 Here | · | is the Euclidean norm of vectors and operators in the Euclidean space.

AUTOMATION AND REMOTE CONTROL Vol. 70 No. 7 2009

1122 LEONOV

Proof. We suppose that (8) does not hold. Then from the continuity of y(t) and the inequality|y(0)| ≤ ρ the existence of the number τ ∈ (0, T ) follows, for which

|x(t)| ≤ 2ρ, ∀ t ∈ [0, τ ], (10)|x(τ)| = 2ρ. (11)

We will introduce for consideration the function V (t) = |x(t)−y(t)|. This function is continuousin [0, τ ]. Therefore, V (t) can be different from zero only in the intervals of the form (α, β) ∈ [0, τ ].At the remaining points of the interval [0, τ ], the equality V (t) = 0 is fulfilled.

We will make some estimates in one of the intervals (α, β). It follows from the relation (5) that

(V (t)2)· ≤ 2|P |V (t)2 + 2|q|μεV (t), ∀ t ∈ (α, β).

This inequality can be rewritten in the following form:

(V (t) + εμ|q||P |−1)· ≤ |P |(V (t) + εμ|q||P |−1), ∀ t ∈ (α, β).

From this differential inequality and from the assumption that the interval (α, β) at the left ismaximum (i.e., V (α) = 0), we obtain the estimate

V (t) ≤ εμ|q||P |−1 exp(|P |T ), ∀ t ∈ (α, β). (12)

Because the intervals (α, β) are all possible intervals in which V (t) �= 0, we conclude that theestimate (12) takes place at all t ∈ [0, τ ].

Selecting the ε so as to fulfil the inequality

εμ|q||P |−1 exp(|P |T ) < ρ,

from (12) we obtain the estimate

|x(t)| ≤ |x(t) − y(t)| + |y(t)| < 2ρ, ∀ t ∈ [0, τ ].

The latter contradicts the equality (11). The obtained contradiction proves the estimate (8).It follows from the estimate (8) that the inequality (12) is fulfilled at t ∈ [0, T ]. The latter

signifies the fulfilment of the inequality (9), which thus proves the Lemma 1.It follows from the Lemma 1 that the solutions of the systems (1) and (6) differ by no more

than O(ε). To construct the Poincare mapping, for which finer estimates are used, we will bringthe system (1) by the nonspecial linear transformation to the following canonical form:

x1 = −ω0x2 + b1ϕ(x1 + c∗x3),x2 = ω0x1 + b2ϕ(x1 + c∗x3), (13)x3 = Ax3 + bϕ(x1 + c∗x3).

Here, A is the constant (n − 2) × (n − 2)-matrix, all eigenvalues of which have negative realparts, b and c are the constant (n− 2)-dimensional vectors, b1 and b2 are some numbers.

Without diminishing generality, here it can be taken that for the matrix A there exists a positivenumber λ, for which

x∗3(A+A∗)x3 ≤ −2λ|x3|2, ∀x3 ∈ Rn−2. (14)



We will introduce the following set in the phase space of the system (13):

Ω = {|x3| ≤ Dε2, x1 + c∗x3 = 0, x2 ∈ [−a1,−a2]}, a1 > a2.

Here a1, a2, and D are some positive numbers that will be defined below.It follows from the Lemma 1 that for solutions of the system (13) with initial data x1(0), x2(0),

x3(0) from the set Ω, the following relations are valid:

x1(t) = − sin(ω0t)x2(0) +O(ε),x2(t) = cos(ω0t)x2(0) +O(ε), (15)

x3(t) = O(ε)

for all t ∈ [0, 4π/ω0]. Hence, it follows that for such solutions there exist numbers

0 < τ1 < τ2 < τ3 < τ4 < T,

for which σ(t) = x1(t) + c∗x3(t) ∈ (0, ε), ∀ t ∈ (0, τ1) and σ(τ1) = ε, σ(t) > ε, ∀t ∈ (τ1, τ2) andσ(τ2) = ε, σ(t) ∈ (−ε, ε), ∀t ∈ (τ2, τ3) and σ(τ3) = −ε, σ(t) < −ε, ∀t ∈ (τ3, τ4) and σ(τ4) = −ε,σ(t) ∈ (−ε, 0), ∀t ∈ (τ4, T ) and σ(T ) = 0. In which case

τ1 =ε

ω0|x2(0)| +O(ε2),

τ2 − τ1 =π

ω0+O(ε),

τ3 − τ2 =2ε

ω0|x2(0)| +O(ε2), (16)

τ4 − τ3 =π

ω0+O(ε),

T − τ4 =ε

ω0|x2(0)| +O(ε2).

