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UDC 517.916
A NONLINEAR ANALOG OF THE LYAPUNOV TRANSFORMATION
Yu. S. Bogdanov, M. P. Bogdanova
Lyapunov transformations, i.e., linear substitutions with bounded essentially nonsingular matrices of coefficients ([1], p. 42), are the basic transformations of the qualitative theory of linear differential systems. With the aid of these transformations one establishes the asymptotic equivalence of linear differential systems to one another [7] and determines the asymptotic invariants of a system—quantities that remain unchanged under Lyapunov transformations. By virtue of Erugin’s theorem [2], a complete set of independent invariants of a reducible linear differential system contains a finite number of elements.
Using Lyapunov functions, one can construct a nonlinear analog of the transformations indicated above—a vd-transformation. The totality of all vd-transformations forms a group. With the aid of vd-transformations one determines both individual (pertaining to separate solutions) and collective (taking into account the mutual arrangement of solutions) vd-invariants of a system. An example of individual invariants of a system is furnished by the small vd-numbers of solutions [8, 9]. In the definition of the concept “asymptotic stability of the zero solution of a system,” the entire totality of solutions of the system with sufficiently small initial values participates as a whole; therefore the property of such invariance serves as an example of a collective invariant of the system. However, the analysis carried out in [8—10] shows that, with the aid of small vd-numbers, both necessary and sufficient criteria for the asymptotic stability of the zero solution of a system can be constructed. Examples of vd-invariants that are essentially collective are the various adjacent vd-numbers of a pair of sets from the space of initial values of solutions of the system [11]. Using adjacent vd-numbers, one can estimate from above the influence of variations of a system on the magnitude of the vd-numbers of its solutions. In particular, one can derive sufficient conditions under which perturbations of the system do not enlarge the characteristic totality of the set of solutions of the system.
The cycle of works described, including the present note, is based on ideas borrowed from the monographs of A. M. Lyapunov [1] and N. P. Erugin [2].
§ 1. VD-TRANSFORMATIONS
In the \((n+1)\)-dimensional real Euclidean space \(E\) with points
\(\eta=(\xi,\tau)=(\xi_1,\ldots,\xi_n,\tau)\), consider the cylinder
\[ H=G\times(-\infty,+\infty), \tag{1} \]
where \(G\) is a domain of the \(n\)-dimensional space \(E_0=\{\xi\}\), containing the origin \(O_0\). Denote by \(T\) the \(\tau\)-axis in \(E\)
\[ T=\{(0,\ldots,0,\tau)\mid \tau\in(-\infty,+\infty)\}. \tag{2} \]
From (1) and (2) it follows that \(T\subset H\).
Below, by the symbol \(l\) we denote a one-to-one transformation of \(H\) onto itself:
\[ l:H \to H,\qquad \eta' = l(\eta),\qquad \eta = l^{-1}(\eta'), \]
where it is assumed that the vector functions \(l(\eta)\) and \(l^{-1}(\eta')\) have continuous partial derivatives with respect to all the arguments \(\xi_1,\ldots,\xi_n,\tau\) (for sufficient conditions for the existence of derivatives of \(l^{-1}(\eta')\), see, for example, [3]) everywhere on \(H\), except for isolated planes \(\tau=\tau^*\), on which both \(l\) and \(l^{-1}\) are continuous, while the partial derivatives may have discontinuities of the first kind. In addition, it is assumed that the axis \(T\) is invariant with respect to the transformation \(l\),
\[ l(T)=T \tag{3} \]
and that each section of the cylinder \(H\) by the plane \(\tau=\tau_1\),
\[ H_{\tau_1}=\{\eta\mid \eta=(\xi,\tau),\ \xi\in G,\ \tau=\tau_1\}, \]
is also invariant with respect to \(l\),
\[ l(H_{\tau_1})=H_{\tau_1}. \tag{4} \]
Thus, for all real \(\tau\),
\[ l(O_0,\tau)=(O_0,\tau),\qquad l(\xi,\tau)=(\xi',\tau). \]
The transformations \(l\), taken together, form a group with respect to composition of transformations. We denote this group by \(L\).
