ISSN 1064�5624, Doklady Mathematics, 2010, Vol. 82, No. 1, pp. 540–542. © Pleiades Publishing, Ltd., 2010.Original Russian Text © G.A. Leonov, V.O. Bragin, N.V. Kuznetsov, 2010, published in Doklady Akademii Nauk, 2010, Vol. 433, No. 2, pp. 163–166.

540

Consider the system

(1)

where P is a constant n × n matrix, q and r are constantn�dimensional vectors, the star denotes the transpose,and ϕ(σ) is a continuous piecewise differentiable sca�lar function. Assume that ϕ(0) = 0 and, at differentia�bility points,

(2)

Here, μ1 and μ2 are numbers. This form of writingnonlinear dynamical systems with a single nonlinear�ity is traditional for the absolute stability theory ofnonlinear control systems [1].

Kalman put forward the following conjecture [2](which is stronger that Aizerman’s conjecture [3]): Ifthe linear system

(3)

is asymptotically stable for all k ∈ (μ1, μ2), then system (1)is globally stable (i.e., the trivial solution is Lyapunovstable and any solution of system (1) tends to zero as t →∞; in other words, the trivial solution is a global attractorof system (3)).

It is well known that this conjecture holds for n = 2and 3 [4].

The only counterexample to this conjecture widelycited in the literature is that of Fitts [5], who simulatedsystem (1) on a computer with n = 4, the transfer func�tion

,

ϕ(σ) = 10σ3, and β ∈ (0.01; 0.75). This computerexperiment revealed periodic solutions of system (1).

For some of the parameters, namely, for β ∈ (0.572;0.75), it was shown in [6] that the results of Fitts’

dxdt���� Px qϕ r*x( ), x �

n,∈+=

μ1 ϕ' σ( ) μ2.< <

dxdt���� Px kqr*x+=

W p( ) p2

p β+( )2 0.92+[ ] p β+( )2 1.12+[ ]������������������������������������������������������������������=

experiment are invalid. Presented in [6], the proof ofthe existence of system (1) for which Kalman’s con�jecture does not hold is an existence theorem andneeds to be carefully checked. For example, errors in[6] were indicated in [7, 8], and attempts were made in[8] to overcome them.

In this paper, we proposed an algorithm for con�structing classes of systems (1) for which Kalman’sconjecture does not hold. The algorithm is consider�ably simpler than the approaches suggested in [6, 8].

First, we assume that μ1 = 0, μ2 > μ > 0, and thematrix P has two purely imaginary eigenvalues ±iω0(ω0 > 0), while the other eigenvalues of P have negativereal parts. In this case, system (1) can be written as

(4)

Here, x1, x2 ∈ �1; x3 ∈ �n – 2, A is a constant (n – 2) ×(n – 2) matrix all of whose eigenvalues have negativereal parts; b and c are constant (n – 2)�dimensionalvectors; and b1 and b2 are numbers.

The transfer function of system (4) from the inputϕ to the output r*x is

Here, α = –b1, β = b2ω0, and

Define a function ϕ0(σ) of special form:

(5)

Here, μ < μ2, M are positive numbers, and ε is a smallpositive parameter.

The following result holds for system (4) with ϕ = ϕ0.Theorem. If α > 0 and

x· 1 ω0x2– b1ϕ x1 c*x3+( ),+=

x· 2 ω0x1 b2ϕ x1 c*x3+( ),+=

x· 3 Ax3 bϕ x1 c*x3+( ).+=

W p( ) r* P pI–( ) 1– q αp β+

p2 ω02+

�������������� c* A pI–( ) 1– b.+= =

c*b b1+ r*q pW p( ).p ∞→

lim–= =

ϕ0 σ( )Mε3

, σ∀ ε>

μσ, σ∀ ε– ε,[ ]∈

Mε3, σ∀– ε.–<⎩

⎪⎨⎪⎧

=

μβr*q αω02,>

Algorithm for Constructing Counterexamples to the Kalman Problem

Corresponding Member of the RAS G. A. Leonov, V. O. Bragin, and N. V. KuznetsovReceived March 16, 2010

DOI: 10.1134/S1064562410040101

Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504 Russiae�mail: leonov@math.spbu.ru

MATHEMATICS

DOKLADY MATHEMATICS Vol. 82 No. 1 2010

ALGORITHM FOR CONSTRUCTING COUNTEREXAMPLES 541

then, for sufficiently small ε, system (4) with nonlinearity(5) has an orbitally stable periodic solution satisfying therelations

(6)

Proof techniques for this theorem were developedin [4, 9–11].

