St. Petersburg Math. J. 13 (2001),.1 Vol. 13 (2002), No. 4

Transcription

1 St Petersburg Math J 13 (21), 1 Vol 13 (22), No 4 BROCKETT S PROBLEM IN THE THEORY OF STABILITY OF LINEAR DIFFERENTIAL EQUATIONS G A Leonov Abstract Algorithms for nonstationary linear stabilization are constructed Combined with a nonstabilizabiity criterion, these algorithms result in the solution of the Brockett problem in a number of cases 1 Introduction In the book [1], R Brockett formulated the following problem For a triplet of matrices A, B, and C, what conditions ensure the existence of a matrix K(t) such that the system dx (1) dt = Ax + BK(t)Cx, x Rn, is asymptotically stable The problem of stabilizing system (1) with the help of a constant matrix K is classical for automatic control theory [2, 3] From this point of view, Brockett s problem can be reformulated as follows To what extent are the possibilities of classical stabilization extended by introducing matrices K(t) that depend on time t? Stabilizing mechanical systems often necessitates the invocation of a special class of stabilizing matrices K(t) These matrices must be periodic and have zero mean on the period [, T ]: (2) T K(t) dt = For example, consider a linear approximation near an equilibrium point for the pendulum with vertically oscillating suspension point: (3) θ + α θ + ( K(t) ω 2 ) θ =, where α and ω are positive numbers Here, the most common choice for the function K(t) is either β sin ωt (see [4]), or { β, t [, T/2), (4) K(t) = β, t [T/2, T ) (see [5, 6]) For such functions K(t), the effect of stabilization of the upper equilibrium point is well known for large ω and, consequently, small T In this paper, we present certain algorithms enabling us to construct periodic piecewise constant functions K(t) that solve the Brockett problem in a number of cases, and also periodic functions K(t) that satisfy (2) and solve the stabilization problem Moreover, we show that low-frequency stabilization (T 1) is possible for the pendulum equation (3) with K(t) of the form (4) 2 Mathematics subject Classification Primary 34D5 Key words and phrases Stability, transfer function, controllability, observability, invariant manifold c 22 American Mathematical society 1

2 2 Conditions sufficient for stabilization Suppose we have two matrices K j (j = 1, 2) such that the systems (5) dx dt = (A + BK jc)x, x R n, possess stable linear manifolds L j and invariant linear manifolds M j We assume that M j L j = {} and dim M j +dim L j = n, and that λ j, κ j, α j, and β j are positive numbers satisfying the inequalities (6) x(t) α j e λ jt x(), x() L j, (7) x(t) β j e κjt x(), x() M j, In what follows We also assume that there exists a matrix U(t) and a number τ > such that (8) Y (τ)m 1 L 2, where Y (t) is the fundamental matrix (Y () = I) of the following system: (9) dy dt = (A + BU(t)C)y Theorem 1 Suppose that (1) λ 1 λ 2 > κ 1 κ 2 and that (8) is true Then there exists a periodic matrix K(t) such that system (1) is asymptotically stable Proof Condition (1) implies that for every T > there exist two numbers t 1 and t 2 such that (11) λ 1 t 1 + κ 2 t 2 < T, λ 2 t 2 + κ 1 t 1 < T We define the periodic matrix K(t) as follows: K 1, t [, t 1 ), (12) K(t) = U(t t 1 ), t [t 1, t 1 + τ), K 2, t [t 1 + τ, t 1 + t 2 + τ) The period of K(t) is equal to t 1 = t 2 + τ We show that if T is sufficiently large, then system (1) with such matrix K(t) is asymptotically stable For this, we introduce nonsingular matrices S j bringing system (5) to a canonical form: (13) dz j dt = Q jz j, dim z j = dim L j, dw j dt = P j w j, dim w j = dim M j 2