On the domain \(G\), consider a continuous function \(v=v(\xi)\), positive on \(\dot G=G\setminus\{O_0\}\), vanishing at \(\xi=O_0\), and increasing without bound when approaching the boundary of \(G\) or when \(\xi\) increases without bound in norm (the latter in the case where \(G\) is unbounded). Extend the definition of \(v\) to the domain
\[ \dot H=\dot G\times(-\infty,+\infty)=H\times T, \]
by putting, for all \(\eta=(\xi,\tau)\in \dot H\) (with the notation preserved), \(v(\eta)=v(\xi)\). In addition, consider a real function \(d(\gamma_1,\gamma_2)\), continuous for all positive values of the arguments, and such that, for all \(\gamma>0\), \(\gamma_3>\gamma_2>\gamma_1>0\), the following hold:
\[ d(\gamma_2,\gamma_1)=-d(\gamma_1,\gamma_2),\qquad d(\gamma_2,\gamma)>d(\gamma_1,\gamma), \]
\[ d(\gamma_3,\gamma_2)+d(\gamma_2,\gamma_1)>d(\gamma_3,\gamma_1), \]
\[ \bigcup_{\gamma}\{d(\gamma,\gamma_1)\}=(-\infty,+\infty). \]
We shall call the transformation \(l\) a vd-transformation if all quantities \(d[v(\eta),v(\eta')]\) are bounded in the aggregate,
\[ \sup_{\eta\in\dot H}\left|d\{v(\eta),\ v[l(\eta)]\}\right|<+\infty. \tag{5} \]
Let us note that from (5) and the existence of \(l^{-1}\) it follows that
\[ \sup_{\eta'\in\dot H}\left|d\{v(\eta'),\ v[l^{-1}(\eta')]\}\right| = \sup_{\eta\in\dot H}\left|d[v(\eta'),\ v(\eta)]\right|<+\infty. \tag{6} \]
If \(v_0'(\xi)\) denotes the norm of \(\xi\), and \(d_0(\gamma_1,\gamma_2)\) the function \(\ln(\gamma_1/\gamma_2)\), then a transformation \(l_0\) of the space \(E\), homogeneous linear in each plane \(H_\tau\), will be a \(v_0d_0\)-transformation if and only if \(l_0\) is a Lyapunov transformation.
Let us denote by \(L_{vd}\) the totality of all \(vd\)-transformations of the domain \(H\). Obviously \(L_{vd}\subset L\). Take two transformations \(l\) and \(l_1\) from \(L_{vd}\). Let us estimate the quantity \(d\{v(\eta), v[l_1^{-1}l(\eta)]\}\). If we use the properties of \(d\) and formula (13) from [9], then we obtain
\[ \left|d\{v(\eta),\ v[l_1^{-1}l(\eta)]\}\right| = \left|d\{v[l^{-1}(\eta')],\ v[l_1^{-1}(\eta')]\}\right|\leq \]
\[ \leq \left|d\{v[l^{-1}(\eta')],\ v[l_1^{-1}(\eta')]\} - d\{v(\eta'),\ v[l_1^{-1}(\eta')]\}\right| + \left|d\{v(\eta'),\ v[l_1^{-1}(\eta')]\}\right|\leq \]
\[ \leq 2\left|d\{v[l^{-1}(\eta')],\ v(\eta')\}\right| + \left|d\{v(\eta'),\ v[l_1^{-1}(\eta')]\}\right| = \]
\[ = 2\left|d\{v(\eta),\ v[l(\eta)]\}\right| + \left|d\{v(\eta'),\ v[l_1^{-1}(\eta')]\}\right|. \]
On the basis of (5), (6), and the last inequality,
\[ \sup_{\eta\in H}\left|d\{v(\eta),\ v[l_1^{-1}l(\eta)]\}\right|<+\infty . \tag{7} \]
It follows from (7) that the composition of the transformations \(l\) and \(l_1^{-1}\) belongs to \(L_{vd}\). On the basis of the known criterion (see, for example, [12], p. 28), the set \(L_{vd}\) is a subgroup of \(L\).
