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Stability analysis of quasilinear uncertain dynamical systems
Nonlinear Analysis: Real World Applications 9 (2008) 1090 – 1102 www.elsevier.com/locate/nonrwa
Stability analysis of quasilinear uncertain dynamical systems A.A. Martynyuk∗ , Yu.A. Martynyuk-Chernienko Department of Processes Stability, Institute of Mechanics, National Academy of Sciences of Ukraine, Nesterov, str. 3, 03057, MSP-680, Kiev-57, Ukraine Received 31 January 2007; accepted 31 January 2007
Abstract In this paper, we establish new conditions of uniform asymptotic stability for uncertain quasilinear systems. Also, a new class of Lyapunov functions: canonical matrix-valued Lyapunov functions is presented and used for stability analysis. Illustrative examples are given. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Quasilinear uncertain system; Canonical matrix-valued Lyapunov function; Uniform asymptotic stability
1. Introduction For stability analysis of solutions to uncertain systems [1,3] main theorem of the method of matrix-valued Lyapunov functions are proved in [5,6]. Generally, these results are formulated in terms of existence of matrix-valued functions [4] which possess specific properties. For the investigation of particular problems of dynamics an explicit expression of the auxiliary function taking into account peculiarities of the system under consideration is required. This fact necessitates the construction of appropriate auxiliary functions satisfying general theorems of generalized direct Lyapunov method as applied to uncertain systems of a particular class. In this paper conditions of uniform asymptotic stability are established for uncertain quasilinear systems. The paper is arranged as follows. In Section 2, a quasilinear system is transformed to the canonical form and special variables are introduced. In Section 3, a new class of matrix-valued Lyapunov functions is introduced and applied in the investigation. In Sections 4 and 5, main results of the paper are formulated. In Sections 6 and 7, some applications of the obtained results are presented and in Section 8 the directions of further development of the proposed approach are discussed. 2. Description and transformation of uncertain quasilinear system The source of ‘uncertainties’ appearing in the modelling of a real system is the impossibility to take into account all forces affecting the system under investigation. ∗ Corresponding author.
E-mail address: [email protected] (A.A. Martynyuk). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.01.019
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Besides, the procedure of physical or mathematical decomposition of initial perturbed motion equations with further linearization also introduces some ‘uncertainties’. These facts are taken into account in the adequate description of real system functioning. As known, uncertain quasilinear systems attract attention of many investigators (see [1] and references therein). We consider perturbed motion equations in the form: dz = P z + Q(z, w, ), dt dw = H w + G(z, w, ). dt
(2.1)
Here z ∈ R n , w ∈ R m , ∈ S ⊆ R d is an uncertainty parameter of system (2.1), P and H are constant matrices of appropriate dimensions, Q : R n × R m × R d → R n and G : R n × R m × R d → R m have as their components the series in positive integer powers z and w, starting with the terms of not lower than the second order and absolutely convergent in the product of arbitrary large open connected neighbourhoods Nz and Nw of states z = 0 and w = 0 for any values of uncertainty parameter ∈ S ⊆ R d . System of (2.1) type appears as a result of decomposition with further linearization of the initial nonlinear system of differential equations. By means of nonsingular linear transformation z = T x and w = Ry (det T = 0, det R = 0) we transform linear part of system (2.1) to the diagonal form dx = Ax + f1 (x, y, ), dt dy = By + f2 (x, y, ), dt
(2.2)
where x ∈ R n , y ∈ R m , A = diag {1 , . . . , n }, B = diag {1 , . . . , m }, f1 (x, y, ) = Q(T z, Rw, ), f2 (x, y, ) = G(T z, Rw, ). For the components xs and yk of vectors x and y new variables [8] xs = rs exp(is ),
x s = rs exp(−is ),
yk = k exp(ik ),
y k = k exp(−ik ),
s = 1, 2, . . . , n, k = 1, 2, . . . , m, s = 0, , k = 0, ,
(2.3)
are considered. Hence, it is easy to obtain rs = xs exp(−is ), k = yk exp(−ik ),
rs = x s exp(is ),
s = 1, 2, . . . , n,
k = y k exp(ik ),
k = 1, 2, . . . , m,
and consequently, drs 1 dxs −is dx s is = e e , s = 1, 2, . . . , n, + dt 2 dt dt dk 1 dyk −ik dy k ik , k = 1, 2, . . . , m. + = e e dt 2 dt dt In view of system (2.2) we obtain drs 1 = Re s rs + (f1s e−is + f 1s eis ), dt 2 dk 1 = Re k k + (f2k e−ik + f 2k eik ), dt 2
s = 1, 2, . . . , n, k = 1, 2, . . . , m.
(2.4)
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In system (2.4) the equations corresponding to complex-conjugated roots are repeated, since the corresponding variables have similar module. Therefore, the number of different equations in systems (2.4) is n1 < n and m1 < m, respectively. Remark 2.1. If in system (2.2) functions f1 and f2 are presented in the form f1 (x, y, ) = f1∗ (x, y, 0) + f1 (·, ), f2 (x, y, ) = f2∗ (x, y, 0) + f2 (·, ), where f1 = f1 (x, y, ) − f1∗ (x, y, 0), f2 = f2 (x, y, ) − f2∗ (x, y, 0), ∈ S ⊆ R d , then system (2.4) is reduced to 1 1 ∗ −is drs ∗ + f 1s eis + f1s e f1s e−is + f 1s eis , = Re s rs + 2 2 dt 1 1 ∗ −ik dk ∗ i f2k e f2k e−ik + f 2k eik . + f 2k e k + = Re k k + 2 2 dt In this system the effect of uncertainties ∈ S ⊆ R d is ‘concentrated’ in the last additives in the right-hand side of equations. Sometimes, such transformation of the system can simplify the estimation of ‘uncertainty’ effect on dynamics of system (2.1). For system (2.4) we shall consider a moving set A∗ () defined by (cf. [3]) A∗ () = {r ∈ R n , ∈ R m : r + = ()}.
(2.5)
Here it is assumed that () > 0 and, moreover, () → 0 (0 = const > 0) as → 0 and () → ∞ as → ∞. The aim of this paper is to establish conditions of uniform asymptotic stability for solutions of system (2.1) with respect to moving invariant set (2.5). To this end generalized direct Lyapunov method is applied. 3. Application of canonical matrix-valued function Dynamical behaviour of solutions to system (2.4) is investigated by means of the matrix-valued function u11 (r) u12 (r, ) , U (r, ) = u21 (r, ) u22 ()
(3.1)
whose elements uij (·), i, j = 1, 2 are defined by the variables r and of system (2.4) as u11 (r) =
n1
s rs2 ,
s = 1, 2, . . . , n1 ,
s=1
u22 () =
m1
k 2k ,
k = 1, 2, . . . , m1 ,
k=1
u12 (r, ) = u21 (r, ) =
min(n 1 ,m1 )
r s k .
(3.2)
s=k
Here s , k are some positive constants and is an arbitrary constant. Function (3.1) with elements (3.2) is called a canonical matrix-valued function in view of the fact that its elements are constructed in terms of transformation of linear approximation of system (2.2) to the canonical form. We denote m = mins ( s ), M = maxs ( s ), m = mink ( k ), M = maxk ( k ). It is easy to verify that for components (3.2) m r2 u11 (r) M r2 ,
m 2 u22 () M 2 ,
for all r ∈ R n1 , for all ∈ R m1 ,
− ru12 (r, ) r,
for all (r, ) ∈ R n1 × R m1 .