From the formula

x3(t) = eAtx3(0) +t∫

0

eA(t−s)bϕ(x1(s) + c∗x3(s))ds,

the relations (14)–(16), and the form of the function ϕ(σ), the estimates sequentially emerge:

|x3(τ1)| ≤ e−λτ1 |x3(0)| +O(ε2),

|x3(τ2)| ≤ e−λτ2 |x3(0)| +O(ε2),

|x3(τ3)| ≤ e−λτ3 |x3(0)| +O(ε2),

|x3(τ4)| ≤ e−λτ4 |x3(0)| +O(ε2),

|x3(T )| ≤ e−λT |x3(0)| +O(ε2).

In the last inequality, we have O(ε2) ≤ Eε2, where the number E depends on |b|, μ and M . (Wenote that from (14) it follows that

|eA(t−s)| ≤ 1, ∀ t ≥ s.)

Therefore, selecting the number D so that

e−λTD + E < D,


1124 LEONOV

we find that the Poincare mapping

F

⎛⎜⎝x1(0)x2(0)x3(0)

⎞⎟⎠ =

⎛⎜⎝x1(T )x2(T )x3(T )

⎞⎟⎠

of the set Ω satisfies the inclusion

FΩ ⊂ {|x3| ≤ Dε2, x1 + c∗x3 = 0, x2 ∈ [−a1 +O(ε),−a2 +O(ε)]}.

We will now define more exactly this inclusion, showing that at some values of the parameters,we have FΩ ⊂ Ω. For this, we will introduce into consideration the function

w(t) = x1(t)2 + x2(t)2,

where xj(t) (j = 1, 2, 3) is the solution of the system (1) with the initial data xj(0) from Ω.We will estimate the increment of the function w:

w(T ) − w(0) =T∫

o

w(t)dt

=τ1∫

0

w(t)dt +τ2∫

τ1

w(t)dt +τ3∫

τ2

w(t)dt +τ4∫

τ3

w(t)dt +T∫

τ4

w(t)dt

=ε∫

−ε

(dw1

dσ− dw2

dσ

)dσ − 8Mε3b1x2(0)

ω0+O(ε4). (17)

Here

w1(σ) = w(t(σ)),

where t(σ) is the function inverse to σ(t) in [0, τ1] and [τ4, T ],

w2(σ) = w(t(σ)),

where t(σ) is the function inverse to σ(t) in [τ2, τ3].The existence of the functions t(σ) follows from the monotonicity of σ(t) in these intervals, which

stems from the relations (15) and (16).The following refinement of the estimates (15) emerges from the first two equations of the

system (13), estimates (5) and (16): in the intervals [0, τ1], [τ4, T ]

x1(t) = − sin(ω0t)x2(0) +O(ε2) = −ω0tx2(0) +O(ε2),x2(t) = cos(ω0t)x2(0) +O(ε2) = x2(0) +O(ε2)

(18)

and in the interval [τ2, τ3]

x1(t) = sin(ω0t)x2(0) +O(ε2) = ω0tx2(0) +O(ε2),x2(t) = − cos(ω0t)x2(0) +O(ε2) = −x2(0) +O(ε2).

(19)

In the intervals [0, τ1] and [τ4, T ] the following relation is fulfilled:

σ(t) = −ω0x2(0) +O(ε),



and in [τ2, τ3] the following estimate is defined:

σ(t) = ω0x2(0) +O(ε).

From here and from (18) and (19) we will obtain the relations

x1(t(σ)) = σ +O(ε2),x2(t(σ)) = x2(0) +O(ε2)

(20)

at t ∈ [0, τ1] and [τ4, T ],

x1(t(σ)) = σ +O(ε2),x2(t(σ)) = −x2(0) +O(ε2),

(21)

at t ∈ [τ2, τ3].It follows from here that

ε∫

−ε

(dw1

dσ− dw2

dσ

)dσ

= 2ε∫

−ε

(b1σ + b2x2(0)

−ω0x2(0) + c∗bϕ(σ) + b1ϕ(σ)− b1σ − b2x2(0)ω0x2(0) + c∗bϕ(σ) + b1ϕ(σ)

