Doklady Physics, Vol. 46, No. 4, 2001, pp. 268–270. Translated from Doklady Akademii Nauk, Vol. 377, No. 5, 2001, pp. 626–628.Original Russian Text Copyright © 2001 by Leonov.

MECHANICS

On the Brockett Stabilization ProblemG. A. Leonov

Presented by Academician N.F. Morozov October 8, 2000

Received October 25, 2000

In the book by P. Brockett [1], the following prob-lem was formulated.

Let three matrices A, B, and C be given. Under whatconditions does there exist a matrix K(t) such that thesystem

(1)

is asymptotically stable?

It is worth noting that the problem of stabilizing sys-tem (1) by means of a constant matrix K is well knownin the automatic-control theory [2, 3]. From this pointof view, the Brockett problem can be reformulated inthe following manner.

How much would introduction of a time-dependentmatrix K(t) improve the capabilities of the conventionalstabilization?

In solving problems of stabilization of mechanicalsystems, a more narrow class of stabilizing matricesK(t) should often be considered. These matrices mustbe periodic and have a zero mean value over theirperiod [0, T]:

(2)

Such stabilizing actions are realized in the problems ofstabilization of pendulum systems.

We consider a pendulum with its point of supportvibrating in the vertical direction. In the linear approx-imation, the pendulum vibrations in the vicinity of theupper equilibrium position are described by the equa-tion

(3)

where α and ω0 are positive numbers. As a rule, thefunctions K(t) are assumed to take either form

dxdt------ Ax BK t( )Cx, x Rn∈+=

K t( ) td

0

T

∫ 0.=

θ αθ K t( ) ω02–( )θ+ + 0,=

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1028-3358/01/4604- $21.00 © 20268

βsinωt [4] or

(4)

[5, 6]. For these functions K(t), the effect of stabiliza-tion of the pendulum upper equilibrium position is wellknown for large ω or small T.

In this paper, we present algorithms for constructingperiodic piecewise-constant functions K(t), whichallow us to solve the Brockett problem in a number ofcases.

Theorem 1. Let α2 < 4(β – ). Then, for any τ > 0,there exists a number T > τ such that Eq. (3) with afunction K(t) having form (4) is asymptotically stable.

In particular, the possibility of stabilizing the upperequilibrium position for a pendulum point of supportunder low-frequency vibrations follows from this theo-rem.

We here outline the proof of Theorem 1. ForK(t) = –β, Eq. (3) has two linear manifolds in its phasespace, namely, a stable (η = L1θ) and an unstable (η =

L2θ) one (with η = ). In this case, the rate of solutionconvergence to the unstable manifold is higher than thedivergence rate along this manifold. After a change atK(t) = β, this unstable manifold may turn along the tra-jectories of Eq. (3) and can then attain coincidence withthe straight line η = L1θ before the next change. (Largevalues of T imply that the pendulum vibrates manytimes during time T.) Since the above-mentioned con-vergence of the solution predominates over its diver-gence after the change at the moment t = T, any solutionas a whole can be embedded into a sphere of arbitrarilysmall radius.

We now describe the similar algorithm for system (1).We assume that there exists a matrix K1 such that the

system

(5)

with scalar parameter µ has a stable linear invariantmanifold L(µ). Here, µ ≥ µ0 and µ0 is a certain number.

K t( )β at t [0, T /2)∈β– at t [T /2, T )∈

=

ω02

θ

dxdt------ A µBK1C+( )x=

001 MAIK “Nauka/Interperiodica”

ON THE BROCKETT STABILIZATION PROBLEM 269

We also assume that

(6)

and, for any given number δ > 0, there exists a numberµ1 ≥ µ0 such that

(7)

Here, x(0, x0) = x0 and the existence of limit (6) impliesthat the set L(µ) ∩ {|x | ≤ 1} is in the ε-neighborhoodL0 ∩ {|x | ≤ 1}, where ε 0 as µ +∞.