(3.3)
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The function V (r, ) = T U (r, ) ,
2 ∈ R+ , > 0,
(3.4)
satisfies the inequalities (a) v T H T A1 H v V (r, ) for r + (), (b) V (r, )v T H T A2 H v
(3.5)
for r + (),
where v T = (r, ), H = diag ( 1 , 2 ), m − M , A2 = . A1 = − m M In order to apply generalized direct Lyapunov method (see [5,6]) for the investigation of behaviour of solutions to system (2.4) in terms of function (3.4) we shall determine first the structure of it total derivative by virtue of system (2.4). To this end we introduce some assumptions. Assumption 3.1. There exist constants j l , j = 1, 2; l = 1, 2, 3, 4 such that for r + > () (a)
n1
s (f1s e−is + f 1s eis )rs 11 r2 + 12 r,
s=1
(b)
m1
k (f2k e−ik + f 2k eik )k 21 2 + 22 r,
k=1
(c)
(f1s e−is + f 1s eis )k 13 r + 14 2 ,
s=k
(d)
(f2k e−ik + f 2k eik )rs 23 r + 24 r2 .
s=k
Assumption 3.2. There exist constants j l , j = 1, 2; l = 1, 2, 3, 4 such that r + < () inequalities (a)–(d) from Assumption 3.1 are satisfied with the reversed sign ‘ ’ instead of ‘ .’ We designate a=maxs (Re s ), b=maxk (Re k ), a=mins (Re s ), b=mink (Re k ). If all conditions of Assumptions 3.1 and 3.2 are satisfied, then for the total derivative elements of matrix-valued function (3.2) along solutions of system (2.4) the following estimates are true: (a) for all r + > (), ∈ S ⊆ R d u˙ 11 (r)(2 M a + 11 )r2 + 12 r, u˙ 22 ()(2 M b + 21 )2 + 22 r, u˙ 12 (r, ) 21 24 r2 + [ (a + b) +
1 2
( 13 + 23 )]r +
1 2
14 2 ,
(3.6)
1 2
14 2 .
(3.7)
(b) for all r + < (), ∈ S ⊆ R d u˙ 11 (r)(2 m a + 11 )r2 + 12 r, u˙ 22 ()(2 m b + 21 )2 + 22 r, u˙ 12 (r, ) 21 24 r2 + [ (a + b) +
1 2
( 13 + 23 )]r +
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Lemma 3.1. Let all conditions of Assumptions 3.1 and 3.2 be satisfied. Then for the total derivative of function (3.4) by virtue of system (2.4) the following estimates are true: (a) for r + > () and for all ∈ S ⊆ R d dV (r, ) v T Cv, dt (2.4) c11 c12 T , where v = (r, ) and C = c21 c22
(3.8)
c12 = c21 , c11 = 21 (2 M a + 11 ) + 1 2 24 , c22 = 22 (2 M b + 21 ) + 1 2 14 , c12 =
1 2
21 12 +
1 2
22 22 + 1 2 [ (a + b) + 21 ( 13 + 23 )],
(b) for r + < () and for all ∈ S ⊆ R d dV (r, ) v T Dv, dt (2.4) d11 d12 , where D = d21 d22
(3.9)
d12 = d21 , d11 = 21 (2 m a + 11 ) + 1 2 24 , d22 = 22 (2 m b + 21 ) + 1 2 14 , d12 =
1 2
21 12 +
1 2
22 22 + 1 2 [ (a + b) + 21 ( 13 + 23 )].
Proof of Lemma 3.1. Is carried out by the direct substitution by estimates (3.6) and (3.7) into the expression of the total derivative of function (3.4) with almost obvious transformations. 4. Uniform asymptotic stability conditions Estimates (3.5) and inequalities (3.8) and (3.9) allow one to establish sufficient conditions of uniform asymptotic stability of solutions to system (2.4) with respect to the moving invariant set A∗ (). Theorem 4.1. In system (2.2) assume that functions f1 (z, w, ) and f2 (z, w, ) are continuous on R n × R m × R d and (1) (2) (3) (4) (5)
for any ∈ S ⊆ R d there exists a function () > 0 such that the set A∗ () is nonempty for all ∈ S ⊆ R d ; for the transformed system (2.4) matrix-valued function (3.1) with elements (3.2) is constructed; all conditions of Assumptions 3.1 and 3.2 are satisfied; in inequalities (3.5)(a,b) the matrices A1 and A2 are positive definite; the following conditions are satisfied: dV (r, ) (a) < 0 for r + > (), dt (2.4) dV (r, ) (b) = 0 for r + = (), dt (2.4)
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dV (r, ) > 0 for r + < (); dt (2.4) (6) for functions a(r + ) = m (H T A1 H )(r, )T (r, ) and b(r + ) = M (H T A2 H )(r, )T × (r, ) the correlation (c)
a(()) = b(()) is satisfied for any () > 0. Then solutions (x(t, ), y(t, )) of system (2.2) are uniformly asymptotically stable with respect to the moving invariant set A∗ (). Proof. According to condition (1) of Theorem 4.1 for all ∈ S ⊆ R d there exists nonempty moving set A∗ () for system (2.2). If conditions (2)–(3) of Theorem 4.1 are satisfied, then under condition (4) the function V (r, ) : R+ × R+ → R+ , determined by (3.4) is positive definite in the domain r + > (), ∈ S ⊆ R d and decreasing in the domain r + (), ∈ S ⊆ R d . The invariance of the set A∗ () with respect to system (2.1) follows by Theorem 2.3 from [5], since all conditions of this theorem are satisfied under conditions of Theorem 4.1. Uniform asymptotic stability of motions of system (2.1) with respect to the set A∗ () is proved if under all conditions of Theorem 4.1 for any ε ∈ (0, H ), H = const > 0 a T = T (ε) > 0 is found such that the condition () − 0 < r(t0 , ) + (t0 , ) < () + 0 ,
(4.1)
where 0 = 0 (), implies the existence of t ∗ ∈ [t0 , t0 + T ] for which () − < r(t ∗ , ) + (t ∗ , ) < () + ,
(4.2)
where = (ε). Under conditions of Theorem 4.1 let no t ∗ ∈ [t0 , t0 + T (ε)] exist for which inequality (4.2) holds. Then, two cases are possible: (a) r(t, ) + (t, ) > +
for all t ∈ [t0 , t0 + T ]
or (b) r(t, ) + (t, ) < − for all t ∈ [t0 , t0 + T ]. Consider case (a). Since by condition (5)(a) dV (r, ) <0 dt
for r + > (), t0 t < ∞,
(4.3)
the function V (r(t), (t)) monotonically decreases along solutions of system (2.1) lim V (r(t), (t)) = inf V (r(t), (t)) = ().
t→∞
t
Condition (5)(a) implies that the matrix C is negative definite, i.e. Re M (C) < 0. Then, dV (r, ) < − M (C)(r + ) for r + > (), dt
(4.4)
where (r + ) (r, )T (r, ), (0) = 0 and ∈ K. By conditions of Theorem 4.1 system (2.1) is uniformly stable and hence, one may assume that r(t, ) + (t, ) H < + ∞ for all ∈ S ⊆ R d . We calculate on the set () + r + H () =
inf
+ r+ H
(r + ),
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it is clear that () 0. Inequality (4.4) yields V (r(t0 + T (ε)), (t0 + T (ε))) < V (r(t0 ), (t0 )) − M (C)
t0 +T (ε)
(r() + ()) d.
t0
Since −(r + ) − () for () + r + H , then
V (r(t0 + T (ε)), (t0 + T (ε))) < V (r(t0 ), (t0 )) − M (C)
t0 +T (ε)
() d
t0
< V (r(t0 ), (t0 )) − M (C) ()T (ε). Hence, it follows that for T = T (ε) being large enough V (r(T ), (T )) < 0
for r + > ().
