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Invariance and asymptotic stability
Journal of Applied Mathematics and Mechanics 75 (2011) 317–322
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Invariance and asymptotic stability夽 A.M. Kovalev Donetsk, Ukraine
a r t i c l e
i n f o
Article history: Received 16 October 2010
a b s t r a c t The stability of the zero solution of an autonomous non-linear system is considered. The problem of finding the variables in relation to which the solution is asymptotically stable if the Lyapunov function with a sign-definite derivative is known, is formulated and solved. The maximality of the set in relation to which the solution is asymptotically stable is established. The investigation is based on the method of auxiliary functions and clarifies the relation between the properties of invariance and the asymptotic stability of dynamical systems. The constructiveness of the results obtained is demonstrated by an example. © 2011 Elsevier Ltd. All rights reserved.
The first Lyapunov theorem on stability1 introduces two functions to the researcher: V(x) and V˙ (x). The question arises as to whether, knowing these functions, it is possible to obtain additional information about the solution (apart from its stability). The Barbashin–Krasovskii theorem2 is such an attempt to reinforce the Lyapunov theorem by analysing the set M on which the derivative V˙ (x) vanishes, and the possibility of trajectories of the system being contained in this set. In this context, mention must be made of the Rumyantsev–Oziraner theorem3 and the Risito theorem,4 which extend the Barbashin–Krasovskii theorem to problems of stability for part of the variables. The relation between these results and the Samoilenko theorem,5 which establishes the stability and unilateral invariance of a set described by a sign-definite function having a sign-definite derivative, is interesting. The development of the Barbashin–Krasovskii theorem consists of constructing6 a Lyapunov function with a sign-definite derivative for Barbashin–Krasovskii systems. It involves an analysis of the set M for invariance, i.e., the presence in it of invariant sets, including trajectories. Another important result of this investigation is the creation of the method of auxiliary functions, which opens the way to the construction of a structural stability theory. The next step in this direction that is taken in the present paper consists of applying the method of auxiliary functions to systems for which the conditions of Lyapunov’s first theorem are satisfied, with the aim of establishing the properties of asymptotic stability of their solutions. The formulation of the problem and the preliminary material are given in the first three sections. The scheme for constructing the Lyapunov function with a sign-definite derivative is given in Section 4. The main result is the proof of the theorem on partial asymptotic stability. The property of maximality of the set in relation to which the zero solution is asymptotically stable is established, and the limitation of its dimension is also established. This is the focus of Section 5, in the conclusion of which the results obtained are used to investigate the stability of the zero solution of a fourth-order non-linear system.
1. Stability and partial asymptotic stability Consider the stability of the zero solution of the system (1.1) where D is a certain vicinity of zero, and the function f(x) for x ∈ D is assumed to be continuously differentiable a sufficient number of times. The dot with respect to time t of the dependent variable x, and also of the function V(x) by virtue of system denotes differentiation (1.1): V˙ (x) = ∇ V (x), f (x) , Here ∇ is the differentiation operator: when applied to a scalar function it gives the gradient, and when applied to a vector function it give the Jacobi matrix; the symbol , denotes a scalar product.
夽 Prikl. Mat. Makh. Vol. 75, No. 3, pp. 449-456, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.07.009
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In order to study partial asymptotic stability, we will introduce the notation
and rewrite system (1.1) in the form (1.2) We will assume that the zero solution of system (1.1) is isolated and stable, and also that the Lyapunov function V(x) satisfying Lyapunov’s first theorem is known.1 Problem 1. Suppose that, for system (1.1), the function V(x) satisfying Lyapunov’s first theorem is known. It is necessary to find the variables y = g(x) in relation to which the zero solution of system (1.1) is asymptotically stable. To solve this problem, the following theorems can be used. Theorem 1. Suppose that, for system (1.1), a sign-definite function V(x) exists, the derivative of which V˙ (x), by virtue of system (1.1), is a sign-definite function with respect to y, and its sign is opposite to the sign of the function V(x). Then, the zero solution of system (1.1) is stable and asymptotically stable with respect to y. Theorem 2. Suppose that, for system (1.1), a sign-definite function V(x) exists, which vanishes on the set M, the derivative of which V˙ (x), by virtue of system (1.1), is a sign-constant function, and its sign is opposite to the sign of the function V(x). Furthermore, the set {x: y = 0} is invariant, and the set M\{x: y = 0} does not contain whole half-trajectories. Then the zero solution of system (1.1) is asymptotically stable with respect to y. Theorem 1. is the corollary of Theorem 5 of Ref. 7 for autonomous systems. Theorem 2 is actually a reformulation, for the case under consideration, of the Risito theorem,4 which in turn extends the Barbashin–Krasovskii theorem to the case of partial stability. Note that Theorem 1 indicates that the problem formulated in the present paper has already been examined by Rumyantsev.7 Theorem 2 identifies the influence of the property of invariance on the asymptotic stability: the set {x: y = 0} should be invariant. The relation between the invariance and the asymptotic stability was established for the first time by Barbashin and Krasovskii, and was notable by the absence of whole half-trajectories, apart from the zero solution, on the set M. Subsequent analysis shows that both of the above properties reflect the essence of the situation. Therefore, to solve the problem formulated above, certain information on invariance theory is required. 2. Invariant relations In the qualitative theory of differential equations, concepts of the invariant set and invariant relation are connected with the property of invariance. Definition 1. The set G ⊂ D is termed the invariant set (IS) of system (1.1) if any solution x(t) having, with G, a common point x(t*), belongs entirely to this set: x(t) ∈ G, t ∈ [t0 , ∞). Definition 2. The relation (x) = 0 is called the invariant relation (IR) of system (1.1) if the set defined by it contains the IS of system (1.1). ISs play an important role in describing the dynamics of systems and are described by Levi-Civita equations.8 Theorem 3. In order for the equations Vi (x) = 0 (i = 1,..., l) to determine the IS of system (1.1), it is necessary and sufficient for the functions Vi (x) to satisfy the system of linear partial differential equations
(2.1) where the functions ij (x) do not have singularities in the region considered. The following theorem is a convenient tool for checking whether the given relation will be the invariant relation of system (1.1). Theorem 4.