+O(ε2))ϕ(σ)dσ

= 4ε∫

−ε

(b2(c∗b+ b1)ϕ(σ) + b1ω0σ)ω2

0|x2(0)| ϕ(σ)dσ +O(ε4)

=8με3

3ω20|x2(0)| (b2(c∗b+ b1)μ+ b1ω0) +O(ε4). (22)

Thus, if the next inequality is fulfilledμ

3ω0a2(b2(c∗b+ b1)μ+ b1ω0) +Mb1a2 > 0, (23)

then it follows from the relations (17) and (22) that at x2(0) = −a2 w(T ) > w(0); but if theinequality given below is fulfilled

μ

3ω0a1(b2(c∗b+ b1)μ+ b1ω0) +Mb1a1 < 0, (24)

then it follows from the relations (17) and (22) that at x2(0) = −a1 w(T ) < w(0).It follows from here that in the fulfilment of the inequalities b1 < 0, (23) and (24) there exists

an inclusion

FΩ ⊂ Ω. (25)

From the relation (25) and the Brouwer theorem it follows that there exists a fixed point F ofthe mapping and, hence, there exists a periodic solution of the system (13) with initial data fromthe set Ω.

It stems from the inequalities b1 < 0, (23) and (24) that these initial data satisfy the relations

x1(0) = O(ε2), x3(0) = O(ε2),

x2(0) = −√μ(b2(c∗b+ b1)μ+ b1ω0)

3ω0M(−b1) +O(ε).

In this case, such a periodic solution is stable in the sense of the inclusion (23).


1126 LEONOV

3. ALGORITHM FOR DEFINITION OF STABLE PERIODIC SOLUTIONS

We will write the transfer function of the system (1) (or (13)) from the “input u” to the “out-put r∗x”

W (p) = r∗(P − pI)−1q =αp+ β

p2 + ω20

+ c∗(A− pI)−1b. (26)

Here α = −b1, β = b2ω0,

c∗b+ b1 = r∗q = − limp→∞pW (p).

We will now formulate in terms of the transfer function W (p) the results obtained in the pre-ceding section.

Theorem 1. If the inequalities α > 0 and

μβr∗q > αω20 ,

are fulfilled, then the system (1) with the nonlinearity (4) has a periodic solution satisfying therelations (15) and

x1(0) = O(ε2), x3(0) = O(ε2),

x2(0) = −√μ(μβr∗q − αω2

0)3ω2

0Mα+O(ε).

(27)

This solution is stable in the sense of the inclusion (23).

It follows from the relations (16) that for the period of this solution the estimate 2π/ω0 +O(ε)takes place.

In keeping with the approach suggested in [7], the periodic solution with the initial data satis-fying (27) can be considered as a certain “support” periodic oscillation, while the system (13), asa “generating” start system in the algorithm of the search for the periodic solution of the othersystem

dx

dt= Px+ qψ(r∗x), (28)

satisfying the generalized Routh–Hurwitz condition. In this case it is possible to organize a finitesequence of the functions ϕj(σ), such that the plots of each pair ϕj and ϕj+1 are close to eachother. Here, for the system

dx

dt= Px+ qϕj(r∗x) (29)

with the ϕ1(σ) = ϕ(σ) and the small parameter ε, the periodic solution g1(t) in the form of (15),(27) is taken. All points of this periodic solution are either located in the domain of attraction ofthe stable periodic solution of g2(t) of the system (29) with j = 1 or, in the transition from thesystem (29) with j = 2, the bifurcation of the loss of stability and the disappearance of the periodicsolution is observed. In the first case, it is possible to define the g − 2(t) numerically by releasingthe trajectory of the system (29) with j = 2 from the initial point x(0) = g1(0).

Starting from the point g1(0), the computational procedure, after the transition process, getsout for the periodic solution of the g2(t) and calculates it. For this, the interval [0, T ] in which thecalculation takes place must be sufficiently large.

After the calculation of the g2(t), it is possible to pass on to the next system (29) with j = 3and set up a similar procedure of the calculation of the periodic solution of the g3(t), drawing from



0

–0.5

–1.0 –0.5 0

x

2

x

1

–1.00.5 1.0

0.5

1.0

0

–0.5

0 500 1000

t

x

2

(

t

)

–1.01500

0.5

1.0

Fig. 1. Transient process at ε2 = 0.1.

the initial point x(0) = g2(T ) the trajectory that with an increase of t approaches the periodictrajectory of the g3(t).