However, this contradicts conditions (4) of Theorem 4.1. Therefore, case (a) is not possible, i.e. a value t ∗ ∈ [t0 , t0 + T ] must exist for which r(t ∗ , ) + (t ∗ , ) () + .
(4.5)
We shall show now that r(t, ) + (t, ) → () as t → ∞. Actually, let ε > 0 be an arbitrary small number and l() = inf M (C)(r + ) > 0
for () + ε r + H .
It follows from (4.4) that there exists a T (ε) > t0 such that V (r(T (ε)), (T (ε))) > l().
(4.6)
Meanwhile, by conditions (4.3) the function V (r(t), (t)) decreases monotonically and so V (r(t), (t)) < l() for all t > T (ε),
(4.7)
for r + > (). Consequently, we have for t > T (ε) r(t, ) + (t, ) < () + ε. Let this be false and there exists a t1 > T (ε) for which r(t1 , ) + (t1 , ) > () + ε. Then, (4.6) and (4.7) imply l() > V (r(t1 ), (t1 ))M (C)(r + )l(), which is a contradiction. Therefore, lim (r(t, ) + (t, )) = ().
t→∞
Further we consider case (b). Condition (5)(b) implies that the matrix D is positive definite. Then for the function dV (r, )/dt it is easy to obtain the estimate dV (r, ) > m (D)W (r + ) dt
for r + < (),
(4.8)
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where W (r + ) (r, )T (r, ), W (0) = 0, W ∈ K, m (D) is a minimal eigenvalue of the matrix D. By condition (3.5)(b) we have V (r, )b(r + )
for r + ().
Hence, it follows that the function V (r, ) is bounded, i.e. there exists M > 0 such that |V (r, )|M for t0 t < ∞ and r + () < H , where M and H are some positive numbers. Let > 0 ( < () ∀ ∈ S ⊆ R d ) be an arbitrary small number. By condition (4) of Theorem 4.1 there exists a point r0 , 0 : 0 < r0 + 0 < (), such that V (r0 , 0 ) = > 0.
(4.9)
Let the solution (r T (t, ), T (t, ))T of uncertain system (2.1) be defined by the initial conditions r(t0 , )=r0 , (t0 , )= 0 and 0 < r(t, ) + (t, ) < () − for t ∈ [t0 , t0 + T ].
(4.10)
According to conditions (5)(b) the function V (r(t), (t)) monotonically increases for r + () together with t and therefore, we have for t t0 V (r(t), (t)) > V (r0 , 0 ) = > 0. Let us show that for some t ∗ ∈ [t0 , t0 + T ] the inequality r(t ∗ , ) + (t ∗ , ) > () − ,
(4.11)
is satisfied. Let this be not true and r(t, ) + (t, ) < () − for all t t0 . The solution (r T (t, ), T (t, ))T is infinitely extendable to the right and a < () − is found for all ∈ S ⊆ R d so that 0 < r(t, ) + (t, ) () −
for t0 t < ∞.
We calculate () =
inf
r+ ()−
W (r + ).
(4.12)
It is clear that () > 0 and in the domain we have r + () dV (r, ) > m (D) () dt
for t0 t < ∞.
Consequently, for t0 t < ∞ we get V (r(t), (t)) > V (r0 , 0 ) + m (D) ()(t − t0 ),
(4.13)
which contradicts the boundedness of the function V (r, ) in the domain r + (). Thus, we proved that there exists a t ∗ ∈ [t0 , t0 + T ] for which inequality (4.11) holds true. Together with inequality (4.5) the estimate () − < r(t ∗ , ) + (t ∗ , ) < () +
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is satisfied for t ∗ ∈ [t0 , t0 + T ]. This proves the uniform asymptotic stability of motions of uncertain system (2.1) with respect to the set A∗ (). 5. Corollary Let perturbed motion equations represent a system of quasilinear equations dz = P z + Q(z, ), dt
(5.1)
where z ∈ R n , P is a constant real n × n matrix and Q : R n × S → R n , S ⊆ R d . Similarly to Section 2 we reduce system (5.1) to the form dx = P ∗ x + Q∗ (x, ), dt
(5.2)
where x ∈ R n , P ∗ =A−1 P A=[sr r +s−1,r r ], s, r =1, 2, . . . , n, Ais an n×n matrix of nonsingular transformation z = Ax (i.e. det A = 0), Q∗ = A−1 Q(Ax, ), r are eigenvalues of the matrix P and r are sufficiently small positive values. We define the moving set
n ∗ n |xs | = () , (5.3) A () = x ∈ R : s=1
where () → 0 as → 0. The behaviour of solutions to system (5.2) with respect to set (5.3) is investigated by means of the positive definite function V = x =
n s=1
Since
|xs | =
n
rs .
(5.4)
s=1
1 drs 1 ¯ ∗s eis , = Re s rs + s−1 xs−1 e−is + x¯s−1 eis + Q∗s e−is + Q dt 2 2
(5.5)
the estimates for time-derivative of function V in the domains ext A∗ () and int A∗ () are (a) for x > (), ∈ S ⊆ R d dV ( + )V + f (V , ), dt
(5.6)
where = maxs {Re s }, r , r = 1, 2, . . . , n; f (V , ) is a polynomial with respect to V estimating the last additive in the right-hand side of (5.5), f (0, ) = 0 for any ∈ S ⊆ R d ; (b) for x < (), ∈ S ⊆ R d dV (∗ − ∗ )V − f (V , ), dt
(5.7)
where ∗ = mins {Re s }, ∗ r , r = 1, 2, . . . , n. Estimates (5.6) and (5.7) enable us to formulate the following assertion. Theorem 5.1. In system (5.1) assume that the function Q(z, ) is continuous on R n × S and (1) for any ∈ S ⊆ R d there exists a function () > 0 for any ∈ S ⊆ R d such that set (5.3) is nonempty for all ∈ S ⊆ Rd ;
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(2) there exist functions W1 (x) and W2 (x) such that for all ∈ S ⊆ R d (a) ( + )V + f (V , ) W1 (x), for x > (), (b) (∗ − ∗ )V − f (V , ) W2 (x), for x < (), ¯ ∗s eis ) = 0 for x = (); (c) Re s rs + 21 s−1 (xs−1 e−is + x¯s−1 eis ) + 21 (Q∗s e−is + Q (3) the following inequalities are true: (a) W1 (x) < 0 for x > (); (b) W2 (x) > 0 for x < (). Then the motions of system (5.1) are uniformly asymptotically stable with respect to the moving set A∗ (). The proof of this theorem is omitted, since it follows directly from the proof of Theorem 4.1. 6. Generalized oscillating system Consider a system of perturbed motion equations dx = x + y − g(x, y, )x(x 2 + y 2 ), dt dy = y − x − g(x, y, )y(x 2 + y 2 ), dt
∈ S ⊆ Rd ,
(6.1)
where = const > 0, g(x, y, ) > 0 is a function characterizing uncertainties in system (6.1). In the case when function g(x, y, ) = g ∗ , where g ∗ is a constant value independent of uncertainties, system (6.1) was studied in [2]. The following results were obtained: (i) if < 0 and g ∗ < 0, then the solution x = y = 0 is stable for any x0 , y0 ∈ R; (ii) if 0 and g ∗ > 0, then the solution x = y = 0 is unstable; (iii) If > 0 and g ∗ < 0, then for the values x0 , y0 inside the circle x 2 + y 2 = − /g ∗ , the motion is unstable and for x0 , y0 outside this circle the motion is stable; (iv) if > 0 and g ∗ < 0, then the motions with the initial values x0 , y0 inside or on the circle with the radius R = (− /g ∗ )1/2 are stable and for any x0 , y0 outside this circle the motions are unstable. Further system (6.1) is considered for > 0 and g(x, y, ) > 0. We perform the change of variables x = −r cos ,