3 Here (14) S j x = There is no loss of generality in assuming that ( zj w j ) (15) z j (t) e λ jt z j (), w j (t) e κ jt w j () Relations (12) (14) show that ( ) z2 (t (16) 1 + τ) w 2 (t 1 + τ) ( ) = S 2 Y (τ)s1 1 z1 (t 1 ) w 1 (t 1 ) The inclusion (8) implies that the matrix S 2 Y (τ)s1 1 has the following structure: ( ) S 2 Y (τ)s1 1 R11 (τ) R = 12 (τ) R 21 (τ) There fore, by (11) and (15), z 2 (t 1 + t 2 + τ) R 11 (τ) e 2T z 1 () + R 12 (τ) e T w 1 (), w 2 (t 1 + t 2 + τ) R 21 (τ) e T z 1 (), which implies that for all sufficiently large values of T and for the initial values in the ball x() 1, we have x(t 1 + t 2 + τ) 1 2 Since the matrix K(t) is periodic, it follows that system (1) is asymptotically stable Now, we assume that the matrix K(t) in (1) is a scalar function, K 1 = K 2 = K, λ 1 = λ 2 = λ, κ 1 = κ 2 = κ, U(t) U, K U <, the function Y (t) is uniformly bounded on the interval (, + ), and there exists a sequence τ j such that (17) Y (τ j )M 1 L 2 Theorem 2 If λ > κ and (17) is fulfilled, then there exists a T -periodic function K(t) such that (2) is true and system (1) is asymptotically stable Proof We define K, t [, U τ j /2K ), (18) K(t) = U, t [ U τ j /2K, τ j + U τ j /2K ), K, t [τ j + U τ j /2K, τ j + U τ j /K ) The period of K(t) is equal to T = τ j (1 + U /K ) Here, τ j is a sufficiently large number satisfying condition (17) The rest of the proof repeats the arguments used in the proof of Theorem 1 We apply Theorem 2 to equation (3) with a function K(t) of the form (4) Suppose that (19) α 2 < 4(β ω 2 ) 3

4 In this case, without loss of generality, we may assume that β ω 2 α 2 /4 = 1 We set K = β and U = β Condition (19) implies that the characteristic polynomial of equation (3) with K(t) = U has complex zeros, and, consequently, condition (17) is fulfilled for some τ 1 > Clearly, here we have τ j = τ 1 + 2jπ Since the zeros in question of the characteristic polynomial have negative real parts, we easily see that Y (t) is uniformly bounded on (, + ) For K(t) = K = β, the quantities λ and κ can easily be calculated: λ = α 2 + α 2 κ = α 2 + α (β + ω2 ), 4 + (β + ω2 ) Thus, all assumptions of Theorem 2 are fulfilled, and equation (3) with K(t) of the form (18) is asymptotically stable for sufficiently large j This result can be stated as follows Proposition 1 Under condition (19), for every τ there exists T > τ such that equation (3) with a function K(t) of the form (4) is asymptotically stable In particular, this implies the possibility of stabilizing the upper equilibrium point of the pendulum for low-frequency vertical oscillations of the suspension point Naturally, here the amplitude a of the oscillations is large: a = lt 2 β 8 where l is the length of the pendulum and β is the absolute value of the acceleration divided by l The stabilization effect is well known for high-frequency oscillations (for small T ); see [5, 6] The following lemmas are often useful for checking condition (8) Consider the system (2) ż = Qz, z R n, where Q is a constant nonsingular (n n)-matrix and h is a vector in R n Lemma Suppose that the solution z(t) of system (2) has the form z(t) = v(t) + w(t), where v(t) is a periodic vector-valued function such that h v(t), and w(t) is a vector-valued function for which + w(τ) dτ < +, Then there exist two numbers τ 1 and τ 2 such that, lim w(t) = t + (21) h z(τ 1 ) > and h z(τ 2 ) < Proof Assuming the contrary, we see that either h z(t) for any t, or h z(t) for any t For definiteness, suppose that h z(t) for any t Then the relation h v(t) implies that (22) lim t + t h z(τ) dτ = + 4