Consider the finite sequence of functions

(7)

where j = 1, 2, …, m and

Let m be such that the graphs of ϕj and ϕj + 1 differslightly from each other outside small neighborhoodsof the discontinuity points.

Based on the theorem, we can find a close�to�har�monic stable periodic solution x(t) = x0(t) of thesystem

(8)

with j = 0. All the points of this periodic solutioneither lie in the domain of attraction of the stable solu�tion x1(t) to system (8) with j = 1 or the transition fromsystem (8) with j = 0 to system (8) with j = 1 is accom�panied by a stability loss bifurcation and the vanishingof the periodic solution. In the former case, x1(t) canbe numerically determined by issuing the trajectory ofsystem (8) with j = 1 from the initial point x0(0).

Starting from the point x0(0), after a transient pro�cess, the computational procedure produces a peri�odic solution. For this purpose, the computationinterval (0, T) has to be sufficiently long.

After x1(t) has been computed, we can pass to thenext system (8) with j = 2 and organize a similar pro�cedure for computing the periodic solution x2(t) byissuing a trajectory from the initial point x(0) = x1(T)that approaches x2(t) with increasing t (or a stabilityloss bifurcation occurs and the periodic solution van�ishes).

Continuing this procedure and sequentially calcu�lating the periodic solutions x j(t) with the use ofthe trajectories of system (8) with initial data x j(0) =x j – 1(T), we either compute the periodic solution ofsystem (8) with j = m or a vanishing�of�periodic�solu�

x1 t( ) ω0t( )sin– x2 0( ) O ε( ),+=

x2 t( ) ω0t( )cos x2 0( ) O ε( ),+=

x3 t( ) O ε( ),=

x1 0( ) O ε2( ), x3 0( ) O ε2( ),= =

x2 0( )μ μβr*q αω0

2–( )

3ω02Mα

��������������������������������� O ε( ).+–=

ϕj σ( )

Mεj3, σ∀ εj>

μσ, σ∀ εj– εj,[ ]∈

Mεj3, σ∀– εj,–<⎩

⎪⎨⎪⎧

=

εjj

m��� μ

M���� .=

dxdt���� Px qϕj r*x( )+=

tion bifurcation occurs at some step and the algorithmterminates.

Assume that we have calculated a periodic solutionxm(t) of system (8) with a monotone continuous func�tion ϕm(σ). In this case, a similar computational pro�cedure is organized for the sequence of systems

(9)

where i = 0, 1, …, h; ψ0(σ) = ϕm(σ); and

(10)

Here, N is a positive parameter such that hN < μ2.Finding the periodic solutions xi(t) of system (9),

we obtain a counterexample to Kalman’s conjecturefor each i = 1, 2, …, h. Below are the correspondingexamples.

Example 1. Consider the fourth�order system

(11)

For ϕ(σ) = kσ, linear system (11) is stable for k ∈ (0;9.9) and, for a piecewise continuous nonlinearityϕ(σ) = ϕ0(σ) with sufficiently small ε, the theoremimplies that system (11) has a periodic solution.