The above-formulated assumption implies that thetrajectories rapidly converge on manifold L(µ) for largevalues of parameter µ.

We denote by M(µ) the linear invariant manifold ofsystem (5) such that for this manifold,

We also assume that M(µ) is the manifold of slowmotions; i.e., there exists a number R such that the ine-quality

(8)

is valid for arbitrary µ ≥ µ0 . We then assume that amatrix K2 exists such that for the system

, (9)

the fundamental matrix Y(t) can be found with Y(0) = I,which satisfies the condition

(10)

for a certain number τ.

We now define the matrix K(t) with period (2 + τ) inthe following manner:

(11)

Theorem 2. System (1) with a matrix K(t) havingthe form of (11) is asymptotically stable for sufficientlylarge µ.

The following corollary of Theorem 2 takes place.

Theorem 3. Let the matrices K1 and K2 exist andsatisfy the following conditions:

(i) The matrix BK1C has (n – 1) eigenvalues withnegative real parts, and detBK1C = 0.

L µ( )µ +∞→lim L0=

x 1 x0,( ) δ,≤x0∀ x 1={ } L µ( ), µ µ1.≥∩∈

M µ( )µ +∞→lim M0,=

dimM µ( ) dimL µ( )+ n, M µ( ) L µ( )∩ 0{ } .= =

x 1 x0,( ) R,≤x0∀ x 1={ } M µ( )∩∈

dydt------ A BK2C+( )y=

Y τ( )M0 L0⊂

K t( )µK1 for t [0, 1)∈K2 for t [1, 1 τ )+∈µK1 for t [1 τ+ 2 τ ).+,∈

=

DOKLADY PHYSICS Vol. 46 No. 4 2001

(ii) For a certain number λ and a vector u ≠ 0 satis-fying the equality BK1Cu = 0, the vector-function

is periodic.

Then, there exists a periodic matrix K(t) such thatsystem (1) is asymptotically stable.

In a two-dimensional case, Theorem 3 has the fol-lowing simple formulation.

Theorem 4. Let n = 2 and there exist matrices sat-isfying the following conditions:

(I) detBK1C = 0 and TrBK1C ≠ 0.

(II) The matrix A + BK2C has complex-valuedeigenvalues.

Then, there exists a periodic matrix K(t) havingform (11) such that system (1) is asymptotically stable.

A more complicated algorithm for constructing thepiecewise stabilizing function K(t) allows us to provethe following theorem.

Theorem 5. Let n = 3, B be a column, C be a row,and the following conditions be satisfied:

(1) det(B, AB, A2B) ≠ 0.

(2) CB ≠ 0.

(3) There exists a number k1 such that the matrix A +k1BC has two complex-valued and one negative eigen-value.

(4) There exists a number k2 such that the function

has at least one zero in the interval (−∞, 0).

Then, there exists a periodic function K(t) such thatthe system of Eqs. (1) is asymptotically stable.

We then consider system (1) provided that B is avector column, C is a vector row, and K(t) is a scalarpiecewise continuous function R1 R1. Assumingcomplete controllability of pair (A, B) and completeobservability of pair (A, C), we reduce system (1) to theform [7]

Here, aj and cj are certain numbers. In the case of cn ≠ 0,without loss of generality, we can set cn = 1.

Theorem 6. Let the following inequalities be satis-fied:

(i) for n > 2, c1 ≤ 0, …, cn – 2 ≤ 0;

A BK2C λ I+ +( )t[ ]uexp

C A k2BC+( )t[ ]exp B

x1 x2,=

…………

xn 1– xn,=

xn – anxn … a1x1+ +( ) K t( ) cnxn … c1x1+ +( ).–=

270 LEONOV

(ii)

Then, there are no functions K(t) for which system (1)is asymptotically stable.