5 On the other hand, we have int t h z(τ) dτ = h Q 1( z(t) z() ) Since z(t) is uniformly bounded on (, + ), the function t h z(τ) dτ is uniformly bounded, which contradicts (22)This proves the lemma Lemma 2 Let n = 2 and let the matrix Q have complex eigenvalues Then for any two nonzero vectors h, u R 2 there exist numbers τ 1 and τ 2 such that (23) h e Qτ 1 u > and h e Qτ 2 u < This obvious assertion can be viewed as a consequence of Lemma 1 Lemma 3 Suppose that the matrix Q has two complex eigenvalues λ ± iω, and that the remaining eigenvalues λ j (Q) of Q satisfy the condition Re λ j (Q) < λ Let h, u R n be two vectors such that (24) det(h, Q h,, (Q ) n 1 h), (25) det(u, Qu,, Q n 1 u) Then there exist numbers τ 1 and τ 2 such that (26) h e Qτ 1 u > and h e Qτ 2 u < We recall that conditions (24) and (25) are controllability conditions for the pair (Q, u) and observability conditions for the pair (Q, h) Proof It suffices to observe that the solution z(t) = e Qt u can be written as z(t) = e λ t (v(t) + w(t)), where v(t) and w(t) satisfy the assumptions of Lemma 1 The relation h v(t) follows from the observability of (Q, h) and the controllability of (Q, u) Theorem1 and Lemma 2 readily imply the following statement Theorem 3 Let n = 2 Suppose there exist matrices K and U satisfying the following conditions: 1) det BK C =, Tr BK C ; 2) the matrix A + BU C has complex eigenvalues Then there exists a periodic matrix K(t) such that system (1) is asymptotically stable Proof It suffices to set K 1 = K 2 = µk, where µ is a sufficiently large number, and Tr µbk C < In this case, obviously, the assumptions of Theorem 1 are fulfilled Now we consider the case where B is a column vector, C is a row vector, and K(t) : R 1 R 1 is a piecewise continuous function 5

6 We introduce the transfer function of system (1): W (p) = C(A pi) 1 B, where p is a complex variable WE assume that the function W (p) is nondegenerate This means that the pair (A, B) is controllable and the pair (A, C) is observable Lemma 4 If the hyperplane {h z = } is an invariant manifold for the system (27) ẋ = (A + µbc)x, µ, then the pair (A, h) is observable Proof Suppose that (A, h) is not observable In this case (see [7]), there exists a vector q and a number γ such that h q =, Aq = γq, q The observability of the pair (A, C) implies the inequality Cq Since {h z = } is invariant with respect to (27), for all z {h z = } we have h (A + µbc) k z =, k = 1, 2, Putting z = q and k = 1, we obtain h BCq =, whence h B = For z = q and k = 2, using the preceding relation, we obtain h AB = Continuing in this way, we obtain h A k 1 B = The controllability of the pair (A, B) implies that h =, which contradicts our assumption that the pair (A, h) is not observable The lemma is proved Lemma 5 Let u R n If the line {αu α R 1 } is invariant with respect to system (27), then the pair (a, u) is controllable Proof Invariance yields (A + µbc) k u = γ k u, k =, 1,, where the γ k are some numbers The observability of (A, C) yields Cu (see [7]) Therefore, for z R n satisfying z u =, z Au =, and z A n 1 u = we have z B = z AB = = z A n 1 B =, whence z = because the pair (A, B) is controllable Thus, the relations z u = = z A n 1 u = imply that z = Therefore, the pair (A, u) is controllable The following result is a consequence of Theorem 1 and Lemmas 3-5 Theorem 4 Suppose that B, C R n, dim M 1 = 1, dim L 2 = n 1, and the inequality (1) is fulfilled Also, we assume that for some number U K j the matrix A + U BC has two complex eigenvalues λ ± iω, and that the remaining eigenvalues λ j satisfy the condition Re λ j < λ Then there exists a periodic function K(t) such that system (1) is asymptotically stable Proof Combining Lemma 4 with the controllability of the pair (A, B), the observability of the pair (A, C), and the fact that U K j, we deduce that the pair (A + U BC, h) is observable, where h is a normal to L 2 Lemma 5 shows that the pair (A + U BC, u) is controllable Here u and u M 1 Consequently, by Lemma 3, there is a number τ such that h exp[(a + U BC)τ]u = 6