§ 2. \(vd\)-SIMILARITY
Suppose that in the cylindrical domain \(H\) a differential system is given
\[ dx/d\tau=f(x,\tau). \tag{8} \]
We assume the vector-function \(f(\xi,\tau)\) to be measurable in \(\tau\) for each fixed \(\xi\) and continuous in \(\xi\) for each fixed \(\tau\), and moreover, for each point \((\xi_0,\tau_0)\in H\) one can specify a neighborhood \(u\) and a summable function \(m_u(\tau)\) such that for all \((\xi,\tau)\in u\) the following holds:
\[ |f(\xi,\tau)|\leq m_u(\tau). \]
System (8) satisfies conditions ensuring the uniqueness of the solution of any Cauchy problem:
\[ x(\tau_0)=\xi_0,\qquad (\xi_0,\tau_0)\in H \]
(for the subsequent arguments, in essence, right-hand uniqueness is important). In addition, we assume that all solutions of (8) can be continued without bound in both directions [5, 6]. Finally, we suppose that for any real \(\tau\)
\[ f(O_0,\tau)=O_0. \]
By a solution \(x(\tau)\) of system (8) we understand an absolutely continuous vector-function defined on the entire real axis and satisfying almost everywhere the identity
\[ dx(\tau)/d\tau=f[x(\tau),\tau]. \]
Denote by \(X\) the family of solutions \(x=x(\tau)\) of system (8):
\[ X=\{x(\tau,\xi_0,\tau_0)\},\qquad x(\tau_0,\xi_0,\tau_0)=\xi_0 . \]
By virtue of the assumptions made, \(X\) contains the zero vector-function \(O_0(\tau)\); the graphs \(x\in X\) in \(E\) fill \(H\) completely, and the graphs of two vector-functions intersect only when the vector-functions coincide. In \(X\) there is a continuous dependence of the solution on the initial data.
Let us denote by \(y\) the graph of the vector-function \(x\in X\) in \(E\):
\[ y(\tau)=(x(\tau),\tau),\qquad \tau\in(-\infty,+\infty). \]
The totality of all \(y\) for the given family \(X\) will be denoted by \(Y\). If all graphs from \(Y\) are subjected to the transformation \(l\), then we obtain the family of vector-functions
\[ Y'=\{y'\},\qquad y'(\tau)=(x'(\tau),\tau), \]
defined by the system of equations
\[ dx'/d\tau=f'(x',\tau) \tag{9} \]
of the same type as (8).
Two systems of equations of type (8) will be called vd-similar if there exists a vd-transformation \(l\) taking one of these systems into the other. The vd-transformations form a group \(L_{vd}\); therefore the relation of vd-similarity has the properties of reflexivity, symmetry, and transitivity. Consequently, the entire set of systems of type (8), defined on \(H\), splits into mutually disjoint classes of systems that are vd-similar to one another.
§ 3. vd-INVARIANTS
A property \(A\) of the family of solutions of system (8) will be called a vd-invariant if the same property is possessed by the family of solutions \(X'\) of any differential system (9) vd-similar to system (8).
Theorem 1. The vd-number \(\overset{*}{\Omega}vdx\) (see [9]) is a vd-invariant, i.e. from
\[ x'=l(x),\qquad l\in L_{vd}, \]
it always follows that
\[ \overset{*}{\Omega}vdx'=\overset{*}{\Omega}vdx . \tag{10} \]
Proof. On the basis of the definition of \(d\)
\[ \begin{aligned} d\{v[x'(\tau_0+\tau)],\ v[x'(\tau_0)]\} &=d\{v[l(x(\tau_0+\tau))],\ v[l(x(\tau_0))]\}\\ &=d\{v[x(\tau_0+\tau)],\ v[x(\tau_0)]\} +d\{v[l(x(\tau_0+\tau))],\ v[l(x(\tau_0))]\}\\ &\quad -d\{v[x(\tau_0+\tau)],\ v[l(x(\tau_0))]\}\\ &\quad +d\{v[x(\tau_0+\tau)],\ v[l(x(\tau_0))]\}\\ &\quad -d\{v[x(\tau_0+\tau)],\ v[x(\tau_0)]\}. \end{aligned} \tag{11} \]
By (13) of [9],
\[ d\{v[x'(\tau_0+\tau)],\ v[x'(\tau_0)]\} = \]
\[ =d\{v[x(\tau_0+\tau)],\ v[x(\tau_0)]\}+A+B, \]
\[ |A|=\left|d\{v[l(x(\tau_0+\tau))],\right. \]
\[
v[l(x(\tau_0))]\} - d\{v[x(\tau_0+\tau)],\ v[l(x(\tau_0))]\}\leq
\]
\[
\leq 2\left|d\{v[l(x(\tau_0+\tau))],\ v[x(\tau_0+\tau)]\}\right|,
\tag{12}
\]
\[
|B|=\left|d\{v[x(\tau_0+\tau)],\ v[l(x(\tau_0))]\} -\right.