9
The IS G of system (1.1), that is generated by the IR (x) = 0, is defined by the equations (2.2)
where l is the number of functionally independent functions in the sequence (2.3) / 0 for x ∈ G. Here, ∇ (x) = Using Theorem 4, an important property that is necessary to construct Lyapunov’s function with a sign-definite derivative is established.10,11 Lemma 1. If the set N = {x: (x) = 0, ∇ (x) = / 0} ⊂ D does not contain whole half-trajectories, then, for each point x0 ∈ N,a k will be found such that (k) (x0 ) = / 0. This property made it possible to obtain6,10,11 auxiliary functions, the addition of which to the initial Lyapunov functions, when the conditions of the Barbashin–Krasovskii theorem are satisfied, successively narrows the set on which its derivative vanishes, from the initial set M down to the zero point, retaining the sign-definite nature of the function itself and of its derivative at the remaining points.
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3. Auxiliary functions To construct auxiliary functions, the structure of the set M, defined by its geometric and differential singularities, is important. First (geometric singularities), the set M may be the sum of subsets
Furthermore, paired intersections Mk ∩ Mm may contain non-zero points for certain k, and m, which must also be taken into account. Second (differential singularities), for certain sets Mi , the problem of the existence of ISs may not be solved by the first two terms of sequence (2.3), i.e. in, Lemma 1 for the points x0 ∈ Mi we have k > 1. We will give two types of auxiliary function6,10,11 that can be used to construct a Lyapunov function with a sign-definite derivative. In the simplest case, when the set M, on which V˙ (x) = 0, is described by a single function (x), M = {x: (x) = 0, ∇ (x) = / 0}, and the problem of the existence of an IS is solved by the first two terms of sequence (2.3); the following function is adopted as an auxiliary function: (3.1) A function of the type
(3.2) is adopted as an auxiliary function for the set Mi when the set M consists of certain sets: M1 ∪ M2 ... ∪ Ms , for each of which the problem of the existence of an IS is solved by the first two terms of sequence (2.3). Remark 1. To simplify the functions (3.1) and (3.2), instead of the function f(x) it is possible to use its value fN (x) calculated for x ∈ M, x ∈ Mi .6 4. Construction of the sign-definite derivative The scheme for setting up an auxiliary function, which ensures that a sign-definite derivative is obtained, is as follows. 1◦ . The set M = {x: (x) = 0, ∇ (x) = / 0} is represented in the form M = 2◦ . Each set Mi is represented in the form of the sum of subsets:
s
Mi , Mk ∩ Mm = ∅, k, m = 1,..., s.
i=1
such that
3◦ . For each subset Mij , an auxiliary function Vaij (x) of the second type of (3.2) is constructed. 4◦ . The values of parameters mij and ␣ij are selected, and the sign-definite function
with a sign-definite derivative is constructed on a set extended compared with the initial set M for the function V(x). This scheme was used10,11 to construct a sign-definite function with a sign-definite derivative for systems satisfying the Barbashin–Krasovskii theorem. Construction is carried out in stages by the successive addition to the initial Lyapunov function of auxiliary functions with the selection of appropriate values of the parameters mij and ␣ij . The proof of sign-definiteness at each stage is identical with the proof6 for an auxiliary function of the first type. 5. The fundamental theorem We will return to the solution of Problem 1. We will assume that the set M on which the derivative V˙ (x) vanishes is represented in the form of the sum of sets:
Using Theorem 4, we investigate the sets Mi for invariance. Two cases are possible. In the first case, Eqs (2.2) for i = 1,..., s allow of only a zero solution, i.e., none of the sets Mi contains an IS, and for system (1.1) the conditions of the Barbashin–Krasovskii theorem are satisfied. Then we establish that the zero solution is asymptotically stable with respect
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to all the variables, and for system (1.1) a Lyapunov function with a sign-definite derivative is constructed by the method of auxiliary functions.6,10,11 In the second case, besides the sets Mi not containing ISs, there are sets M1 ,..., Mc containing ISs that are described by the functions p1 (x),..., pc (x): pi (x) = 0 (i = 1,..., c). Suppose that, among the functions pi (x), there are k independent ones. We will adopt them as new variables yi = pi (x) (i = 1,..., k). Then, the auxiliary functions retain the sign-definiteness of the initial Lyapunov function, and here the functions corresponding to the first group ensure the sign-definiteness of the derivative on the corresponding sets, while functions of the second group lead to a sign-definiteness only with respect to the variables yi . Therefore, the function V(x), plotted by the proposed scheme, will be sign-definite, and its derivative V˙ (x) will be sign-definite. Thus, in the present case, the zero solution will be asymptotically stable with respect to y. More specifically, the following theorem holds. Theorem 5. Suppose that, for system (1.1), a sign-definite function V(x) exists whose derivative, by virtue of system (1.1), is sign-definite, and its sign is opposite to the sign of the function V(x). The set M = {x: V˙ (x) = 0} is represented by the sum of sets:
(5.1) We will assume that V(x) and i (x) are functions, differentiated a sufficient number of times, the sign-definiteness of the function V(x) (j) is defined by a form of finite order, and the sign-definiteness of V˙ (x) and the inequality i (x) = 0 are defined by expansion terms in the vicinity of finite-order zero. Then numbers mij and ˛ij exist such that the function