7 Since this implies (8, theorem 4 is proved Lemmas 1-5 were obtained for checking relation (8) in the case where the rotation of the subspace M 1 is caused by complex eigenvalues Another approach involves an impulse action U(t) = µ with large µ on a small time interval In this case, the velocity vector ẋ is often close to the vector γb, where γ is a number We describe this approach in more detail We consider the system (27) with large parameter µ: µ 1 Lemma 6 Suppose that CB = and that h, u are two vectors satisfying h B and Cu Then there exist numbers µ and τ(µ) > such that h x(τ, u) = and lim τ(µ) = µ Proof Introducing h u t = µh BCu, R = we fix µ so as to have t > and It is easily seen that (1 + 2 µ B C t ) u 1 (2 A t + 4 µ A B C t 2 ), 2 A t + 4 µ A B C t 2 < 1 Cẋ(t, u) = CAx(t, u) A C max x(t, u) t [,2t ] for all t [, 2t ] Hence, for t [, 2t ] we have Combining this with (27), we obtain whence Cx(t, u) Cu 2 A C t max t [,2t ] x(t, u) x(t, u) u µbcut (2 A t + 4 µ A B C t 2 ) max x(t, u), t [,2t ] x(t, u) R, t [, 2t ], h x(t, u) h u µh BCut (2 A t + 4 µ A B C t 2 )R h, t [, 2t ] It is easy to check that (2 A t + 4 µ A B C t 2 )R h = O ( ) 1 µ Therefore, for large µ there exists τ [, 2t ] such that h x(τ, u) = The lemma is proved Lemma 7 Suppose that CB and that h and u are two vectors satisfying h B, Cu, and h ucb h BCu < 1 7

8 Then there exist numbers µ and τ(µ) > such that h x(τ, u) = and lim τ(µ) = µ Proof We define t = 1 ( ) µcb log 1 h ucb h, BCu R = (1 + B C CB 1 (e 2µCBt + 1) u 1 (2 A t + 2 A B C t CB 1 (e 2µCBt + 1)) The number µ is taken in such a way that t > and It is easily seen that 2 A t + 2 A B C t CB 1 (e 2µCBt + 1) < 1 Cẋ(t, u) µcbx(t, u) A C max x(t, u) t [,2t ] for all t [, 2t ] Hence, for t [, 2t ] we have Cx(t, u) e µcbt Cu 1 e2µcbt µcb Combining this with (27), we obtain BCu x(t, u) u CB (eµcbt 1) whence A C max x(t, u) t [,2t ] ( 2 A t + 2 A B C t CB 1 (e 2µCBt + 1) ) max x(t, u), t [,2t ] x(t, u) R, t [, 2t ], h x(t, u) h u h BCu CB (eµcbt 1) ( 2 A t + 2 A B C t CB 1 (e 2µCBt + 1) ) R h, t [, 2t ] It is easy to check that ( 2 A t + 2 A B C t CB 1 (e 2µCBt + 1) ) ( ) 1 R h = O µ There fore, for large µ there exists τ [, 2t ] such that h x(τ, u) = The lemma is proved Theorem 5 Suppose that B, C R n, CB =, dim M 1 = 1, dim L 2 = n 1, and inequality (1) is fulfilled Then there exists a periodic function K(t) such that system (1) is asymptotically stable Proof By Lemmas 4 and 5, the controllability of (A, B) and the observability of (A, C) imply the observability of ((A+KBC), h) for any K K 2 and the controllability of ((A+KBC), u) 8

