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TRANSFORMATION OF TIME IN PROBLEMS OF STABILITY WITH RESPECT TO THE FIRST APPROXIMATION
B. F. BYLOV
Many problems of qualitative theory are connected with the consideration of a system of \(n\) differential equations, written in vector form as
\[ \frac{dx}{dt}=A(t)x+F(t,x), \tag{1} \]
where \(A(t)\) is a square \((n\times n)\) matrix, \(x\) is an \(n\)-dimensional vector, and \(F(t,x)\) is a vector function. In this connection, the usual assumptions include boundedness of the norm \(\|A(t)\|\) and the fulfillment of a Lipschitz condition, or of some analogue of it, for the function \(F(t,x)\).
In the present note we give a method, based on a transformation of time, which makes it possible to extend many of the known results to a class of systems of the form (1) subject to less restrictive limitations. To be definite, we shall assume that the matrix \(A(t)\), continuous on the positive half-axis \(t\ge 0\), satisfies the condition
\[ \int_t^{t+1} \|A(t)\|\,dt \le K; \tag{2} \]
the vector function \(F(t,x)\), defined and continuous in the cylinder \(\|x\|<H\le\infty,\ t\ge 0\), satisfies the condition
\[ \|F(t,x)\|\le g(t)\|x\|^m, \tag{3} \]
where \(g(t)\) is such that
\[ \int_t^{t+1} g(t)\,dt \le \beta \tag{4} \]
and \(K,m,\beta\) are some positive constants.
The norms \(\|x\|\) and \(\|A\|\) of a vector and a matrix are defined, as desired, in any suitable way. Thus, instead of boundedness of \(\|A\|\) and \(g(t)\), we assume boundedness on average of these quantities.
We now proceed to the construction of the transformation that is of interest to us. Let the numbers \(q_1,q_2,q\) satisfy the relations
\[ q_1\ge 0,\quad q_2\ge 0,\quad 1\ge q\ge q_1+q_2, \tag{5} \]
and let us agree to regard \(q_1=0\) only in the case \(A(t)\equiv 0\), and \(q_2=0\) in the case \(F(t,x)\equiv 0\).
Denote by \(\delta_n\) the difference
\[ \delta_n=q-\frac{q_1}{K}\int_n^{n+1}\|A(t)\|\,dt-\frac{q_2}{\beta}\int_n^{n+1}g(t)\,dt. \]
From conditions (2) and (4) it follows that the quantities subtracted on the right-hand side of the preceding equality do not exceed, respectively, \(q_1\) and \(q_2\), and therefore for any \(n\) the inequality
\[ 0\leq \delta_n\leq q \]
is satisfied.
Let us choose a discontinuous and nonnegative function \(\delta(t)\) such that the requirements
\[ \delta(n)=0,\qquad \int_n^{n+1}\delta(t)\,dt=\delta_n\quad (n=0,1,2,\ldots) \]
are satisfied, and otherwise arbitrary. The existence of such a function is obvious. We now define the function \(M(t)\) by the equality
\[ M(t)=\frac{q_1}{k}\|A(t)\|+\frac{q_2}{\beta}\,g(t)+\delta(t) \tag{6} \]
and set
\[ \tau=(1-q)t+\int_0^t M(t)\,dt=\tau(t). \tag{7} \]
Since for every \(t\geq 0\) there exists a positive derivative
\[ \frac{d\tau}{dt}=(1-q)+M(t), \]
the function \(\tau(t)\) increases monotonically and therefore has an inverse \(t=t(\tau)\). Let us examine in somewhat more detail the relation between the corresponding values of \(t\) and \(\tau\). Let \(t\geq 0\) be arbitrary, and let the integer \(n\) be such that \(n\leq t<n+1\). According to our construction, for any integer \(k=1,2,\ldots,n\) we have
\[ \int_{k-1}^k M(t)\,dt=q \]
and therefore
\[ \int_0^n M(t)\,dt=nq. \]
With the aid of these calculations and the equality
\[ \tau=(1-q)t+\int_0^n M(t)\,dt+\int_n^t M(t)\,dt \tag{8} \]
we obtain the estimate
\[ \tau<(1-q)t+tq+q=t+q \]
and, analogously,
\[ \tau>(1-q)t+nq=t+(n-t)q>t-q. \]
Thus, for the corresponding values \(t=t(\tau)\) and \(\tau=\tau(t)\), the inequalities
\[ t-q<\tau<t+q;\qquad \tau-q<t<\tau+q \tag{9} \]
are satisfied.