Continuing this procedure and calculating gj(t), using the trajectories of the system (29) withinitial data x(0) = gj−1(T ), we either come to the calculation of the periodic solution of thesystem (29) with j = m, ϕm(σ) = ψ(σ), or observe at a certain step the bifurcation of the loss ofstability and the disappearance of the periodic solution.

We will give an example of such a procedure of the calculation of periodic oscillations and thebifurcation of the loss of stability.

Example 1. We will consider the system (29) with the transfer function

W (p) =p− 1p2 + 1

+1

p+ 1.

Here r∗q = −2, ω0 = 1, b1 = −1, b2 = −1, A = −1, c = 1, b = −1.The stability of the linear system (29) with ϕj(σ) = kσ here takes place at all k ∈ (0,+∞).Let ϕj(σ) take the form

ϕj(σ) = μσ, ∀σ ∈ (−εj , εj),ϕj(σ) = Mε3j , ∀σ > εj ,

ϕj(σ) = −Mε3j , ∀σ < −εj .Here μ = 2, M = 1, ε1 = ε, ε2 = 0.1, ε3 = 0.2, . . . , ε8 = 0.7, ε is a small positive parameter.

The initial data of the stable periodic oscillation at the first step j = 1 according to the Theo-rem 1 will take the form

x1(0) = O(ε), x3(0) = O(ε), x2(0) = −√

23

+O(ε).

Therefore, at j = 2 we put out the trajectory from the point x1(0) = x3(0) = 0, x2(0) =−√2/3 ≈ −0.8165. The projection of this trajectory on the plane (x1, x2) and its coordinate x2(t)are displayed in Fig. 1. It is seen here that after the transient process, the passage to the stableperiodic solution occurs.


1128 LEONOV

0

–1.0

–2 –1 0

x

2

x

1

–1.51 2

0.5

1.5

0

–0.5

0 500 1000

t

x

2

(

t

)

–1.51500

0.5

1.5

–0.5

1.0

–1.0

1.0


0

–1

–2 –1 0

x

2

x

1

–21 2

1

2

0

–1

0 500 1000

t

x

2

(

t

)

–21500

1

2


0

–1

–2 –1 0

x

2

x

1

–21 2

1

2

0

–1

0 500 1000

t

x

2

(

t

)

–21500

1

2


Continuing this procedure at j = 3, . . . , 7, we find periodic solutions, while at ε8 = 0.7 weobserve the vanishing of the periodic solution and the attraction to the stable equilibrium state(Figs. 2–7).



0

–2

–4 –2 0

x

2

x

1

–32 4

1

3

0

–1

0 500 1000

t

x

2

(

t

)

–31500

1

3

–1

2

–2

2


0

–2

–4 –2 0

x

2

x

1

–32 4

1

3

0

–1

0 500 1000

t

x

2

(

t

)

–31500

1

3

–1

2

–2

2


0

–2

–4 –2 0

x

2

x

1

–32 4

1

3

0

–1

0 500 1000

t

x

2

(

t

)

–31500

1

3

–1

2

–2

2


We note that at the εj =√

2, the nonlinearity of ϕj(σ) is monotone. For the system (29) withthis nonlinearity at n = 3, the fact of the absence of periodic solutions is well known [5, 6].


1130 LEONOV

Therefore, here we observe the bifurcation of the loss of stability and the vanishing of the periodicsolution.

An important problem is the search for systems of the higher order, for which the algorithmdescribed above of the search for periodic solutions leads to the system (28) with the nonlinear-ity ψ(σ) from the Hurwitz sector, which is monotone relative to both bounds of the Hurwitz sectorand has a stable periodic solution.

In the next example, we will show that the condition μβr∗q ≤ αω20 is necessary, while the

inequality

μβr∗q < αω20 (30)

is the sufficient condition of the absolute stability of the system (1) at n = 3.

Example 2. We will examine the system (1) of the third order with the transfer function (25)of the form

W (p) =αp+ β

p2 + ω20

+κ

p+ γ, ω0 > 0, γ > 0.

The necessary and sufficient conditions of the fact that the system (1) at ϕ(σ) = kσ and at smallpositive k be stable on the whole are the relations α > 0 or α = 0, κβ > 0. These are the so-calledconditions of limit stability [10].