(5.2) will be sign-definite, while its derivative V˙ f (x) will be sign-definite with respect to y, having a sign opposite to the sign of the function V(x), and the zero solution of system (1.1) will be stable with respect to all the variables and asymptoticaly stable with respect to y. The functions Vaij (x) are defined by formula (3.2), yT = (y1 ,..., yk ); yi = pi (x) are independent functions by means of which the ISs contained in sets Mij are described. Proof. By virtue of the assumptions made, to establish the sign-definiteness of the functions Vf (x) and V˙ f (x), it is sufficient to analyse the expansions of the functions V(x), V˙ (x) and Vaij (x), V˙ aij (x). Since the sign-definiteness of the function V(x) is defined by a form of finite order, sign-definiteness of the function Vf (x) is achieved by selecting the numbers mij in such a way that the additional terms for the functions Vaij (x) have a higher order than the order of the form for the function V(x). The function Vf (x), ensuring sign-definiteness of V˙ f (x) (and accordingly a proof) is plotted in stages by the successive addition of auxiliary functions constructed for the sets Mij . At the initial stage, the sets Mij are divided into two groups: (1) not containing ISs, (2) containing k ISs. Among the functions pi describing the sets of the second group, the independent functions yi = pli (x) (i = 1,..., k) are selected, in terms i of which the two functions pi (x) used in forming the sets Mij are expressed. By virtue of the properties of functions yi , each set Mij is described by the equations (5.3) (j−1) (j) where ijl (y1 ,..., yk ) are independent functions of the sequence i (x), ˙ i (x),..., i (x). Here, by virtue of i (x) = / 0, we have
(5.4) Taking the above into account, by means of formula (3.2) we obtain
(5.5) where
It must be pointed out that Vaij = Vaij (y), as the quantity ∇ ij (y), f (y, z) consists of the derivatives of the function ϕi (x), and, by virtue (s)
of the assumptions made, are expressed in terms of the variable y. The same remark holds as regards the derivatives Vaij by virtue of system (1.1). For the sets Mij of the first group, according to the Scheme 10,11 used for the Barbashin–Krasovskii system, a sign-definite function Vf1 (x) is constructed, the derivative of which vanishes only on sets of the second group but retains its sign at the remaining points of region D. For the sets Mij of the second group, from formulae (5.3) to (5.5) it follows that V˙ aij (y) > 0 for y ∈ Mij . Repeating for the given case the proof6 of the sign-definiteness of the functions V˙ f = V˙ s + ˛V˙ a , we establish that, through the selection of the quantities ␣ij and mij , we can aim to keep the function V˙ f 1 (x) + ˛ij V˙ aij (y) sign-definite with respect to y on the initial set, and furthermore, to keep it y-sign-definite of the same sign on the set Mij . Also repeating this construction for all sets of the second group, we will find that, with the selected values
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of ␣ij and mij , the function (5.2) will be sign-definite, and its derivative V˙ f will be y-sign-definite, having an opposite sign to that of the function Vf . Thereby, the conditions of the theorem concerning stability and asymptotic stability with respect to y are satisfied. Remark 2. 1.
As auxiliary functions Vaij , instead of those defined by formula (5.5), we can use the simplified functions indicated in remark
Remark 3. stable.
Theorem 5 generalizes Theorem 1, extending the set of variables with respect to which the zero solution is asymptotically
Remark 4. When k = n, Theorem 5 gives the construction6,10,11 of the Lyapunov function for systems satisfying the Barbashin–Krasovskii theorem, and when k < n it generalizes known extensions of the Barbashin–Krasovskii theorem to the case of partial asymptotic stability. Remark 5. In Theorem 5, partial asymptotic stability is established with respect to the variable y possessing the same property as in Risito’s theorem: the set {x: y = 0} is invariant. Theorem 5 solves all questions related to the stability and asymptotic stability with respect to the all and to part of the variables for non-linear autonomous systems if a positive-definite function having a negative-definite derivative is known, thereby excluding the use of other theorems based on the Lyapunov function method, Here, it generalizes the Rumyantsev–Oziraner theorem and works when the set on which the derivative vanishes is not invariant, and Risito’s theorem cannot be applied. In both cases, Theorem 5 extends the set of variables with respect to which the zero solution is asymptotically stable. A new and important factor for stability theory is the formulation and solution, using Theorem 5, of Problem 1, which is the foundation for the development of the coordinate approach in stability theory. We will examine the question as to whether it is possible to extend the set of partial asymptotic stability obtained in Theorem 5, by constructing, for example, another Lyapunov function. The answer to this question is negative, by virtue of which the constructed set Mp = {x: y = 0} of partial asymptotic stability is invariant and, for its points, the condition V˙ f (x) = 0 is satisfied. Theorem 6. Suppose the conditions of Theorem 5 are satisfied. Then, the set with respect to which the zero solution of system (1.1) is asymptotically stable is the maximum possible, i.e., does not allow of extension. Proof. For the proof, we will show that all solutions of system (1.1) with initial data from the set Mp will not be asymptotically stable but only stable. Changing to the variables y, z(xT = (yT , zT )), by virtue of the invariance of Mp we conclude that, for these solutions, y = 0 ∀ t ≥ t0 , while for the variable z with y = 0 we obtain the system z˙ = f2 (z) with the Lyapunov function Vf (z) > 0, V˙ f (z) = 0, which proves the theorem. Additional information