\]
\[
\left.- d\{v[x(\tau_0+\tau)],\ v[x(\tau_0)]\}\right|\leq
\]
\[
\leq 2\left|d\{v[l(x(\tau_0))],\ v[x(\tau_0)]\}\right|.
\tag{13}
\]
The quantities \(A\) and \(B\), on the basis of \(l\in L_{vd}\), turn out to be bounded; therefore
\[
\overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}
\sup_{-\infty<\tau_0<+\infty}
d\{v[x'(\tau_0+\tau)],\ v[x'(\tau_0)]\} =
\]
\[
= \overline{\lim_{\tau\to+\infty}}\frac{1}{\tau}
\sup_{-\infty<\tau_0<+\infty}
d\{v[x(\tau_0+\tau)],\ v[x(\tau_0)]\}.
\tag{14}
\]
Equality (14), on the basis of the definition of the \(vd\)-number \(\Omega^{*}vdx\) (see [9]), also means the coincidence of the \(vd\)-numbers of the solutions \(x\) and \(x'\), i.e. (10).
Theorem 2. The lower \(vd\)-number \(\overline{\Omega}vdx\) (see [9]) is a \(vd\)-invariant.
Proof. Indeed, suppose that \(l\), \(x\), \(x'\), \(A\), and \(B\) have the same meaning as in the proof of the preceding theorem. If we use formula (11) and the boundedness of the quantities \(A\) and \(B\), following from (12) and (13), then we obtain
\[
\underline{\lim_{\tau\to+\infty}}\frac{1}{\tau}
d\{v[x'(\tau_0+\tau)],\ v[x'(\tau_0)]\} =
\]
\[
= \underline{\lim_{\tau\to+\infty}}\frac{1}{\tau}
d\{v[x(\tau_0+\tau)],\ v[x(\tau_0)]\}.
\tag{15}
\]
On the other hand, replacing \(\tau\) by \(-\tau\) in the preceding arguments, we obtain
\[
d\{v[x'(\tau_0-\tau)],\ v[x'(\tau_0)]\} =
\]
\[
= d\{v[x(\tau_0-\tau)],\ v[x(\tau_0)]\}+C+D,
\]
where \(C\) and \(D\) are bounded for all values of \(\tau\), \(\tau_0\), and all vector-functions \(x\in X\) for fixed \(l\in L_{vd}\). Consequently,
\[
\underline{\lim_{\tau\to+\infty}} d\{v[x'(\tau_0-\tau)],\ v[x'(\tau_0)]\} =
\]
\[
= \underline{\lim_{\tau\to+\infty}} d\{v[x(\tau_0-\tau)],\ v[x(\tau_0)]\}.
\tag{16}
\]
From the definition of the lower \(vd\)-number \(\overline{\Omega}vdx\) (see [9]) it follows that formulas (15) and (16), taken together, mean
\[
\overline{\Omega}vdx'=\overline{\Omega}vdx.
\]
Theorem 3. Stability in the sense of Lyapunov in the positive direction of the zero vector-function \(O_0(\tau)\) is a \(vd\)-invariant.
Proof. Suppose that \(O_0(t)\) is stable with respect to perturbations from \(X\), i.e.
\[
\lim_{\varepsilon\to+0}\sup_{v(\xi)\leq\varepsilon,\ \tau\geq 0} v[x(\tau,\xi,0)]=0.