At α > 0, we will resort to the Popov frequency criterion1μ

+R(1 + θiω)W (iω) > 0, (31)

while at α = 0, κβ > 0, the criterion

Re [(iω)W (iω)] �= 0, ∀ω �= 0 (32)

[5, 6].It is easy to see that the inequality (32) is fulfilled if there exist relations α = 0, κβ > 0. In this

case, the sector (0,+∞) is the sector of absolute stability. The inequality (31) will take the form

1μ

+β

ω20

+κ(γ + θω2)ω2 + γ2

> 0, θ =β

αω20

. (33)

Because the functionγ + θω2

ω2 + γ2

is monotone, the inequality (33) is fulfilled if

1μ

+β

ω20

+κ

γ> 0,

1μ

+β

ω20

+κβ

αω20

> 0.

Ifκ

γ<

κβ

αω20

,β

ω20

+κ

γ> 0,

then the absolute stability takes place in the sector (0,+∞), if

κ

γ<

κβ

αω20

,β

ω20

+κ

γ< 0

—in the sector (0,−

(β

ω20

+κ

γ

)−1).



This sector coincides with the maximum Hurwitz sector and, hence, in this case the Aizermanhypothesis is valid.

Ifκ

γ>

κβ

αω20

, (34)

then the absolute stability exists at

μβ(α+ κ) + αω20 > 0. (35)

The condition (35) coincides with the inequality (30). Thus, it follows from the Theorem (1)that the condition μβr∗q ≤ αω2

0 is necessary and it follows from (35) that (30) is sufficient for theabsolute stability of the system (1) at n = 3. From here it follows that if (34) is fulfilled, then theAizerman hypothesis is unjustifiable.

The book [11] is devoted to this analysis of the system (1) with n = 3.

4. CONCLUSIONS

The development of the method of harmonic linearization for the system satisfying the gener-alized Routh–Hurwitz condition made it possible to set up the analytic-numerical algorithm fordefining periodic oscillations. The application of this algorithm is displayed on specific systems ofthe third order.

REFERENCES

1. Aizerman, M.A., On One Problem Concerning the Stability in “Large” of Dynamic Systems, Usp. Mat.Nauk , 1949, vol. 4, issue 4, pp. 186–188.

2. Aizerman, M.A. and Gantmacher, F.R., Absolyutnaya ustoichivost’ reguliruemykh sistem (Absolute Sta-bility of Adjustable Systems), Moscow: Akad. Nauk SSSR, 1963.

3. Liberzon, M.R., Essays on the Absolute Stability Theory, Autom. Remote Control , 2006, no. 10,pp. 1610–1644.

4. Leonov, G.A., On Necessity of the Frequency Condition of Absolute Stability of Stationary Systems inthe Critical Case of the Pair of Pure Imaginary Roots, Dokl. Akad. Nauk SSSR, 1970, vol. 193, no. 4,pp. 756–759.

5. Leonov, G.A., Burkin, I.M., and Shepelyavyi, A.I., Frequency Methods in Oscillation Theory, Dordrect:Kluver, 1993.

6. Leonov, G.A., Ponomarenko, D.V., and Smirnova, V.B., Frequency Methods for Nonlinear Analysis.Theory and Applications , Singapore: World Scientific, 1996.

7. Leonov, G.A., On the Method of Harmonic Linearization, Autom. Remote Control , 2009, no. 5, pp. 800–810.

8. Malkin, I.G., Ustoichivost’ dvizheniya (Motion Stability), Moscow: Nauka, 1966.9. Gelig, A.Kh., Leonov, G.A., and Yakubovich, V.A., Ustoichivost’ nelineinykh system s needinstven-

nym sostoyaniem ravnovesiya (Stability of Nonlinear Systems with the Nonunique Equilibrium State),Moscow: Nauka, 1978.

10. Metody issledovaniya nelineinykh sistem avtomaticheskogo upravleniya (Methods of Investigation of Non-linear Automatic Control Systems), Nelepin, R.A., Ed., Moscow: Nauka, 1975.

11. Pliss, V.A., Nekotorye problemy teorii ustoichivosti dvizheniya (Some Problems of Theory of MotionStability), Leningrad: Leningr. Gos. Univ., 1958.

This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board


on the aizerman problem

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