concerning the maximum set of asymptotic stability is given by the following theorem. Theorem 7. Suppose the zero solution of system (1.1) is an isolated stable solution. Then the dimension of the set of partial asymptotic stability cannot be greater than n – 2: dim Mp ≤ n – 2 (or the solution is asymptotically stable with respect to all the variables). Proof. We will show that, if, despite the statement of the theorem, dim Mp = n – 1, then the zero solution will not be an isolated zero solution. In fact, by virtue of the invariance of Mp , for solutions with an initial condition y = 0, we have the equation z˙ = azk +..., z ∈ R1 . Then, with a = / 0, the solution of z(t) is either asymptotically stable or unstable.1 Therefore, by virtue of the condition of stability of the solution considered, we obtain a = 0, i.e., z˙ =0, which signifies the isolated nature of the zero solution and proves the theorem. Example. We will use the results obtained to investigate the stability of the zero solution of the system (5.6) / 0 for system (5.6), special point. To analyse the stability of the zero solution, we will consider the When 1 2 ac = zero is an isolated function V = x12 + x22 + x32 + x42 ; V˙ = 2 1 x12 + 2 x22 x32 ). On the basis of Theorem 1 we conclude that, when 1 < 0 and 2 < 0, the zero solution of system (7.1) is stable (with respect to all the variables) and asymptotically stable with respect to x1 . We will use Theorem 5 to find the maximum set with respect to which the zero solution of system (7.1) is asymptotically stable. We will consider two cases separately:
We will begin the investigation by analising the set M on which V˙ (x) = 0. In the first case, the set M consists of three sets: M = M1 ∪ M2 ∪ M3 , where
Here it is assumed that the points of the set M1 ∩ M2 are excluded from the sets M1 and M2 . Let us consider the behaviour of the derivatives of the functions i (x) on sets Mi . On the set M1 we have ˙ 11 = bx32 and ˙ 12 = 0, and therefore ˙ 11 = / 0, as the points of the set M1 for which x3 = 0 are attributed to M3 . On the set M2 we have ˙ 21 = ax22 and ˙ 22 = x4 , and
/ 0. By Theorem 5 numbers m1 , m2 , therefore ˙ 221 + ˙ 222 > 0. On the set M3 we have ˙ 31 = 0, ˙ 32 = 0 and ˙ 33 = x4 , and therefore ˙ 33 = m3 , ␣1 , ␣2 and ␣3 exist such that the function
(5.7)
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will be positive-definite, and its derivative, by virtue of system (5.6), will be negative-definite. Thus, in the first case the zero solution of system (5.6) is asymptotically stable. Note that, when constructing function (5.7), in accordance with Remark 2, simplified auxiliary functions are used. The second case differs from the first solely in the fact that, on the set M1 , we have ˙ 11 = ˙ 12 = 0 (likewise, all higher derivatives are equal to zero). By Theorem 4 we conclude that the set M1 is invariant. This is also confirmed by Theorem 3. From the form of system (5.6) it is possible to obtain the Levi-Civita equation (2.1) directly for the set M1 :
The function Vf will have the form (5.7), where it must be assumed that b = 0, and will remain positive-definite, while its derivative, unlike the first case, will be negative-definite and will vanish only on the set M1 , i.e., will be negative-definite with respect to x1 and x2 . Thus, by Theorem 5, in the second case the zero solution of system (5.6) is stable (with respect to all the variables) and asymptotically stable with respect to x1 and x2 . The maximality of the set of partial asymptotic stability is demonstrated by the fact that, for x1 = 0 and x2 = 0, the system takes the form x˙ 3 = x4 , x˙ 4 = x3 . Motion with respect to x3 and x4 is stable by virtue of the presence of the integral Vf 0 = x32 + x42 . Motion in a small neighbourhood of zero can be characterized as rotational, approaching, as t → ∞, rotation about a circle x1 = 0, x2 = 0, x32 + x42 = c 2 . We note, in conclusion, that the relation between two antagonist properties of dynamical systems – asymptotic stability and invariance – has been established and investigated. On the one hand the absence of invariant sets leads to the asymptotic stability established by the Barbashin–Krasovskii theorem, while on the other hand the invariant set is limiting for the partially asymptotic stable motions established by Theorem 5, proof of which has been obtained in this paper. References 1. Lyapunov AM. The General Problem of the Stability of Motion. Moscow/Leningrad: Gostekhizdat; 1950; Collected Papers Vol. 1. Moscow/Leningrad: Izd Akad Nauk SSSR; 1956. 2. Barbashin YeA, Krasovskii NN. The stability of motion as a whole. Dokl Akad Nauk SSSR 1952;86(3):453–6. 3. Rumyantsev VV, Oziraner AS. Stability and Stabilization of Motion with respect to part of the Variables. Moscow: Nauka; 1987. 4. Risito C. Sulla stabilita asintotica parziale. Ann Math Pure Applied 1970;84:279–92. 5. Samoilenko AM. Elements of the Mathematical Theory of Multifrequency Vibrations. In: Invariant Tori. Moscow: Nauka; 1987. 6. Kovalev AM. Construction of the Lyapunov function with a sign-definite derivative for systems satisfying the Barbashin–Krasovskii theorem. Prikl Mat Mekh 2008;72(2):266–72. 7. Rumyantsev VV. The asymptotic stability and instability of motion with respect to part of the variables. Prikl Mat Mekh 1971;35(1):138–43. 8. Levi-Civita T, Amaldi U. Lezioni di Meccanica Razionale. Vol. 2. Bologana: Zanichelli; 1952. 9. Kharlamov PV. Invariant relations of a system of differential equations. In: Mechanics of Solids. No. 6. Kiev: Naukova Dumka; 1974: 15-24. 10. Kovalev AM, Suikov AS. Construction of the Lyapunov function when the Barbashin–Krasovskii theorem is satisfied. Dop Nats Akad Nauk Ukraini 2008;(12):22–7. 11. Kovalev AM, Suikov AS. Lyapunov functions for systems satisfying the conditions of the Barbashin–Krasovskii theorem. Problemy Upravl i Informatiki 2008;(6):5–15.