y = r sin
and reduce system (6.1) to the form dr = r − g(r, , )r 3 , dt
d = 1, dt
(6.2)
Here, g m g(r, , )g M
(6.3)
for all (r, , ) ∈ R+ × [0, 2] × S, g m < g M are some given values. Note that the solution r = 0 of first approximation equations (6.2) is unstable by Lyapunov, since the linear approximation dr/dt = r possesses the eigenvalue = > 0. Together with system (6.2) we consider the function V = r 2 and its total derivative dV /dt by virtue of system (6.2) dr dV = 2r 2 [ − g(r, , )r 2 ], ∈ S ⊆ R d . = 2r (6.4) dt (6.2) dt
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Expression (6.4) implies that for any function g(r, , ), satisfying condition (6.3) the inequalities below are satisfied dV
, t 0, < 0 for r 2 > g(r, , ) dt (6.2)
dV = 0 for r 2 = , t 0, dt (6.2) g(r, , )
dV , t 0, (6.5) > 0 for r 2 < dt (6.2) g(r, , ) and the moving set ∗ A () = r : r 2 =
, g(r, , )
∈ S ⊆ Rd
is uniformly asymptotically stable. We study motions of system (6.1) with respect to the domains: S1 = {r : r 2 < H },
0 < H < ∞,
S2 = {r : r 2 },
=
S3 = {r : r 2 },
=
gM
gm
1/2 , 1/2
under restriction (6.3). Let the motion of system (6.2) start outside the ring with the radius r0 + , where r0 = ( /g m )1/2 , is an arbitrary small constant value. Since dV = 2 V − 4g(r, , )V 2 , dt (6.2) the estimate of the time interval during which the solutions of system (6.2) get on the moving surface r2 =
g(r, , )
is found by the inequality dc , 2 c − 4g m c2 1
(6.6)
where 1 < , 1 = 21 r 2 , = 21 (r0 + )2 . Estimate (6.6) implies (r + )2 (r 2 − r 2 ) 1 0 0 ln . 2 r2 2r0 + 2 We estimate in the same way the time interval sufficient for the solutions of system (6.1) starting in the domain r ∗ − 0, where r ∗ = ( /g M )1/2 , to get on the moving surface r 2 = /g(r, , ). Note that the function g(r, , ), ∈ S ⊆ R d is not assumed continuously differentiable. Therefore, solutions of system (6.1) are studied efficiently by the qualitative method, while its investigation via the immediate integration is difficult.
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7. Uncertain nonconservative Hamiltonian systems We apply now our approach to a perturbed (forced and damped) uncertain Hamiltonian system [7] x˙1 =
jH + 1 (t, x1 , x2 , 1 ), jx2
x˙2 = −
jH + 2 (t, x1 , x2 , 2 ). jx1
(7.1)
Here H : R 2 → R, i : R × R 2 × S → R, i = 1, 2, . . ., H (x1 , x2 ) is a positive definite C 1 -function, H (0, 0) = 0, 1 (t), 2 (t) ∈ [0, 1] are uncertain scalar parameters, 1 and 2 are nonpotential dissipative forces. Let E0 be a constant energy of system (7.1) for 1 = 2 = 0. Given function r() > 0, we shall consider the motions of system (7.1) with respect to the moving set A(r) = {x ∈ R 2 : x = r()},
∈ S ⊆ Rd ,
(7.2)
where lim r() = E0 as → 0. It is known that H˙ (x1 (t), x2 (t)) =
2 jH i (t, x1 , x2 , ). jxi i=1
Let functions Wi (x1 , x2 ), i = 1, 2, exist, such that (a)
2 jH i=1
(b)
2 jH i=1
(c)
jxi
jxi
2 jH i=1
jxi
i (t, x1 , x2 , ) < W1 (x1 , x2 ) for x > r(),
i (t, x1 , x2 , ) = 0
for x = r(),
i (t, x1 , x2 , ) > W2 (x1 , x2 )
∈ S ⊆ Rd ,
∈ S ⊆ Rd ,
for x < r(),
∈ S ⊆ Rd .
If, together with conditions (a)–(c), the inequalities (a) W1 (x1 , x2 ) < 0
for x > r(), ∈ S ⊆ R d ,
(b) W2 (x1 , x2 ) > 0
for x < r(), ∈ S ⊆ R d ,
are satisfied, then by Theorem 5.1 the solutions of system (7.1) converge to the moving surface (7.2). Conditions (a) and (b) demonstrate that outside the moving surface A(r) system (7.1) is dissipative, meanwhile inside this surface the energy accumulation takes place. Moreover, the moving surface A(r) is a surface of energy unloading. 8. Concluding remarks Alongside the investigation of behaviour of solutions to quasilinear systems with respect to a moving surface it is of interest to analyse stability and instability of zero solution of systems of equations under consideration. The approach proposed in this paper is applicable in this case as well. Besides, uncertain quasilinear systems with persistent perturbations can be studied. In this case the auxiliary function can be either the canonical matrix-valued function (3.1) or an appropriate vector Lyapunov function [9].
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References [1] F. Garofalo, L. Glielmo (Eds.), Robust Control via Variable Structure and Lyapunov Techniques, Springer, Berlin, 1996. [2] G.V. Kamenkov, Stability of Motion, Oscillation, Aerodynamics, vol. 1, Nauka, Moscow, 1971, Selected Papers. [3] V. Lakshmikantham, A.S. Vatsala, Stability of moving invariant sets, in: S. Sivasundaram, A.A. Martynyuk (Eds.), Advances in Nonlinear Dynamics, vol. 5, Gordon and Breach Scientific Publishers, Amsterdam, 1997, pp. 79–83. [4] A.A. Martynyuk, Qualitative Methods in Nonlinear Dynamics, Novel Approaches to Lyapunov Matrix Functions, Marcel Dekker Inc., New York, 2002. [5] Yu.A. Martynyuk-Chernienko, Stability analysis of uncertain systems via matrix-valued Lyapunov functions, Nonlinear Anal. 63 (2005) 388–404. [6] Yu.A. Martynyuk-Chernienko, Exponential convergence and instability of solutions of uncertain dynamical systems, Nonlinear Anal. 66 (2006) 1318–1328. [7] J.M. Skowronski, Parameter and state identification in non-linearizable uncertain systems, Int. J. Non-linear Mech. 19 (1984) 421–429. [8] A.A. Tikhonov, On stability of motion under constantly acting disturbances, Vestn. Leningrad. Un-t 1 (1) (1965) 95–101. [9] Y. Wang, Y. Shi, H. Sasaki, A criterion for stability of nonlinear time-varying dynamic system, Nonlinear Dyn. Syst. Theory 4 (2) (2004) 217–229.