Let us note that in the case \(t=n\), from equality (8) it follows that
\[ \tau=(1-q)n+nq=n=t, \]
i.e., the integer values of \(t\) are invariants of the time transformation. Figuratively speaking, transformation (7) on each of the intervals \([n,n+1]\) carries out a “stretching” of time without changing the positions of the endpoints of the interval.
From inequalities (9) it follows that
\[ \lim_{t\to\infty}\frac{\tau(t)}{t} = \lim_{\tau\to\infty}\frac{t(\tau)}{\tau} =1. \tag{10} \]
Consider the system of equations
\[ \frac{dz}{d\tau}=A_{1}(\tau)z+F_{1}(\tau,z), \tag{11} \]
where we set
\[ A_{1}(\tau)=\frac{A(t(\tau))}{1-q+M(t(\tau))}, \tag{12} \]
\[ F_{1}(\tau,z)=\frac{F(t(\tau);z)}{1-q+M(t(\tau))}. \tag{13} \]
Replacing here \(M(t(\tau))\), in accordance with the definition, and taking into account the nonnegativity of the terms on the right-hand side of (6), we easily verify that \(A_{1}(\tau)\) and \(F_{1}(\tau,z)\) satisfy the conditions
\[ \|A_{1}(\tau)\|\leq K_{1};\qquad \|F_{1}(\tau,z)\|\leq \beta_{1}\|z\|^{m}, \tag{14} \]
where we have put
\[ K_{1}=\frac{K}{q_{1}} \quad\text{and}\quad \beta_{1}=\frac{\beta}{q_{2}}. \]
Here the last inequality is fulfilled for all \(\tau\geq 0\) and \(z\) belonging to the domain \(\|z\|<H\). Thus, system (11) satisfies the usual conditions of boundedness of the norm \(\|A_{2}(\tau)\|\) and a Lipschitz-type condition for \(F_{1}(\tau,z)\).
Let \(z(\tau)\) be some solution of system (11). Put \(x(t)=z(\tau(t))\). By direct substitution into (1) we verify that, in view of equalities (12), (13) and the rule for differentiating a composite function, \(x(t)\) is a solution of equations (1). Conversely, if \(x(t)\) is some solution of system (1), then the vector-function \(z(\tau)=x(t(\tau))\) is a solution of system (11), which is checked analogously. In view of equalities (10), to a solution of one of the systems (1) or (11), provided it is continuable to the entire half-axis \([0,\infty)\), there corresponds a solution of the other, likewise continuable to the entire half-axis. Let us establish the relation between the characteristic exponents of such solutions. Let \(x(t)\) be a nontrivial solution of system (1), defined on the half-axis \([0,\infty)\), and let \(z(\tau)=x(t(\tau))\) be the corresponding solution of system (11), defined for all \(\tau\geq 0\). Then, according to the definition of the characteristic exponent and equalities (10), we have
\[ \chi(z(\tau))=\lim_{\tau\to\infty}\frac{1}{\tau}\ln\|z(\tau)\|= \]
\[ =\lim_{\tau\to\infty}\frac{t(\tau)}{\tau}\cdot \frac{1}{t(\tau)}\ln\|x(t(\tau))\|= \]
\[ =\lim_{t\to\infty}\frac{1}{t}\ln\|x(t)\|=\chi(x(t)). \]
Thus, the characteristic exponents of the corresponding solutions of systems (1) and (11) coincide.
Let us consider, in particular, the systems of the first approximation
\[ \frac{dy}{dt}=Ay, \tag{15} \]
\[ \frac{dw}{d\tau}=A_1w, \tag{16} \]
where the matrices \(A\) and \(A_1\) are the same as in (1) and (11). The connection between the solutions of these systems obviously remains the same as between the solutions of systems (1) and (11).