Translated by P.S.C.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Invariance and asymptotic stability夽 A.M. Kovalev Donetsk, Ukraine
a r t i c l e
i n f o
Article history: Received 16 October 2010
a b s t r a c t The stability of the zero solution of an autonomous non-linear system is considered. The problem of finding the variables in relation to which the solution is asymptotically stable if the Lyapunov function with a sign-definite derivative is known, is formulated and solved. The maximality of the set in relation to which the solution is asymptotically stable is established. The investigation is based on the method of auxiliary functions and clarifies the relation between the properties of invariance and the asymptotic stability of dynamical systems. The constructiveness of the results obtained is demonstrated by an example. © 2011 Elsevier Ltd. All rights reserved.
The first Lyapunov theorem on stability1 introduces two functions to the researcher: V(x) and V˙ (x). The question arises as to whether, knowing these functions, it is possible to obtain additional information about the solution (apart from its stability). The Barbashin–Krasovskii theorem2 is such an attempt to reinforce the Lyapunov theorem by analysing the set M on which the derivative V˙ (x) vanishes, and the possibility of trajectories of the system being contained in this set. In this context, mention must be made of the Rumyantsev–Oziraner theorem3 and the Risito theorem,4 which extend the Barbashin–Krasovskii theorem to problems of stability for part of the variables. The relation between these results and the Samoilenko theorem,5 which establishes the stability and unilateral invariance of a set described by a sign-definite function having a sign-definite derivative, is interesting. The development of the Barbashin–Krasovskii theorem consists of constructing6 a Lyapunov function with a sign-definite derivative for Barbashin–Krasovskii systems. It involves an analysis of the set M for invariance, i.e., the presence in it of invariant sets, including trajectories. Another important result of this investigation is the creation of the method of auxiliary functions, which opens the way to the construction of a structural stability theory. The next step in this direction that is taken in the present paper consists of applying the method of auxiliary functions to systems for which the conditions of Lyapunov’s first theorem are satisfied, with the aim of establishing the properties of asymptotic stability of their solutions. The formulation of the problem and the preliminary material are given in the first three sections. The scheme for constructing the Lyapunov function with a sign-definite derivative is given in Section 4. The main result is the proof of the theorem on partial asymptotic stability. The property of maximality of the set in relation to which the zero solution is asymptotically stable is established, and the limitation of its dimension is also established. This is the focus of Section 5, in the conclusion of which the results obtained are used to investigate the stability of the zero solution of a fourth-order non-linear system.
1. Stability and partial asymptotic stability Consider the stability of the zero solution of the system (1.1) where D is a certain vicinity of zero, and the function f(x) for x ∈ D is assumed to be continuously differentiable a sufficient number of times. The dot with respect to time t of the dependent variable x, and also of the function V(x) by virtue of system denotes differentiation (1.1): V˙ (x) = ∇ V (x), f (x) , Here ∇ is the differentiation operator: when applied to a scalar function it gives the gradient, and when applied to a vector function it give the Jacobi matrix; the symbol , denotes a scalar product.
夽 Prikl. Mat. Makh. Vol. 75, No. 3, pp. 449-456, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.07.009
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In order to study partial asymptotic stability, we will introduce the notation
and rewrite system (1.1) in the form (1.2) We will assume that the zero solution of system (1.1) is isolated and stable, and also that the Lyapunov function V(x) satisfying Lyapunov’s first theorem is known.1 Problem 1. Suppose that, for system (1.1), the function V(x) satisfying Lyapunov’s first theorem is known. It is necessary to find the variables y = g(x) in relation to which the zero solution of system (1.1) is asymptotically stable. To solve this problem, the following theorems can be used. Theorem 1. Suppose that, for system (1.1), a sign-definite function V(x) exists, the derivative of which V˙ (x), by virtue of system (1.1), is a sign-definite function with respect to y, and its sign is opposite to the sign of the function V(x). Then, the zero solution of system (1.1) is stable and asymptotically stable with respect to y. Theorem 2. Suppose that, for system (1.1), a sign-definite function V(x) exists, which vanishes on the set M, the derivative of which V˙ (x), by virtue of system (1.1), is a sign-constant function, and its sign is opposite to the sign of the function V(x). Furthermore, the set {x: y = 0} is invariant, and the set M\{x: y = 0} does not contain whole half-trajectories. Then the zero solution of system (1.1) is asymptotically stable with respect to y. Theorem 1. is the corollary of Theorem 5 of Ref. 7 for autonomous systems. Theorem 2 is actually a reformulation, for the case under consideration, of the Risito theorem,4 which in turn extends the Barbashin–Krasovskii theorem to the case of partial stability. Note that Theorem 1 indicates that the problem formulated in the present paper has already been examined by Rumyantsev.7 Theorem 2 identifies the influence of the property of invariance on the asymptotic stability: the set {x: y = 0} should be invariant. The relation between the invariance and the asymptotic stability was established for the first time by Barbashin and Krasovskii, and was notable by the absence of whole half-trajectories, apart from the zero solution, on the set M. Subsequent analysis shows that both of the above properties reflect the essence of the situation. Therefore, to solve the problem formulated above, certain information on invariance theory is required. 2. Invariant relations In the qualitative theory of differential equations, concepts of the invariant set and invariant relation are connected with the property of invariance. Definition 1. The set G ⊂ D is termed the invariant set (IS) of system (1.1) if any solution x(t) having, with G, a common point x(t*), belongs entirely to this set: x(t) ∈ G, t ∈ [t0 , ∞). Definition 2. The relation (x) = 0 is called the invariant relation (IR) of system (1.1) if the set defined by it contains the IS of system (1.1). ISs play an important role in describing the dynamics of systems and are described by Levi-Civita equations.8 Theorem 3. In order for the equations Vi (x) = 0 (i = 1,..., l) to determine the IS of system (1.1), it is necessary and sufficient for the functions Vi (x) to satisfy the system of linear partial differential equations
(2.1) where the functions ij (x) do not have singularities in the region considered. The following theorem is a convenient tool for checking whether the given relation will be the invariant relation of system (1.1). Theorem 4.