Theorem 6 is a consequence of the positive invari-ance for the set

We now apply the above-formulated theorems pro-vided that n = 2, B is a column, C is a row, and K(t) isa scalar function. (This case is important in control the-ory.) To do this, we introduce the transfer function forsystem (1):

Here, p is a complex-valued variable.In what follows, we assume that ρ ≠ 0. For the prob-

lem under consideration, without loss of generality, wecan set ρ = 1. Moreover, we assume that the functionW(p) is nongenerate; i.e., the inequality

is met. It is well known [7] that system (1) can berewritten in this case in the form

(12)

It is easy to see that the stabilization of system (12) bymeans of a constant function K(t) ≡ K0 is possible if andonly if α + K0 > 0 and β + γK0 > 0. For the number K0to meet these inequalities, it is necessary and sufficientthat either the condition γ > 0 or the relationships γ ≤ 0and αγ < β be satisfied.

We now consider the case when the stabilization bymeans of a constant function K(t) ≡ K0 is impossiblesince γ ≤ 0 and αγ > β. To do this, we use Theorem 4. Itis evident that hypothesis (i) of Theorem 3 is metbecause detBK1C = K1detBC = 0 and TrBK1C =K1CB = –K1 ≠ 0.

Hypothesis (II) of Theorem 4 will be met if, for acertain K2 , the polynomial

has complex-valued zeroes. It is easy to see that for

c1 an cn 1––( ) a1,>c1 c2 an cn 1––( ) a2,>+

…………………………

cn 2– cn 1– an cn 1––( ) an 1– .>+

x1 0≥ … xn 1–, , 0, xn cn 1– xn 1– … c1x1+ + + 0≥ ≥{ } .

W p( ) C A pI–( ) 1– Bρp γ+

p2 αp β+ +----------------------------.= =

γ2 αγ– β+ 0≠

σ η ,=

η –αη βσ– K t( ) η γσ+( ).–=

p2 αp β K2 p γ+( )+ + +

such K2 to exist, it is necessary and sufficient that theinequality

(13)

be satisfied.Hence, if inequality (13) holds true, there exists a

periodic function K(t) such that system (12) is asymp-totically stable.

As is easy to see, the hypotheses of Theorem (6) aremet if the inequality

(14)

is satisfied.Hence, we arrive at the following theorem [8].Theorem 7. If inequality (13) is met, then there

exists a periodic function K(t) such that system (12) isasymptotically stable.

If inequality (14) is valid, there are no functions K(t)for which system (12) is asymptotically stable.

This result was also obtained in [9] by the averagingmethod for another class of stabilizing functions K(t)having the form

REFERENCES1. R. Brockett, Open Problems in Mathematical Systems

and Control Theory. A Stabilization Problem (Springer-Verlag, Berlin, 1999).

2. L. A. Zadeh and C. A. Desoer, Linear System Theory:The Spatial State Approach (McGraw-Hill, New York,1963; Nauka, Moscow, 1970).

3. A. A. Pervozvanskiœ, Course of Automatic Control The-ory (Nauka, Moscow, 1986).

4. Yu. A. Mitropol’skiœ, Averaging Methods in NonlinearMechanics (Naukova Dumka, Kiev, 1971).

5. V. I. Arnol’d, Ordinary Differential Equations (Nauka,Moscow, 1971; MIT Press, Cambridge, 1973).

6. V. I. Arnold, Mathematical Methods of ClassicalMechanics (Nauka, Moscow, 1979; Springer-Verlag,New York, 1989).

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

8. G. A. Leonov, in Proceedings of the International Con-ference on Control of Oscillations and Chaos, St. Peters-burg, 2000, p. 38.

9. L. Morean and D. Aeyels, in Proceedings of the Confer-ence of Decision and Control, Phoenix, 1999, p. 108.

Translated by V. Chechin

γ2 αγ– β+ 0>

γ2 αγ– β+ 0<

K t( ) k0 k1ω ωtcos+( ), ω @ 1.=

DOKLADY PHYSICS Vol. 46 No. 4 2001