9 for any K K 1 Here h is a normal to the hyperplane L 2, and u is a nonzero vector in the subspace M 1 It follows that h B and Cu Then, by Lemma 6, there exist numbers µ and τ(µ) such that if U(t) µ, then system (8) satisfies condition (9) Theorem 6 Suppose that CB and the matrix A has a positive eigenvalue κ and n 1 eigenvalues with real parts less that λ, where λ > κ We also assume that lim p κ CB < 1 (κ p)w (p) Then there exists a periodic function K(t) such that system (1) is asymptotically stable Proof Without loss of generality, we may assume that the matrix A and the vectors B and C are of the form ( ) ( ) κ b1 A =, B =, C = (c A 1, c 2 ), 2 where A 2 is an (n 1) (n 1)-matrix, b 2 R n 1, and c 2 R n 1 In this case, the normal h to the subspace L 1 = L 2 and the vector u M 1 = M 2 are of the form h = u = ( 1 u) Therefore, by Lemma 7, there exist numbers µ and τ(µ) such that if U(t) µ, then condition (9) is fulfilled for system (8) provided that b 2 CB c 1 b 1 < 1 We easily see that c 1 b 1 because (A, B) is controllable and (A, C) is observable, and c 1 b 1 = lim p κ (κ p)w (p) Thus, all assumptions of Theorem 1 are fulfilled, and, consequently, system (1) is stabilizable Lemma 8 Suppose that an (n 2)-dimensional linear subspace L invariant for system (27) lies in the hyperplane {h z = } If h B = CB =, then the pair (A, h) is observable Proof The controllability of the pair (A, B) and the invariance of L with respect to (27) imply that B L (see [7]) Therefore, the linear hull of B and L coincides with the hyperplane {h z = } Suppose that the pair (A, h) is not observable Then there is a vector q and a number γ such that h q =, Aq = γq (see [7]) The observability of the pair (A, C) implies the relation Cq Since L is invariant, we have h (A + µbc) k z =, k =, 1,,, z L The above arguments imply the existence of a number ν and a vector z L such that q = z + νb If ν, then for k = 1 we have νh AB =, whence h AB = Combining this with the relations h B = CB = for k = 2, we obtain νh A 2 B = Successively continuing this procedure for k = 3,, we obtain the relations h A k B = By the controllability of (A, B), these relations imply h =, which contradicts the definition of (A, h) the vector ν Thus, we have proved the observability of (A, h) for ν If ν =, then we use the same arguments as in the proof of Lemma 4 9

10 Theorem 7 Suppose that B, C R n, CB =, dim M 1 = 1, dim L 2 = n 2, and inequality (1) is true If the assumptions of Theorem 4 are fulfilled for some number U K j, then there exists a periodic function K(t) such that system (1) is asymptotically stable Proof Observe that, as µ, the integral manifold Ω(µ) consisting of the trajectories x(t, x ) of system (27) with initial values x inl 2 approaches the hyperplane {h x = } Here h is a normal to the linear hull of L 2 and B Such convergence was described in the proof of Lemma 6 Lemma 7 implies that the pair ((A + U BC), h) is observable, while Lemma 5 implies that the pair ((A + U BC), u) with u M 1, u is controllable By Lemma 3, this shows that for system (9) with U(t) = U the sign of the function h y(t, u) changes for some values of t Therefore, for sufficiently large µ, there exists a number τ (µ) > such that y(τ (µ), u) Ω(µ) Perturbing the right-hand side of (1) slightly, we can ensure that Cy(τ (µ), u) (asymptotic stability is preserved under small perturbations of the right-hand sides of periodic systems) Next, if in (9) we set U(t) = µ (or U(t) = µ) on (τ, τ], then, moving along the manifold Ω(µ), we reach the set L 2 at the moment t = τ Here the sign of µ is chosen so that τ > τ (see the proof of Lemma 6) Now, we have y(τ, u) L 2, and, consequently, condition (8) is fulfilled Thus, all assumptions of Theorem 1 are satisfied 3 Necessary stabilization conditions Now, we pass to conditions necessary for stabilization We consider the case where B is a column vector, C is a row vector, and K(t) : R 1 R 1 is a piecewise continuous function This case in important for control theory As before, W (p) is the transfer function of system (1): W (p) = C (A pi) 1 B = c n p n c 1 p n + a n p n a 1, where the c j and a j are real numbers If the transfer function W (p) is nondegenerate, then system (1) can be written in the following scalar form (see [8]): (28) ẋ 1 = x 2, ẋ n 1 = x n, ẋ n = (a n x n + + a 1 x 1 ) K(t)(c n x n + + c 1 x 1 ) Clearly, here we have x 1 C = (c 1,, c n ), x = x n, 1 A = 1, B = a 1 a n 1 1