Stability analysis of quasilinear uncertain dynamical systems A.A. Martynyuk∗ , Yu.A. Martynyuk-Chernienko Department of Processes Stability, Institute of Mechanics, National Academy of Sciences of Ukraine, Nesterov, str. 3, 03057, MSP-680, Kiev-57, Ukraine Received 31 January 2007; accepted 31 January 2007
Abstract In this paper, we establish new conditions of uniform asymptotic stability for uncertain quasilinear systems. Also, a new class of Lyapunov functions: canonical matrix-valued Lyapunov functions is presented and used for stability analysis. Illustrative examples are given. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Quasilinear uncertain system; Canonical matrix-valued Lyapunov function; Uniform asymptotic stability
1. Introduction For stability analysis of solutions to uncertain systems [1,3] main theorem of the method of matrix-valued Lyapunov functions are proved in [5,6]. Generally, these results are formulated in terms of existence of matrix-valued functions [4] which possess specific properties. For the investigation of particular problems of dynamics an explicit expression of the auxiliary function taking into account peculiarities of the system under consideration is required. This fact necessitates the construction of appropriate auxiliary functions satisfying general theorems of generalized direct Lyapunov method as applied to uncertain systems of a particular class. In this paper conditions of uniform asymptotic stability are established for uncertain quasilinear systems. The paper is arranged as follows. In Section 2, a quasilinear system is transformed to the canonical form and special variables are introduced. In Section 3, a new class of matrix-valued Lyapunov functions is introduced and applied in the investigation. In Sections 4 and 5, main results of the paper are formulated. In Sections 6 and 7, some applications of the obtained results are presented and in Section 8 the directions of further development of the proposed approach are discussed. 2. Description and transformation of uncertain quasilinear system The source of ‘uncertainties’ appearing in the modelling of a real system is the impossibility to take into account all forces affecting the system under investigation. ∗ Corresponding author.
E-mail address: [email protected] (A.A. Martynyuk). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.01.019
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Besides, the procedure of physical or mathematical decomposition of initial perturbed motion equations with further linearization also introduces some ‘uncertainties’. These facts are taken into account in the adequate description of real system functioning. As known, uncertain quasilinear systems attract attention of many investigators (see [1] and references therein). We consider perturbed motion equations in the form: dz = P z + Q(z, w, ), dt dw = H w + G(z, w, ). dt
(2.1)
Here z ∈ R n , w ∈ R m , ∈ S ⊆ R d is an uncertainty parameter of system (2.1), P and H are constant matrices of appropriate dimensions, Q : R n × R m × R d → R n and G : R n × R m × R d → R m have as their components the series in positive integer powers z and w, starting with the terms of not lower than the second order and absolutely convergent in the product of arbitrary large open connected neighbourhoods Nz and Nw of states z = 0 and w = 0 for any values of uncertainty parameter ∈ S ⊆ R d . System of (2.1) type appears as a result of decomposition with further linearization of the initial nonlinear system of differential equations. By means of nonsingular linear transformation z = T x and w = Ry (det T = 0, det R = 0) we transform linear part of system (2.1) to the diagonal form dx = Ax + f1 (x, y, ), dt dy = By + f2 (x, y, ), dt
(2.2)
where x ∈ R n , y ∈ R m , A = diag {1 , . . . , n }, B = diag {1 , . . . , m }, f1 (x, y, ) = Q(T z, Rw, ), f2 (x, y, ) = G(T z, Rw, ). For the components xs and yk of vectors x and y new variables [8] xs = rs exp(is ),
x s = rs exp(−is ),
yk = k exp(ik ),
y k = k exp(−ik ),
s = 1, 2, . . . , n, k = 1, 2, . . . , m, s = 0, , k = 0, ,
(2.3)
are considered. Hence, it is easy to obtain rs = xs exp(−is ), k = yk exp(−ik ),
rs = x s exp(is ),
s = 1, 2, . . . , n,
k = y k exp(ik ),
k = 1, 2, . . . , m,
and consequently, drs 1 dxs −is dx s is = e e , s = 1, 2, . . . , n, + dt 2 dt dt dk 1 dyk −ik dy k ik , k = 1, 2, . . . , m. + = e e dt 2 dt dt In view of system (2.2) we obtain drs 1 = Re s rs + (f1s e−is + f 1s eis ), dt 2 dk 1 = Re k k + (f2k e−ik + f 2k eik ), dt 2
s = 1, 2, . . . , n, k = 1, 2, . . . , m.
(2.4)
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In system (2.4) the equations corresponding to complex-conjugated roots are repeated, since the corresponding variables have similar module. Therefore, the number of different equations in systems (2.4) is n1 < n and m1 < m, respectively. Remark 2.1. If in system (2.2) functions f1 and f2 are presented in the form f1 (x, y, ) = f1∗ (x, y, 0) + f1 (·, ), f2 (x, y, ) = f2∗ (x, y, 0) + f2 (·, ), where f1 = f1 (x, y, ) − f1∗ (x, y, 0), f2 = f2 (x, y, ) − f2∗ (x, y, 0), ∈ S ⊆ R d , then system (2.4) is reduced to 1 1 ∗ −is drs ∗ + f 1s eis + f1s e f1s e−is + f 1s eis , = Re s rs + 2 2 dt 1 1 ∗ −ik dk ∗ i f2k e f2k e−ik + f 2k eik . + f 2k e k + = Re k k + 2 2 dt In this system the effect of uncertainties ∈ S ⊆ R d is ‘concentrated’ in the last additives in the right-hand side of equations. Sometimes, such transformation of the system can simplify the estimation of ‘uncertainty’ effect on dynamics of system (2.1). For system (2.4) we shall consider a moving set A∗ () defined by (cf. [3]) A∗ () = {r ∈ R n , ∈ R m : r + = ()}.
(2.5)
Here it is assumed that () > 0 and, moreover, () → 0 (0 = const > 0) as → 0 and () → ∞ as → ∞. The aim of this paper is to establish conditions of uniform asymptotic stability for solutions of system (2.1) with respect to moving invariant set (2.5). To this end generalized direct Lyapunov method is applied. 3. Application of canonical matrix-valued function Dynamical behaviour of solutions to system (2.4) is investigated by means of the matrix-valued function u11 (r) u12 (r, ) , U (r, ) = u21 (r, ) u22 ()
(3.1)
whose elements uij (·), i, j = 1, 2 are defined by the variables r and of system (2.4) as u11 (r) =
n1
s rs2 ,
s = 1, 2, . . . , n1 ,
s=1
u22 () =
m1
k 2k ,
k = 1, 2, . . . , m1 ,
k=1
u12 (r, ) = u21 (r, ) =
min(n 1 ,m1 )
r s k .
(3.2)
s=k
Here s , k are some positive constants and is an arbitrary constant. Function (3.1) with elements (3.2) is called a canonical matrix-valued function in view of the fact that its elements are constructed in terms of transformation of linear approximation of system (2.2) to the canonical form. We denote m = mins ( s ), M = maxs ( s ), m = mink ( k ), M = maxk ( k ). It is easy to verify that for components (3.2) m r2 u11 (r) M r2 ,
m 2 u22 () M 2 ,
for all r ∈ R n1 , for all ∈ R m1 ,
− ru12 (r, ) r,
for all (r, ) ∈ R n1 × R m1 .