\]
Suppose, moreover, that
\[ \lim_{\varepsilon\to +0}\ \sup_{\upsilon(\xi)<\varepsilon,\ \tau\geq 0}\ \upsilon[x'(\tau,\xi,0)]>0 . \tag{17} \]
Then there exists a number \(\delta>0\) and sequences \(\{\xi_m\}\to O_0\) and \(\{\tau_m\}\), \(\tau_m\geq 0\), such that for all \(m\)
\[ \upsilon[x'(\tau_m,\xi_m,0)]>\delta>0. \]
On the other hand, by assumption
\[ \lim_{m\to\infty}\upsilon[x(\tau_m,\xi_m,0)]=0. \]
Without loss of generality,
\[ \upsilon[x(\tau_m,\xi_m,0)]<\delta, \]
but then, on the basis of the definition of \(d\),
\[ |d\{\upsilon[x'(\tau_m,\xi_m,0)],\ \upsilon[x(\tau_m,\xi_m,0)]\}| = \]
\[ = d\{\upsilon[x'(\tau_m,\xi_m,0)],\ \upsilon[x(\tau_m,\xi_m,0)]\} > \]
\[ > d\{\delta,\ \upsilon[x(\tau_m,\xi_m,0)]\}. \]
Thus,
\[ \lim_{m\to\infty} d\{\delta,\upsilon[x(\tau_m,\xi_m,0)]\}=+\infty, \]
and therefore
\[ \sup |d\{\upsilon[x'(\tau_m,\xi_m,0)],\ \upsilon[x(\tau_m,\xi_m,0)]\}|=+\infty, \]
which contradicts (5). Hence (17) is incompatible with the vd-similarity of the families \(Y\) and \(Y'\), i.e., \(O_0(\tau)\) must be stable also with respect to perturbations from \(Y'\).
On the basis of the three theorems proved, all criteria for stability of the zero solution (8), stated in terms of the generalized characteristic numbers \(\overline{\Omega}\upsilon dx\) and \(\overset{*}{\Omega}\upsilon dx\) (see [7—11]), can be replaced by conditions on the generalized characteristic numbers of solutions of the vd-similar system (9).
§ 4. VD-REDUCIBILITY
We shall call the system of differential equations (8) vd-reducible if there exists a transformation \(l\in L_{vd}\) that carries this system into the stationary differential system
\[ dx'/d\tau=g(x'). \tag{9a} \]
An example of a \(\upsilon_0d_0\)-reducible system is any linear differential system reducible in the sense of Lyapunov (see [1—3]). We obtain another example if we consider the periodic differential system
\[ dx/d\tau=f(x,\tau), \qquad f(\xi,\tau+\omega)=f(\xi,\tau). \tag{8a} \]
There are known (see, for example, [13—18]) cases when there exists a transformation of system (8a), analytic in the components \(\xi\) and periodic in \(\tau\), which carries this system into a stationary system of type (9a). In particular cases the indicated transformation will belong to \(L_{vd}\), and then the system (8a) under consideration is certainly vd-reducible.
From the invariance of the generalized characteristic numbers \(\overline{\Omega}\upsilon dx\) and \(\overset{*}{\Omega}\upsilon dx\), and the invariance of the property of stability with respect to vd-transformations
it follows that all the criteria of stability and instability proved in [7—10] for stationary systems (9a) turn out to be valid also for vd-reducible systems (8).
§ 5. VD-REGULARITY
We shall call a transformation \(\tilde l\) a generalized vd-transformation if, instead of condition (5), the condition
\[ d\{v(\eta),\ v[\tilde l(\eta)]\}=o(\tau) \]
is satisfied as \(\tau\to\pm\infty\) uniformly with respect to \(\eta\). If \(v_0(\xi)=\|\xi\|\), and \(d_0(\gamma_1,\gamma_2)=\ln(\gamma_1/\gamma_2)\), then the transformation \(\tilde l_0\) of the space \(E\), homogeneous and linear in each plane \(H_{\tau_0}\), will be a generalized \(v_0d_0\)-transformation if and only if \(\tilde l_0\) is a generalized Lyapunov transformation, i.e., if the elements of the coefficient matrices of the direct and inverse transformations grow more slowly than \(e^{\alpha t}\) for any positive constant \(\alpha\) [19, 20].
If \(\tilde L_{vd}\) denotes the totality of all generalized vd-transformations of the cylinder \(H\), then \(\tilde L_{vd}\) proves to be a subgroup of \(L\). Systems (8) and (9), similar to one another in the sense of generalized vd-transformations, possess the same property that the vd-numbers and the small vd-numbers of the corresponding solutions of these systems coincide. In other words, the generalized characteristic numbers are invariants of generalized vd-transformations. We note that the property of Lyapunov stability is not an invariant of a generalized vd-transformation.
Along with vd-reducible differential systems one may also consider their generalization—vd-regular differential systems, i.e., such systems (8) which, by means of a suitable generalized vd-transformation, can be transformed into stationary systems of the form (9a).
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Received by the editors
February 18, 1967
Belorussian State University named after V. I. Lenin
Belorussian Polytechnic Institute