9
The IS G of system (1.1), that is generated by the IR (x) = 0, is defined by the equations (2.2)
where l is the number of functionally independent functions in the sequence (2.3) / 0 for x ∈ G. Here, ∇ (x) = Using Theorem 4, an important property that is necessary to construct Lyapunov’s function with a sign-definite derivative is established.10,11 Lemma 1. If the set N = {x: (x) = 0, ∇ (x) = / 0} ⊂ D does not contain whole half-trajectories, then, for each point x0 ∈ N,a k will be found such that (k) (x0 ) = / 0. This property made it possible to obtain6,10,11 auxiliary functions, the addition of which to the initial Lyapunov functions, when the conditions of the Barbashin–Krasovskii theorem are satisfied, successively narrows the set on which its derivative vanishes, from the initial set M down to the zero point, retaining the sign-definite nature of the function itself and of its derivative at the remaining points.
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3. Auxiliary functions To construct auxiliary functions, the structure of the set M, defined by its geometric and differential singularities, is important. First (geometric singularities), the set M may be the sum of subsets
Furthermore, paired intersections Mk ∩ Mm may contain non-zero points for certain k, and m, which must also be taken into account. Second (differential singularities), for certain sets Mi , the problem of the existence of ISs may not be solved by the first two terms of sequence (2.3), i.e. in, Lemma 1 for the points x0 ∈ Mi we have k > 1. We will give two types of auxiliary function6,10,11 that can be used to construct a Lyapunov function with a sign-definite derivative. In the simplest case, when the set M, on which V˙ (x) = 0, is described by a single function (x), M = {x: (x) = 0, ∇ (x) = / 0}, and the problem of the existence of an IS is solved by the first two terms of sequence (2.3); the following function is adopted as an auxiliary function: (3.1) A function of the type
(3.2) is adopted as an auxiliary function for the set Mi when the set M consists of certain sets: M1 ∪ M2 ... ∪ Ms , for each of which the problem of the existence of an IS is solved by the first two terms of sequence (2.3). Remark 1. To simplify the functions (3.1) and (3.2), instead of the function f(x) it is possible to use its value fN (x) calculated for x ∈ M, x ∈ Mi .6 4. Construction of the sign-definite derivative The scheme for setting up an auxiliary function, which ensures that a sign-definite derivative is obtained, is as follows. 1◦ . The set M = {x: (x) = 0, ∇ (x) = / 0} is represented in the form M = 2◦ . Each set Mi is represented in the form of the sum of subsets:
s
Mi , Mk ∩ Mm = ∅, k, m = 1,..., s.
i=1
such that
3◦ . For each subset Mij , an auxiliary function Vaij (x) of the second type of (3.2) is constructed. 4◦ . The values of parameters mij and ␣ij are selected, and the sign-definite function
with a sign-definite derivative is constructed on a set extended compared with the initial set M for the function V(x). This scheme was used10,11 to construct a sign-definite function with a sign-definite derivative for systems satisfying the Barbashin–Krasovskii theorem. Construction is carried out in stages by the successive addition to the initial Lyapunov function of auxiliary functions with the selection of appropriate values of the parameters mij and ␣ij . The proof of sign-definiteness at each stage is identical with the proof6 for an auxiliary function of the first type. 5. The fundamental theorem We will return to the solution of Problem 1. We will assume that the set M on which the derivative V˙ (x) vanishes is represented in the form of the sum of sets:
Using Theorem 4, we investigate the sets Mi for invariance. Two cases are possible. In the first case, Eqs (2.2) for i = 1,..., s allow of only a zero solution, i.e., none of the sets Mi contains an IS, and for system (1.1) the conditions of the Barbashin–Krasovskii theorem are satisfied. Then we establish that the zero solution is asymptotically stable with respect
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to all the variables, and for system (1.1) a Lyapunov function with a sign-definite derivative is constructed by the method of auxiliary functions.6,10,11 In the second case, besides the sets Mi not containing ISs, there are sets M1 ,..., Mc containing ISs that are described by the functions p1 (x),..., pc (x): pi (x) = 0 (i = 1,..., c). Suppose that, among the functions pi (x), there are k independent ones. We will adopt them as new variables yi = pi (x) (i = 1,..., k). Then, the auxiliary functions retain the sign-definiteness of the initial Lyapunov function, and here the functions corresponding to the first group ensure the sign-definiteness of the derivative on the corresponding sets, while functions of the second group lead to a sign-definiteness only with respect to the variables yi . Therefore, the function V(x), plotted by the proposed scheme, will be sign-definite, and its derivative V˙ (x) will be sign-definite. Thus, in the present case, the zero solution will be asymptotically stable with respect to y. More specifically, the following theorem holds. Theorem 5. Suppose that, for system (1.1), a sign-definite function V(x) exists whose derivative, by virtue of system (1.1), is sign-definite, and its sign is opposite to the sign of the function V(x). The set M = {x: V˙ (x) = 0} is represented by the sum of sets:
(5.1) We will assume that V(x) and i (x) are functions, differentiated a sufficient number of times, the sign-definiteness of the function V(x) (j) is defined by a form of finite order, and the sign-definiteness of V˙ (x) and the inequality i (x) = 0 are defined by expansion terms in the vicinity of finite-order zero. Then numbers mij and ˛ij exist such that the function