11 We recall that the transfer function W (p) is nondegenerate if and only if the polynomials c n P n c 1, p n + a n p n a 1 have no common zeros In what follows, we assume that c n In this case, without loss of generality we may put c n = 1 Theorem 8 Assume that the following conditions are satisfied: 1) for n > 2, we have c 1,, c n 2 ; 2) c 1 (a n c n 1 ) > a 1, c 1 + (a n c n 1 )c 2 > a 2, c n 2 + (a n c n 1 )c n 1 > a n 1 Then there is no function K(t) for which system (1) is asymptotically stable Proof Consider the set Ω = {x 1,, x n 1, x n + c n 1 x n c 1 x 1 } R n We prove that Ω is positively invariant, ie, if x(t ) Ω, then x(t) Ω for all t t Observe that if j = 1,, n 1 and τ is such that then x j (τ) =, x i (τ) >, i j, i n 1, x n (τ) + c n 1 x n 1 (τ) + + c 1 x 1 (τ) >, (29) ẋ j (τ) > Indeed, for j = 1,, n 2 we have ẋ j (τ) = x j+1 (τ) > For n = 2, we have and for n > 2 we have ẋ 1 (τ) = x 2 (τ) > c 1 x 1 (τ) =, ẋ n 1 (τ) = x n (τ) > c n 2 x n 2 (τ) c 1 x 1 (τ) Next, we observe that if x n (τ) + c n 1 x n 1 (τ) + + c 1 x 1 (τ) =, and x j (τ) >, j = 1,, n 1, then (3) ( xn (τ) + c n 1 x n 1 (τ) + + c 1 x 1 (τ) ) > Indeed, we have ( xn (τ) + c n 1 x n 1 (τ) + + c 1 x 1 (τ) ) = = ( a n 1 + c n 2 + (a n c n 1 )c n 1 )x n 1 (τ) + + ( a 2 + c 1 + (a n c n 1 )c 2 )x 2 (τ)+ +( a 1 + (a n c n 1 )c 1 )x 1 (τ) 11

12 and it remains to use condition 2) of the theorem Inequalities (29) and (3) imply that the boundary of the set Ω is transversal to the vector field of (28) almost everywhere, and the solutions of (28) pierce the boundary inward Ω almost everywhere Since the solutions of (28) depend on the initial values continuously, we see that the set Ω is positively invariant, whence it easily follows that system (28) is not asymptotically stable The theorem is proved Another condition ensuring that system (1) is nonstable is well known (see [5]): Tr (A + BK(t)C) α >, t R 1 4 Stabilization of systems of order two and three Now we show how the above results can be applied to the case where n = 2, B is a column vector, C is a row vector, and K(t) is a scalar function We introduce the transfer function of system (1): W (p) = C(A pi) 1 B = ρp + γ p 2 + αp + β Here p is a complex variable In what follows, we assume that ρ Furthermore, there is no loss of generality in assuming that ρ = 1 We also assume that the function W (p) is nondegenerate, ie, we have γ 2 αγ + β It is well known (see [8]) that in this case system (1) can be written as follows: (31) σ = η, η = αη βσ K(t)(η + γσ) It is easy to see that stabilization of system (31) with the help of a constant matrix K(t) K is possible if and only if α + K >, β + γk > A number K satisfying these inequalities exists if and only if γ >, or γ and αγ < β We consider the case where stabilization with the help of a constant K(t) K is impossible: γ, αγ > β We apply Theorem 3 Clearly, condition 1) of Theorem 3 is fulfilled, because det BK C = and Tr BK C = K CB = K Condition 2) of Theorem 3 is fulfilled if for some U the polynomial p 2 + αp + β + U (p + γ) has complex zeros We easily see that such a number U sexists if and only if (32) γ 2 αγ + β > Thus, if inequality (32) if fulfilled, then there exists a periodic function K(t) such that system (32) is asymptotically stable The same result can be obtained with the help of Theorem 6 12