Since every solution of system (16) is continued to the entire half-axis \(\tau \ge 0\), every solution of system (15) is also continued to the half-axis \(t \ge 0\). The same remarks are also valid for the systems adjoint to (15) and (16).
Let \(\lambda_1,\lambda_2,\ldots,\lambda_n\) be the collection of characteristic exponents common to systems (15) and (16). Suppose that system (15) is regular, i.e., the equality
\[ \sum_{i=1}^{n}\lambda_i-\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\operatorname{sp} A(t)\,dt=0 \]
holds. Make the substitution \(t=t(\tau)\) in the integral under the limit sign. We then obtain
\[ \sum_{i=1}^{n}\lambda_i-\lim_{t\to\infty}\frac{1}{t}\int_{0}^{\tau(t)} \operatorname{sp}\frac{A(t(\tau))}{1-q+M(t(\tau))}\,d\tau = \]
\[ = \sum_{i=1}^{n}\lambda_i-\lim_{t\to\infty}\frac{\tau(t)}{t}\cdot \frac{1}{\tau(t)}\int_{0}^{\tau(t)}\operatorname{sp} A_1(\tau)\,d\tau = \]
\[ = \sum_{i=1}^{n}\lambda_i-\lim_{\tau\to\infty}\frac{1}{\tau} \int_{0}^{\tau}\operatorname{sp} A_1(\tau)\,d\tau=0. \]
The last equality testifies to the regularity of system (16). In a similar way it is easy to verify that the regularity of system (15) follows from the regularity of system (16).
Let us note that in the definition of the matrix \(A_1(\tau)\) there participates not only the matrix \(A(t)\), but also the function \(g(t)\). Therefore, when the vector function \(F(t,x)\) is varied, the matrix \(A_1(\tau)\), generally speaking, will depend on the choice of \(F(t,x)\). However, in many cases this circumstance does not introduce essential complications into the solution of the problem of stability of the trivial solution in the first approximation. Without claiming originality, we give the following example of a generalization of a well-known theorem of Lyapunov [1].
Suppose that for system (1), in the domain \(\|x\|<H\le \infty,\ t\ge 0\), inequalities (2), (3), and (4) are satisfied, where it is assumed that \(m>1\), and that in the indicated cylinder the conditions ensuring uniqueness of the solution for given initial conditions are provided.
Then, in the case where the system of the first approximation is regular and all its characteristic exponents are negative, the trivial solution is asymptotically stable, uniformly with respect to perturbations
to functions \(F(t,x)\) satisfying conditions (3) and (4) with a constant \(\beta<\beta_0\).
Proof. In the particular case when, for \(A(t)\) and \(F(t,x)\), the conditions
\[ \|A\|<K\quad \text{and}\quad \|F(t,x)\|<\beta \|x\|^m\quad (m>1), \]
are fulfilled, the proof is given in the book of I. G. Malkin [2]. From the proof given there it follows that the number \(\eta>0\), corresponding to a given \(\varepsilon>0\) and such that \(\|x(t)\|<\varepsilon\) and \(\lim_{t\to\infty}\|x(t)\|=0\) for all \(x(t)\) with initial conditions \(\|x(0)\|<\eta\), can be defined as a function
\[ \eta=\eta(\varepsilon,n,m,\lambda_1,\lambda_2,\ldots,\lambda_n,K,\beta_0,B(\alpha)), \tag{17} \]
where the quantities \(n,m,K,\beta_0\) are given by the conditions of the theorem; \(\lambda_1,\lambda_2,\ldots,\lambda_n\) are the exponents of the unperturbed system; and \(B(\alpha)\) is a constant depending only on \(\alpha>0\), and such that, for every \(\alpha>0\) and for arbitrary solutions of system (15) and of the system adjoint to it, an inequality of the form
\[ \|u\|\leq B(\alpha)e^{(\lambda+\alpha)t}, \]
holds, where \(\chi(u)=\lambda\). Otherwise, \(\eta\) does not depend on the structure of \(A(t)\) and \(F(t,x)\). In other words, the quantity \(\eta\) is determined, for the given \(\varepsilon\), uniformly with respect to the set of regular matrices \(A\) and vector-functions \(F(t,x)\) to which the common set of constants occurring on the right-hand side of (17) corresponds.