(5.2) will be sign-definite, while its derivative V˙ f (x) will be sign-definite with respect to y, having a sign opposite to the sign of the function V(x), and the zero solution of system (1.1) will be stable with respect to all the variables and asymptoticaly stable with respect to y. The functions Vaij (x) are defined by formula (3.2), yT = (y1 ,..., yk ); yi = pi (x) are independent functions by means of which the ISs contained in sets Mij are described. Proof. By virtue of the assumptions made, to establish the sign-definiteness of the functions Vf (x) and V˙ f (x), it is sufficient to analyse the expansions of the functions V(x), V˙ (x) and Vaij (x), V˙ aij (x). Since the sign-definiteness of the function V(x) is defined by a form of finite order, sign-definiteness of the function Vf (x) is achieved by selecting the numbers mij in such a way that the additional terms for the functions Vaij (x) have a higher order than the order of the form for the function V(x). The function Vf (x), ensuring sign-definiteness of V˙ f (x) (and accordingly a proof) is plotted in stages by the successive addition of auxiliary functions constructed for the sets Mij . At the initial stage, the sets Mij are divided into two groups: (1) not containing ISs, (2) containing k ISs. Among the functions pi describing the sets of the second group, the independent functions yi = pli (x) (i = 1,..., k) are selected, in terms i of which the two functions pi (x) used in forming the sets Mij are expressed. By virtue of the properties of functions yi , each set Mij is described by the equations (5.3) (j−1) (j) where ijl (y1 ,..., yk ) are independent functions of the sequence i (x), ˙ i (x),..., i (x). Here, by virtue of i (x) = / 0, we have
(5.4) Taking the above into account, by means of formula (3.2) we obtain
(5.5) where
It must be pointed out that Vaij = Vaij (y), as the quantity ∇ ij (y), f (y, z) consists of the derivatives of the function ϕi (x), and, by virtue (s)
of the assumptions made, are expressed in terms of the variable y. The same remark holds as regards the derivatives Vaij by virtue of system (1.1). For the sets Mij of the first group, according to the Scheme 10,11 used for the Barbashin–Krasovskii system, a sign-definite function Vf1 (x) is constructed, the derivative of which vanishes only on sets of the second group but retains its sign at the remaining points of region D. For the sets Mij of the second group, from formulae (5.3) to (5.5) it follows that V˙ aij (y) > 0 for y ∈ Mij . Repeating for the given case the proof6 of the sign-definiteness of the functions V˙ f = V˙ s + ˛V˙ a , we establish that, through the selection of the quantities ␣ij and mij , we can aim to keep the function V˙ f 1 (x) + ˛ij V˙ aij (y) sign-definite with respect to y on the initial set, and furthermore, to keep it y-sign-definite of the same sign on the set Mij . Also repeating this construction for all sets of the second group, we will find that, with the selected values
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of ␣ij and mij , the function (5.2) will be sign-definite, and its derivative V˙ f will be y-sign-definite, having an opposite sign to that of the function Vf . Thereby, the conditions of the theorem concerning stability and asymptotic stability with respect to y are satisfied. Remark 2. 1.
As auxiliary functions Vaij , instead of those defined by formula (5.5), we can use the simplified functions indicated in remark
Remark 3. stable.
Theorem 5 generalizes Theorem 1, extending the set of variables with respect to which the zero solution is asymptotically
Remark 4. When k = n, Theorem 5 gives the construction6,10,11 of the Lyapunov function for systems satisfying the Barbashin–Krasovskii theorem, and when k < n it generalizes known extensions of the Barbashin–Krasovskii theorem to the case of partial asymptotic stability. Remark 5. In Theorem 5, partial asymptotic stability is established with respect to the variable y possessing the same property as in Risito’s theorem: the set {x: y = 0} is invariant. Theorem 5 solves all questions related to the stability and asymptotic stability with respect to the all and to part of the variables for non-linear autonomous systems if a positive-definite function having a negative-definite derivative is known, thereby excluding the use of other theorems based on the Lyapunov function method, Here, it generalizes the Rumyantsev–Oziraner theorem and works when the set on which the derivative vanishes is not invariant, and Risito’s theorem cannot be applied. In both cases, Theorem 5 extends the set of variables with respect to which the zero solution is asymptotically stable. A new and important factor for stability theory is the formulation and solution, using Theorem 5, of Problem 1, which is the foundation for the development of the coordinate approach in stability theory. We will examine the question as to whether it is possible to extend the set of partial asymptotic stability obtained in Theorem 5, by constructing, for example, another Lyapunov function. The answer to this question is negative, by virtue of which the constructed set Mp = {x: y = 0} of partial asymptotic stability is invariant and, for its points, the condition V˙ f (x) = 0 is satisfied. Theorem 6. Suppose the conditions of Theorem 5 are satisfied. Then, the set with respect to which the zero solution of system (1.1) is asymptotically stable is the maximum possible, i.e., does not allow of extension. Proof. For the proof, we will show that all solutions of system (1.1) with initial data from the set Mp will not be asymptotically stable but only stable. Changing to the variables y, z(xT = (yT , zT )), by virtue of the invariance of Mp we conclude that, for these solutions, y = 0 ∀ t ≥ t0 , while for the variable z with y = 0 we obtain the system z˙ = f2 (z) with the Lyapunov function Vf (z) > 0, V˙ f (z) = 0, which proves the theorem. Additional information concerning the