13 For this, without loss of generality we assume that α > This can always be achieved if we properly choose K in the expression (α + K )η (β + γk )σ ( K(t) K ) (η + γσ) and change the notation: α + K α, β + γk β, and K(t) K K(t) The inequality α > implies that λ > κ Here lim p κ CB κ + λ = (κ p)w (p) κ + γ Therefore, all conditions of Theorem 6 are fulfilled if (λ γ)(κ + γ) = γ 2 + αγ β < This inequality coincides with (32) It is easy to see that if (33) γ 2 αγ + β <, then the conditions of Theorem 8 are also fulfilled Thus, the following result is true Theorem 9 [9] If inequality (32) is fulfilled, then there exists a periodic function K(t) such that system (31) is asymptotically stable If inequality (33) is fulfilled, then there are no functions K(t) for which system (31) is asymptotically stable This result was also obtained in [1] for a different class of stabilizing functions K(t) of the form K(t) = (k + k 1 ω cos ωt), ω 1, with the help of averaging Now we consider systems (1) of order three with various transfer functions 41 W (p) = 1 p 3 + ap 2, where a, b, and c are some numbers + bp + c If a > and b >, then stationary stabilization is possible Suppose a > and b In this case, stationary stabilization is impossible; we apply Theorem 7 Clearly, if U is sufficiently large, then the polynomial p 3 + ap 2 + bp + U + c has one negative zero and two complex zeros λ ± iω, λ > We take K 1 so that p 3 + ap 2 + bp + K 1 + c = (p κ)(p 2 + α 1 p + β 1 ) with α 1 = a + κ 1 and β 1 = b + (a + κ 1 )κ 1 For κ 1 large, the polynomial p 2 + α 1 p + β 1 has complex zeros with real part equal to (a + κ 1 )/2 We take K 2 so that p 3 + ap 2 + bp + K 2 + c = (p + λ 2 )(p 2 + α 2 p + β 2 ) with α 2 = a λ 2 and β 2 = b (a λ 2 )λ 2 For λ 2 large, the polynomial p 2 + α 2 p + β 2 has complex zeros with real part equal to (λ 2 a)/2 We have dim M 1 = dim L 2 = 1, dim M 2 = dim L 1 = 2, 13

14 λ 1 = a + κ 1, κ 2 = λ 2 a, 2 2 λ 1 λ 2 κ 1 κ 2 = a(λ 2 + κ 1 ) > Thus, all conditions of Theorem 7 are fulfilled Since Tr (A + BK(t)C) = a, asymptotic stability is impossible for a < So, we can state the following result Theorem 1 For a > system (1) is stabilizable For a < stabilization is impossible 42 W (p) = p p 3 + ap 2 + bp + c If a > and c >, stationary stabilization is possible We consider the case where a > and c <, and apply Theorem 5 with K 1 = K 2, λ 1 = λ 2 = λ, and κ 1 = κ 2 = κ We take K 1 so that (p κ)(p 2 + αp + β) = p 3 + ap 2 + (K 1 + b)p + c with α = a + κ and β = c/κ If κ is small, the polynomial p 2 + αp + β has complex zeros with real part equal to (a + κ)/2 Thus, M 1 = M 2, L 1 = L 2, dim M 1 = 1, dim L 2 = 2, and λ = (a + κ)/2 Clearly, for small κ we have λ > κ Since Tr (A + BK(t)C) = a for a <, asymptotic stability is impossible This leads to the following result Theorem 11 Suppose a and c Then system (1) is stabilizable if and only if a > 43 W (p) = p 2 p 3 + ap 2 = bp + c If b > and c >, stationary stabilization is possible Theorem 8 shows that if b < and c <, then stabilization is impossible We consider the case where b >) and c < and apply Theorem 1 Putting K 1 = K 2, λ 1 = λ 2 = λ, and κ 1 = κ 2 = κ, we take K 1 so that (p κ)(p 2 + αp + β) = p 3 + (a + K 1 )p 2 + bp + c Here (c + κb) α = κ 2, β = c κ Below it is assumed that κ is small In this case, we define (c + κb) λ = 2κ 2 (c + κb) 2 4κ 4 + c κ Obviously, λ > κ for b + κ 2 >, and, consequently, condition (1) of Theorem 1 is fulfilled Now we construct U(t) so as to ensure (8) We introduce a number U such that (p ν)(p 2 + α 1 p + β 1 ) = p 3 + (a + U )p 2 + bp + c 14