Suppose now that, for system (1), the generalized conditions (2), (3), and (4) are fulfilled. Let us find, for the system of first approximation (15), a constant \(B(\alpha)\) possessing the properties indicated above. (The existence of such a constant follows easily from the linearity of system (15).)
Having fixed in some way \(q_1,q_2,q\) in accordance with the requirements (5), and assuming \(F(t,x)\) to be subject to the condition of the theorem and otherwise arbitrary, we carry out the transformation of time (7), which brings system (1) to the form (11). At the same time, as was established above, the regular system (15) passes into the regular system of first approximation (16).
Let \(y(t)\) be a nontrivial solution of system (15), \(w(\tau)=y(t(\tau))\) the corresponding solution of system (16), and \(\chi(y)=\chi(w)=\lambda\). Then, according to the definition of \(B(\alpha)\) and the inequalities (9), we obtain the estimate
\[ \|w(\tau)\|=\|y(t(\tau))\|\leq B(\alpha)e^{(\lambda+\alpha)t(\tau)}\leq \]
\[ \leq B(\alpha)e^{\max/\lambda_i/}e^{(\lambda+\alpha)\tau} = B_1(\alpha)e^{(\lambda+\alpha)\tau}, \]
where we have set
\[ B_1(\alpha)=B(\alpha)e^{\max/\lambda_i/} \tag{18} \]
and the quantity \(\alpha>0\) may be assumed arbitrarily small. The solutions of the system adjoint to (16) will also satisfy an analogous inequality with the same constant \(B_1(\alpha)\). Since system (11) is evidently subject to the conditions of the theorem [2] mentioned at the beginning of the proof, putting
\[ \eta_1=\eta(\varepsilon,n,m,\lambda_1,\lambda_2,\ldots,\lambda_n,K_1,\beta_{01},B_1(\alpha)) \]
we shall have, for all its solutions \(z(\tau)\) such that \(\|z(0)\|<\eta_1\), that \(\|z(\tau)\|<\varepsilon\) for \(\tau\geq 0\) and \(\lim_{\tau\to\infty}\|z(\tau)\|=0\). Hence it easily follows (using the relation between the solutions of systems (1) and (11)) that for any solution \(x(t)\) of system (1) whose initial data are subject to the inequality \(\|x(0)\|<\eta_1\), the relations will hold
\[ \|x(t)\| < \varepsilon \quad \text{and} \quad \lim_{t \to \infty}\|x(t)\| = 0. \]
To complete the proof it remains to note that the constants by means of which the quantity \(\eta_1\) is determined by equality (17) for a given \(\varepsilon\) do not depend on the choice of \(F(t,x)\) subject to conditions (3) and (4). For the constants \(n, m, \lambda_1, \ldots, \lambda_n, K_1, \beta_{0_1}\), the validity of this remark is obvious, and for \(B_1(\alpha)\) the same follows from consideration of equality (18).
The same method of transformation of time can be used in the study of systems of the form (1) satisfying conditions (2), (3), and (4), where the elements of the matrix \(A(t)\) and the function \(g(t)\) are measurable on every finite interval of the half-axis \(t \geqslant 0\), and the integrals are understood in the Lebesgue sense. In this case, a solution of the system is understood to be an absolutely continuous vector-function \(x(t)\) on every finite interval of the half-axis \(t \geqslant 0\), satisfying (1) almost everywhere.
References
- Lyapunov A. M. The General Problem of the Stability of Motion. GITTL, 1950, pp. 73–75.
- Malkin I. G. Theory of Stability of Motion. GITTL, 1952, pp. 360–364.
Received by the editors
February 27, 1965
Moscow Aviation Technological
Institute