(3.3)
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The function V (r, ) = T U (r, ) ,
2 ∈ R+ , > 0,
(3.4)
satisfies the inequalities (a) v T H T A1 H v V (r, ) for r + (), (b) V (r, )v T H T A2 H v
(3.5)
for r + (),
where v T = (r, ), H = diag ( 1 , 2 ), m − M , A2 = . A1 = − m M In order to apply generalized direct Lyapunov method (see [5,6]) for the investigation of behaviour of solutions to system (2.4) in terms of function (3.4) we shall determine first the structure of it total derivative by virtue of system (2.4). To this end we introduce some assumptions. Assumption 3.1. There exist constants j l , j = 1, 2; l = 1, 2, 3, 4 such that for r + > () (a)
n1
s (f1s e−is + f 1s eis )rs 11 r2 + 12 r,
s=1
(b)
m1
k (f2k e−ik + f 2k eik )k 21 2 + 22 r,
k=1
(c)
(f1s e−is + f 1s eis )k 13 r + 14 2 ,
s=k
(d)
(f2k e−ik + f 2k eik )rs 23 r + 24 r2 .
s=k
Assumption 3.2. There exist constants j l , j = 1, 2; l = 1, 2, 3, 4 such that r + < () inequalities (a)–(d) from Assumption 3.1 are satisfied with the reversed sign ‘ ’ instead of ‘ .’ We designate a=maxs (Re s ), b=maxk (Re k ), a=mins (Re s ), b=mink (Re k ). If all conditions of Assumptions 3.1 and 3.2 are satisfied, then for the total derivative elements of matrix-valued function (3.2) along solutions of system (2.4) the following estimates are true: (a) for all r + > (), ∈ S ⊆ R d u˙ 11 (r)(2 M a + 11 )r2 + 12 r, u˙ 22 ()(2 M b + 21 )2 + 22 r, u˙ 12 (r, ) 21 24 r2 + [ (a + b) +
1 2
( 13 + 23 )]r +
1 2
14 2 ,
(3.6)
1 2
14 2 .
(3.7)
(b) for all r + < (), ∈ S ⊆ R d u˙ 11 (r)(2 m a + 11 )r2 + 12 r, u˙ 22 ()(2 m b + 21 )2 + 22 r, u˙ 12 (r, ) 21 24 r2 + [ (a + b) +
1 2
( 13 + 23 )]r +
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Lemma 3.1. Let all conditions of Assumptions 3.1 and 3.2 be satisfied. Then for the total derivative of function (3.4) by virtue of system (2.4) the following estimates are true: (a) for r + > () and for all ∈ S ⊆ R d dV (r, ) v T Cv, dt (2.4) c11 c12 T , where v = (r, ) and C = c21 c22
(3.8)
c12 = c21 , c11 = 21 (2 M a + 11 ) + 1 2 24 , c22 = 22 (2 M b + 21 ) + 1 2 14 , c12 =
1 2
21 12 +
1 2
22 22 + 1 2 [ (a + b) + 21 ( 13 + 23 )],
(b) for r + < () and for all ∈ S ⊆ R d dV (r, ) v T Dv, dt (2.4) d11 d12 , where D = d21 d22
(3.9)
d12 = d21 , d11 = 21 (2 m a + 11 ) + 1 2 24 , d22 = 22 (2 m b + 21 ) + 1 2 14 , d12 =
1 2
21 12 +
1 2
22 22 + 1 2 [ (a + b) + 21 ( 13 + 23 )].
Proof of Lemma 3.1. Is carried out by the direct substitution by estimates (3.6) and (3.7) into the expression of the total derivative of function (3.4) with almost obvious transformations. 4. Uniform asymptotic stability conditions Estimates (3.5) and inequalities (3.8) and (3.9) allow one to establish sufficient conditions of uniform asymptotic stability of solutions to system (2.4) with respect to the moving invariant set A∗ (). Theorem 4.1. In system (2.2) assume that functions f1 (z, w, ) and f2 (z, w, ) are continuous on R n × R m × R d and (1) (2) (3) (4) (5)
for any ∈ S ⊆ R d there exists a function () > 0 such that the set A∗ () is nonempty for all ∈ S ⊆ R d ; for the transformed system (2.4) matrix-valued function (3.1) with elements (3.2) is constructed; all conditions of Assumptions 3.1 and 3.2 are satisfied; in inequalities (3.5)(a,b) the matrices A1 and A2 are positive definite; the following conditions are satisfied: dV (r, ) (a) < 0 for r + > (), dt (2.4) dV (r, ) (b) = 0 for r + = (), dt (2.4)
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dV (r, ) > 0 for r + < (); dt (2.4) (6) for functions a(r + ) = m (H T A1 H )(r, )T (r, ) and b(r + ) = M (H T A2 H )(r, )T × (r, ) the correlation (c)
a(()) = b(()) is satisfied for any () > 0. Then solutions (x(t, ), y(t, )) of system (2.2) are uniformly asymptotically stable with respect to the moving invariant set A∗ (). Proof. According to condition (1) of Theorem 4.1 for all ∈ S ⊆ R d there exists nonempty moving set A∗ () for system (2.2). If conditions (2)–(3) of Theorem 4.1 are satisfied, then under condition (4) the function V (r, ) : R+ × R+ → R+ , determined by (3.4) is positive definite in the domain r + > (), ∈ S ⊆ R d and decreasing in the domain r + (), ∈ S ⊆ R d . The invariance of the set A∗ () with respect to system (2.1) follows by Theorem 2.3 from [5], since all conditions of this theorem are satisfied under conditions of Theorem 4.1. Uniform asymptotic stability of motions of system (2.1) with respect to the set A∗ () is proved if under all conditions of Theorem 4.1 for any ε ∈ (0, H ), H = const > 0 a T = T (ε) > 0 is found such that the condition () − 0 < r(t0 , ) + (t0 , ) < () + 0 ,
(4.1)
where 0 = 0 (), implies the existence of t ∗ ∈ [t0 , t0 + T ] for which () − < r(t ∗ , ) + (t ∗ , ) < () + ,
(4.2)
where = (ε). Under conditions of Theorem 4.1 let no t ∗ ∈ [t0 , t0 + T (ε)] exist for which inequality (4.2) holds. Then, two cases are possible: (a) r(t, ) + (t, ) > +
for all t ∈ [t0 , t0 + T ]
or (b) r(t, ) + (t, ) < − for all t ∈ [t0 , t0 + T ]. Consider case (a). Since by condition (5)(a) dV (r, ) <0 dt
for r + > (), t0 t < ∞,
(4.3)
the function V (r(t), (t)) monotonically decreases along solutions of system (2.1) lim V (r(t), (t)) = inf V (r(t), (t)) = ().
t→∞
t
Condition (5)(a) implies that the matrix C is negative definite, i.e. Re M (C) < 0. Then, dV (r, ) < − M (C)(r + ) for r + > (), dt
(4.4)
where (r + ) (r, )T (r, ), (0) = 0 and ∈ K. By conditions of Theorem 4.1 system (2.1) is uniformly stable and hence, one may assume that r(t, ) + (t, ) H < + ∞ for all ∈ S ⊆ R d . We calculate on the set () + r + H () =
inf
+ r+ H
(r + ),
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it is clear that () 0. Inequality (4.4) yields V (r(t0 + T (ε)), (t0 + T (ε))) < V (r(t0 ), (t0 )) − M (C)
t0 +T (ε)
(r() + ()) d.
t0
Since −(r + ) − () for () + r + H , then
V (r(t0 + T (ε)), (t0 + T (ε))) < V (r(t0 ), (t0 )) − M (C)
t0 +T (ε)
() d
t0
< V (r(t0 ), (t0 )) − M (C) ()T (ε). Hence, it follows that for T = T (ε) being large enough V (r(T ), (T )) < 0
for r + > ().