15 Here (c + κb) α 1 = ν 2, β 1 = c ν Suppose that ν is sufficiently large Without loss of generality, we assume that ( ) ( ) ν b1 A + U )BC =, B =, C = (c Q 1, c 2 ) Here Q is a (2 2)-matrix, and b 2 and c 2 are two-dimensional vectors For ν large, the matrix Q has complex eigenvalues Therefore, for any nonzero vector u M 1 there exists τ 1 > such that the vectors ( ) 1 B,, [exp(a + U BC)τ 1 ]u lie in one plane It follows that for some ρ we have ( ) 1 [exp(a + U BC)τ 1 ]u = + ρ If b 1 >, then ρ [ b 1 1, ], and if b 1 <, then ρ [, b 1 1 ] The relations imply that CB c 1 b 1 = b 2 1 lim (ν p)w (p) = 1 + α 1ν + β 1 ν 2 p ν ( b1 b 2 ) = 1 2c ν 3 b ν 2 < 1 (34) CB(1 + ρb 1 ) c 1 b 1 < 1 for sufficiently large ν By Lemma 8, the above inequality implies the existence of numbers µ and τ(µ) such that ( ) 1 [exp(a + (U + µ)bc)τ(µ)][exp(a + U BC)τ 1 ]u = Since the planes L 2 and { x big( 1 ) x = } intersect, and the matrix Q has complex eigenvalues, it follows that [exp(a + U BC)τ 2 ][exp(a + (U + µ)bc)τ(µ)][exp(a + U BC)τ 1 ]u L 2, for some τ 2 > Thus, the inclusion (8) is valid for the function U, t [, τ 1 ), U(t) = U + µ, t [τ 1, τ 1 + τ(µ)), U, t [τ 1 + τ(µ), τ 1 + τ(µ) + τ 2, where τ = τ 1 + τ(µ) + τ 2 We have proved the following result Theorem 12 Suppose that b and c < Then system (1) is stabilizable if and only if b > 15

16 References 1 R Brockett, A stabilization problem, Open Problems in Mathematical Systems and Control Theory Springer, London, 1999, pp L Zadeh and Ch Desoer, Linear system theory, McGraw-Hill Book Co, Inc, New York etc, A A Pervozvanskii, A course in automatic control theory, Nauka, Moscow, 1986 (Russian) 4 Yu A Mitropol skii, Averaging method in nonlinear mechanics, Naukova Dumka, Kiev, 1971 (Russian) 5 V I Arnol d, Ordinary differential equations, Nauka, Moscow, 1971; English transl Of 3rd ed, Springer-Verlag, Berlin, , Mathematical methods of classical mechanics, Nauka, Moscow, 1979; English transl, 2nd ed, Grad Texts in Math, vol 6, Springer-Verlag, New York- Berlin, G A Leonov, Mathematical problems of control theory Motivation for analysis, S-Peterburg Gos Univ, St Petersburg, 1999 (Russian) 8 S Lefschetz, Stability of nonlinear control systems, Academic Press, New York-London, G A Leonov, The Brockett stabilization problem, International Conference Control of Oscillations and Chaos (July 2, St Petersburg), S-Peterburg Gos Univ, St Petersburg, 2, pp (English) 1 L Moreau and D Aeyels, Stabilization by means of periodic output feedback, Conference of Decision and Control (Phoenix, Arizona USA, December 1999), Phoenix Univ, Phoenix, 1999, pp St Petersburg State University, Department of Mathematics and Mechanics, Bibliotechnaya pl, 2, Stary Peterhof, St Petersburg, 19894, Russia address: mathspburu Received 27/MAR/21 Translated by NYU NETSVETAEV 16