maximum set of asymptotic stability is given by the following theorem. Theorem 7. Suppose the zero solution of system (1.1) is an isolated stable solution. Then the dimension of the set of partial asymptotic stability cannot be greater than n – 2: dim Mp ≤ n – 2 (or the solution is asymptotically stable with respect to all the variables). Proof. We will show that, if, despite the statement of the theorem, dim Mp = n – 1, then the zero solution will not be an isolated zero solution. In fact, by virtue of the invariance of Mp , for solutions with an initial condition y = 0, we have the equation z˙ = azk +..., z ∈ R1 . Then, with a = / 0, the solution of z(t) is either asymptotically stable or unstable.1 Therefore, by virtue of the condition of stability of the solution considered, we obtain a = 0, i.e., z˙ =0, which signifies the isolated nature of the zero solution and proves the theorem. Example. We will use the results obtained to investigate the stability of the zero solution of the system (5.6) / 0 for system (5.6), special point. To analyse the stability of the zero solution, we will consider the When 1 2 ac = zero is an isolated function V = x12 + x22 + x32 + x42 ; V˙ = 2 1 x12 + 2 x22 x32 ). On the basis of Theorem 1 we conclude that, when 1 < 0 and 2 < 0, the zero solution of system (7.1) is stable (with respect to all the variables) and asymptotically stable with respect to x1 . We will use Theorem 5 to find the maximum set with respect to which the zero solution of system (7.1) is asymptotically stable. We will consider two cases separately:
We will begin the investigation by analising the set M on which V˙ (x) = 0. In the first case, the set M consists of three sets: M = M1 ∪ M2 ∪ M3 , where
Here it is assumed that the points of the set M1 ∩ M2 are excluded from the sets M1 and M2 . Let us consider the behaviour of the derivatives of the functions i (x) on sets Mi . On the set M1 we have ˙ 11 = bx32 and ˙ 12 = 0, and therefore ˙ 11 = / 0, as the points of the set M1 for which x3 = 0 are attributed to M3 . On the set M2 we have ˙ 21 = ax22 and ˙ 22 = x4 , and
/ 0. By Theorem 5 numbers m1 , m2 , therefore ˙ 221 + ˙ 222 > 0. On the set M3 we have ˙ 31 = 0, ˙ 32 = 0 and ˙ 33 = x4 , and therefore ˙ 33 = m3 , ␣1 , ␣2 and ␣3 exist such that the function
(5.7)
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will be positive-definite, and its derivative, by virtue of system (5.6), will be negative-definite. Thus, in the first case the zero solution of system (5.6) is asymptotically stable. Note that, when constructing function (5.7), in accordance with Remark 2, simplified auxiliary functions are used. The second case differs from the first solely in the fact that, on the set M1 , we have ˙ 11 = ˙ 12 = 0 (likewise, all higher derivatives are equal to zero). By Theorem 4 we conclude that the set M1 is invariant. This is also confirmed by Theorem 3. From the form of system (5.6) it is possible to obtain the Levi-Civita equation (2.1) directly for the set M1 :
The function Vf will have the form (5.7), where it must be assumed that b = 0, and will remain positive-definite, while its derivative, unlike the first case, will be negative-definite and will vanish only on the set M1 , i.e., will be negative-definite with respect to x1 and x2 . Thus, by Theorem 5, in the second case the zero solution of system (5.6) is stable (with respect to all the variables) and asymptotically stable with respect to x1 and x2 . The maximality of the set of partial asymptotic stability is demonstrated by the fact that, for x1 = 0 and x2 = 0, the system takes the form x˙ 3 = x4 , x˙ 4 = x3 . Motion with respect to x3 and x4 is stable by virtue of the presence of the integral Vf 0 = x32 + x42 . Motion in a small neighbourhood of zero can be characterized as rotational, approaching, as t → ∞, rotation about a circle x1 = 0, x2 = 0, x32 + x42 = c 2 . We note, in conclusion, that the relation between two antagonist properties of dynamical systems – asymptotic stability and invariance – has been established and investigated. On the one hand the absence of invariant sets leads to the asymptotic stability established by the Barbashin–Krasovskii theorem, while on the other hand the invariant set is limiting for the partially asymptotic stable motions established by Theorem 5, proof of which has been obtained in this paper. References 1. Lyapunov AM. The General Problem of the Stability of Motion. Moscow/Leningrad: Gostekhizdat; 1950; Collected Papers Vol. 1. Moscow/Leningrad: Izd Akad Nauk SSSR; 1956. 2. Barbashin YeA, Krasovskii NN. The stability of motion as a whole. Dokl Akad Nauk SSSR 1952;86(3):453–6. 3. Rumyantsev VV, Oziraner AS. Stability and Stabilization of Motion with respect to part of the Variables. Moscow: Nauka; 1987. 4. Risito C. Sulla stabilita asintotica parziale. Ann Math Pure Applied 1970;84:279–92. 5. Samoilenko AM. Elements of the Mathematical Theory of Multifrequency Vibrations. In: Invariant Tori. Moscow: Nauka; 1987. 6. Kovalev AM. Construction of the Lyapunov function with a sign-definite derivative for systems satisfying the Barbashin–Krasovskii theorem. Prikl Mat Mekh 2008;72(2):266–72. 7. Rumyantsev VV. The asymptotic stability and instability of motion with respect to part of the variables. Prikl Mat Mekh 1971;35(1):138–43. 8. Levi-Civita T, Amaldi U. Lezioni di Meccanica Razionale. Vol. 2. Bologana: Zanichelli; 1952. 9. Kharlamov PV. Invariant relations of a system of differential equations. In: Mechanics of Solids. No. 6. Kiev: Naukova Dumka; 1974: 15-24. 10. Kovalev AM, Suikov AS. Construction of the Lyapunov function when the Barbashin–Krasovskii theorem is satisfied. Dop Nats Akad Nauk Ukraini 2008;(12):22–7. 11. Kovalev AM, Suikov AS. Lyapunov functions for systems satisfying the conditions of the Barbashin–Krasovskii theorem. Problemy Upravl i Informatiki 2008;(6):5–15.
Translated by P.S.C.