However, this contradicts conditions (4) of Theorem 4.1. Therefore, case (a) is not possible, i.e. a value t ∗ ∈ [t0 , t0 + T ] must exist for which r(t ∗ , ) + (t ∗ , ) () + .
(4.5)
We shall show now that r(t, ) + (t, ) → () as t → ∞. Actually, let ε > 0 be an arbitrary small number and l() = inf M (C)(r + ) > 0
for () + ε r + H .
It follows from (4.4) that there exists a T (ε) > t0 such that V (r(T (ε)), (T (ε))) > l().
(4.6)
Meanwhile, by conditions (4.3) the function V (r(t), (t)) decreases monotonically and so V (r(t), (t)) < l() for all t > T (ε),
(4.7)
for r + > (). Consequently, we have for t > T (ε) r(t, ) + (t, ) < () + ε. Let this be false and there exists a t1 > T (ε) for which r(t1 , ) + (t1 , ) > () + ε. Then, (4.6) and (4.7) imply l() > V (r(t1 ), (t1 ))M (C)(r + )l(), which is a contradiction. Therefore, lim (r(t, ) + (t, )) = ().
t→∞
Further we consider case (b). Condition (5)(b) implies that the matrix D is positive definite. Then for the function dV (r, )/dt it is easy to obtain the estimate dV (r, ) > m (D)W (r + ) dt
for r + < (),
(4.8)
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where W (r + ) (r, )T (r, ), W (0) = 0, W ∈ K, m (D) is a minimal eigenvalue of the matrix D. By condition (3.5)(b) we have V (r, )b(r + )
for r + ().
Hence, it follows that the function V (r, ) is bounded, i.e. there exists M > 0 such that |V (r, )|M for t0 t < ∞ and r + () < H , where M and H are some positive numbers. Let > 0 ( < () ∀ ∈ S ⊆ R d ) be an arbitrary small number. By condition (4) of Theorem 4.1 there exists a point r0 , 0 : 0 < r0 + 0 < (), such that V (r0 , 0 ) = > 0.
(4.9)
Let the solution (r T (t, ), T (t, ))T of uncertain system (2.1) be defined by the initial conditions r(t0 , )=r0 , (t0 , )= 0 and 0 < r(t, ) + (t, ) < () − for t ∈ [t0 , t0 + T ].
(4.10)
According to conditions (5)(b) the function V (r(t), (t)) monotonically increases for r + () together with t and therefore, we have for t t0 V (r(t), (t)) > V (r0 , 0 ) = > 0. Let us show that for some t ∗ ∈ [t0 , t0 + T ] the inequality r(t ∗ , ) + (t ∗ , ) > () − ,
(4.11)
is satisfied. Let this be not true and r(t, ) + (t, ) < () − for all t t0 . The solution (r T (t, ), T (t, ))T is infinitely extendable to the right and a < () − is found for all ∈ S ⊆ R d so that 0 < r(t, ) + (t, ) () −
for t0 t < ∞.
We calculate () =
inf
r+ ()−
W (r + ).
(4.12)
It is clear that () > 0 and in the domain we have r + () dV (r, ) > m (D) () dt
for t0 t < ∞.
Consequently, for t0 t < ∞ we get V (r(t), (t)) > V (r0 , 0 ) + m (D) ()(t − t0 ),
(4.13)
which contradicts the boundedness of the function V (r, ) in the domain r + (). Thus, we proved that there exists a t ∗ ∈ [t0 , t0 + T ] for which inequality (4.11) holds true. Together with inequality (4.5) the estimate () − < r(t ∗ , ) + (t ∗ , ) < () +
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is satisfied for t ∗ ∈ [t0 , t0 + T ]. This proves the uniform asymptotic stability of motions of uncertain system (2.1) with respect to the set A∗ (). 5. Corollary Let perturbed motion equations represent a system of quasilinear equations dz = P z + Q(z, ), dt
(5.1)
where z ∈ R n , P is a constant real n × n matrix and Q : R n × S → R n , S ⊆ R d . Similarly to Section 2 we reduce system (5.1) to the form dx = P ∗ x + Q∗ (x, ), dt
(5.2)
where x ∈ R n , P ∗ =A−1 P A=[sr r +s−1,r r ], s, r =1, 2, . . . , n, Ais an n×n matrix of nonsingular transformation z = Ax (i.e. det A = 0), Q∗ = A−1 Q(Ax, ), r are eigenvalues of the matrix P and r are sufficiently small positive values. We define the moving set
n ∗ n |xs | = () , (5.3) A () = x ∈ R : s=1
where () → 0 as → 0. The behaviour of solutions to system (5.2) with respect to set (5.3) is investigated by means of the positive definite function V = x =
n s=1
Since
|xs | =
n
rs .
(5.4)
s=1
1 drs 1 ¯ ∗s eis , = Re s rs + s−1 xs−1 e−is + x¯s−1 eis + Q∗s e−is + Q dt 2 2
(5.5)
the estimates for time-derivative of function V in the domains ext A∗ () and int A∗ () are (a) for x > (), ∈ S ⊆ R d dV ( + )V + f (V , ), dt
(5.6)
where = maxs {Re s }, r , r = 1, 2, . . . , n; f (V , ) is a polynomial with respect to V estimating the last additive in the right-hand side of (5.5), f (0, ) = 0 for any ∈ S ⊆ R d ; (b) for x < (), ∈ S ⊆ R d dV (∗ − ∗ )V − f (V , ), dt
(5.7)
where ∗ = mins {Re s }, ∗ r , r = 1, 2, . . . , n. Estimates (5.6) and (5.7) enable us to formulate the following assertion. Theorem 5.1. In system (5.1) assume that the function Q(z, ) is continuous on R n × S and (1) for any ∈ S ⊆ R d there exists a function () > 0 for any ∈ S ⊆ R d such that set (5.3) is nonempty for all ∈ S ⊆ Rd ;
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(2) there exist functions W1 (x) and W2 (x) such that for all ∈ S ⊆ R d (a) ( + )V + f (V , ) W1 (x), for x > (), (b) (∗ − ∗ )V − f (V , ) W2 (x), for x < (), ¯ ∗s eis ) = 0 for x = (); (c) Re s rs + 21 s−1 (xs−1 e−is + x¯s−1 eis ) + 21 (Q∗s e−is + Q (3) the following inequalities are true: (a) W1 (x) < 0 for x > (); (b) W2 (x) > 0 for x < (). Then the motions of system (5.1) are uniformly asymptotically stable with respect to the moving set A∗ (). The proof of this theorem is omitted, since it follows directly from the proof of Theorem 4.1. 6. Generalized oscillating system Consider a system of perturbed motion equations dx = x + y − g(x, y, )x(x 2 + y 2 ), dt dy = y − x − g(x, y, )y(x 2 + y 2 ), dt
∈ S ⊆ Rd ,
(6.1)
where = const > 0, g(x, y, ) > 0 is a function characterizing uncertainties in system (6.1). In the case when function g(x, y, ) = g ∗ , where g ∗ is a constant value independent of uncertainties, system (6.1) was studied in [2]. The following results were obtained: (i) if < 0 and g ∗ < 0, then the solution x = y = 0 is stable for any x0 , y0 ∈ R; (ii) if 0 and g ∗ > 0, then the solution x = y = 0 is unstable; (iii) If > 0 and g ∗ < 0, then for the values x0 , y0 inside the circle x 2 + y 2 = − /g ∗ , the motion is unstable and for x0 , y0 outside this circle the motion is stable; (iv) if > 0 and g ∗ < 0, then the motions with the initial values x0 , y0 inside or on the circle with the radius R = (− /g ∗ )1/2 are stable and for any x0 , y0 outside this circle the motions are unstable. Further system (6.1) is considered for > 0 and g(x, y, ) > 0. We perform the change of variables x = −r cos ,