The algorithm for constructing periodic solutionsis applied as follows. Let μ = M = 1, ε1 = 0.1, ε2 =0.2, …, ε10 = 1. For j = 1, 2, …, 10, we sequentially

dxdt���� Px qψi r*x( ),+=

ψi σ( )

i σ εm–( )N μεm, σ∀+ εm>

μσ, σ∀ εm– εm,[ ]∈

i σ εm+( )N μεm, σ∀– εm.–<⎩⎪⎨⎪⎧

=

x· 1 –x2 10ϕ x1 10.1x3– 0.1x4–( ),–=

x· 2 x1 10.1ϕ x1 10.1x3– 0.1x4–( ),–=

x· 3 x4,=

x· 4 –x3 x4– ϕ x1 10.1x3– 0.1x4–( ).+=

–30

–20–40 0 20 40x1(t)

–20

–10

0

10

20

40

30

x 2(t

)

Fig. 1. Projection of the trajectory with initial data x1(0) =x3(0) = x4(0) = 0, x2(0) = 20 onto the plane (x1, x2).

542

DOKLADY MATHEMATICS Vol. 82 No. 1 2010

LEONOV et al.

construct solutions of system (11) setting ϕ(σ) equal toϕj(σ) (7). Here, for all εj, j = 1, 2, …, 10, there existperiodic solutions.

Finally, at j = 10, for system (11) with the increas�ing continuous nonlinearity

we conclude the existence of a periodic solution (Fig. 1).Note that if the solution is calculated with initialdata (6) at ε = 1 instead of consecutively increasing εj,then the solution breaks off to zero.

The consecutive construction of periodic solutionsto system (11) is continued with the nonlinearity ϕ(σ)replaced by the strictly increasing function ψi(σ) (9),where μ = 1, εm = 1, N = 0.01, and i = 1, 2, 3, 4, 5.

Here, at i = 5, the solution with initial data x1(0) =x3(0) = x4(0) = 0 and x2(0) = 10 tends to a periodicsolution (Fig. 2).

Example 2. System (11) with the strictly increasingsmooth nonlinearity

(12)

has a periodic solution (Fig. 3).

ϕ σ( ) ϕ10 σ( )1, σ∀ 1>

σ, σ∀ 1– 1,[ ]∈

1, σ∀– 1–<⎩⎪⎨⎪⎧

= =

φ σ( ) σ( )tanh eσ e σ––

eσ e σ–+���������������,= =

0 ddσ����� σ( )tanh 1, σ∀≤<

ACKNOWLEDGMENTS

The work was supported by the Federal TargetedProgram “Scientific and Educational HumanResources for Innovation�Driven Russia” for the years2009–2013.

REFERENCES

1. S. Lefschetz, Stability of Nonlinear Control Systems(Academic, New York, 1965; Mir, Moscow, 1967).

2. R. E. Kalman, Trans. ASME 79, 553–566 (1957).

3. M. A. Aizerman and F. R. Gantmakher, Absolute Stabil�ity of Controlled Systems (Akad. Nauk SSSR, Moscow,1963) [in Russian].

4. G. A. Leonov, D. V. Ponomarenko, and V. B. Smirnova,Frequency Methods for Nonlinear Analysis: Theory andApplications (World Scientific, Singapore, 1996).

5. R. E. Fitts, IEEE Trans. Autom. Control 11, 553–556(1966).

6. N. E. Barabanov, Sib. Mat. Zh. 29 (3), 3–11 (1988).

7. “In Moscow Mathematical Society,” Usp. Mat. Nauk53 (2), 169–172 (1998).

8. J. Bernat and J. Llibre, Dyn. Continuous, DiscreteImpulsive Syst. 2, 337–379 (1996).

9. G. A. Leonov, Dokl. Akad. Nauk SSSR 193, 756–759(1970).

10. G. A. Leonov, I. M. Burkin, and A. I. Shepelyavyi, Fre�quency�Domain Methods in Oscillation Theory (Kluwer,Dordrecht, 1993).

11. G. A. Leonov, Avtom. Telemekh., No. 7, 37–49 (2009).

–10–30 10 20 30x1(t)

–20

–10

0

10

20

30x 2

(t)

0–20

Fig. 2. Projection of the trajectory of the system onto theplane (x1, x2).

–30

–20–40 0 20 40x1(t)

–20

–10

0

10

20

40

30

x 2(t

)

Fig. 3. Projection of the trajectory with initial data x1(0) =x3(0) = x4(0) = 0 and x2(0) = 20 system (12) onto theplane (x1, x2).