y = r sin
and reduce system (6.1) to the form dr = r − g(r, , )r 3 , dt
d = 1, dt
(6.2)
Here, g m g(r, , )g M
(6.3)
for all (r, , ) ∈ R+ × [0, 2] × S, g m < g M are some given values. Note that the solution r = 0 of first approximation equations (6.2) is unstable by Lyapunov, since the linear approximation dr/dt = r possesses the eigenvalue = > 0. Together with system (6.2) we consider the function V = r 2 and its total derivative dV /dt by virtue of system (6.2) dr dV = 2r 2 [ − g(r, , )r 2 ], ∈ S ⊆ R d . = 2r (6.4) dt (6.2) dt
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Expression (6.4) implies that for any function g(r, , ), satisfying condition (6.3) the inequalities below are satisfied dV
, t 0, < 0 for r 2 > g(r, , ) dt (6.2)
dV = 0 for r 2 = , t 0, dt (6.2) g(r, , )
dV , t 0, (6.5) > 0 for r 2 < dt (6.2) g(r, , ) and the moving set ∗ A () = r : r 2 =
, g(r, , )
∈ S ⊆ Rd
is uniformly asymptotically stable. We study motions of system (6.1) with respect to the domains: S1 = {r : r 2 < H },
0 < H < ∞,
S2 = {r : r 2 },
=
S3 = {r : r 2 },
=
gM
gm
1/2 , 1/2
under restriction (6.3). Let the motion of system (6.2) start outside the ring with the radius r0 + , where r0 = ( /g m )1/2 , is an arbitrary small constant value. Since dV = 2 V − 4g(r, , )V 2 , dt (6.2) the estimate of the time interval during which the solutions of system (6.2) get on the moving surface r2 =
g(r, , )
is found by the inequality dc , 2 c − 4g m c2 1
(6.6)
where 1 < , 1 = 21 r 2 , = 21 (r0 + )2 . Estimate (6.6) implies (r + )2 (r 2 − r 2 ) 1 0 0 ln . 2 r2 2r0 + 2 We estimate in the same way the time interval sufficient for the solutions of system (6.1) starting in the domain r ∗ − 0, where r ∗ = ( /g M )1/2 , to get on the moving surface r 2 = /g(r, , ). Note that the function g(r, , ), ∈ S ⊆ R d is not assumed continuously differentiable. Therefore, solutions of system (6.1) are studied efficiently by the qualitative method, while its investigation via the immediate integration is difficult.
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7. Uncertain nonconservative Hamiltonian systems We apply now our approach to a perturbed (forced and damped) uncertain Hamiltonian system [7] x˙1 =
jH + 1 (t, x1 , x2 , 1 ), jx2
x˙2 = −
jH + 2 (t, x1 , x2 , 2 ). jx1
(7.1)
Here H : R 2 → R, i : R × R 2 × S → R, i = 1, 2, . . ., H (x1 , x2 ) is a positive definite C 1 -function, H (0, 0) = 0, 1 (t), 2 (t) ∈ [0, 1] are uncertain scalar parameters, 1 and 2 are nonpotential dissipative forces. Let E0 be a constant energy of system (7.1) for 1 = 2 = 0. Given function r() > 0, we shall consider the motions of system (7.1) with respect to the moving set A(r) = {x ∈ R 2 : x = r()},
∈ S ⊆ Rd ,
(7.2)
where lim r() = E0 as → 0. It is known that H˙ (x1 (t), x2 (t)) =
2 jH i (t, x1 , x2 , ). jxi i=1
Let functions Wi (x1 , x2 ), i = 1, 2, exist, such that (a)
2 jH i=1
(b)
2 jH i=1
(c)
jxi
jxi
2 jH i=1
jxi
i (t, x1 , x2 , ) < W1 (x1 , x2 ) for x > r(),
i (t, x1 , x2 , ) = 0
for x = r(),
i (t, x1 , x2 , ) > W2 (x1 , x2 )
∈ S ⊆ Rd ,
∈ S ⊆ Rd ,
for x < r(),
∈ S ⊆ Rd .
If, together with conditions (a)–(c), the inequalities (a) W1 (x1 , x2 ) < 0
for x > r(), ∈ S ⊆ R d ,
(b) W2 (x1 , x2 ) > 0
for x < r(), ∈ S ⊆ R d ,
are satisfied, then by Theorem 5.1 the solutions of system (7.1) converge to the moving surface (7.2). Conditions (a) and (b) demonstrate that outside the moving surface A(r) system (7.1) is dissipative, meanwhile inside this surface the energy accumulation takes place. Moreover, the moving surface A(r) is a surface of energy unloading. 8. Concluding remarks Alongside the investigation of behaviour of solutions to quasilinear systems with respect to a moving surface it is of interest to analyse stability and instability of zero solution of systems of equations under consideration. The approach proposed in this paper is applicable in this case as well. Besides, uncertain quasilinear systems with persistent perturbations can be studied. In this case the auxiliary function can be either the canonical matrix-valued function (3.1) or an appropriate vector Lyapunov function [9].
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References [1] F. Garofalo, L. Glielmo (Eds.), Robust Control via Variable Structure and Lyapunov Techniques, Springer, Berlin, 1996. [2] G.V. Kamenkov, Stability of Motion, Oscillation, Aerodynamics, vol. 1, Nauka, Moscow, 1971, Selected Papers. [3] V. Lakshmikantham, A.S. Vatsala, Stability of moving invariant sets, in: S. Sivasundaram, A.A. Martynyuk (Eds.), Advances in Nonlinear Dynamics, vol. 5, Gordon and Breach Scientific Publishers, Amsterdam, 1997, pp. 79–83. [4] A.A. Martynyuk, Qualitative Methods in Nonlinear Dynamics, Novel Approaches to Lyapunov Matrix Functions, Marcel Dekker Inc., New York, 2002. [5] Yu.A. Martynyuk-Chernienko, Stability analysis of uncertain systems via matrix-valued Lyapunov functions, Nonlinear Anal. 63 (2005) 388–404. [6] Yu.A. Martynyuk-Chernienko, Exponential convergence and instability of solutions of uncertain dynamical systems, Nonlinear Anal. 66 (2006) 1318–1328. [7] J.M. Skowronski, Parameter and state identification in non-linearizable uncertain systems, Int. J. Non-linear Mech. 19 (1984) 421–429. [8] A.A. Tikhonov, On stability of motion under constantly acting disturbances, Vestn. Leningrad. Un-t 1 (1) (1965) 95–101. [9] Y. Wang, Y. Shi, H. Sasaki, A criterion for stability of nonlinear time-varying dynamic system, Nonlinear Dyn. Syst. Theory 4 (2) (2004) 217–229.