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ON STABILITY IN THE LARGE OF THE ZERO SOLUTION OF A COAGULATED SYSTEM*
B. V. SHIROKORAD
The paper proves a theorem, important in applications, on stability in the large of the zero solution, abbreviated as the 0-solution, for comparison of nonautonomous systems of ordinary differential equations, nonlinear and linear, of which the first differs (in the most interesting case) from the second by the introduction of nonlinear couplings between separate groups of linear equations, and effectively determines the bounds of these couplings (not violating stability).
We consider a system of ordinary differential equations
\[ \dot{x}=A(t)x+f(t,x) \tag{1} \]
with a coalesced matrix \(A(t)\), defined and continuous for \(t>\theta\),
\[ A(t)=\operatorname{diag}\{A_1(t),\ldots,A_m(t)\}\quad (1<m\le n), \]
consisting of blocks with square matrices \(A_k(t)\) of order \(r_k\ge 1\) \((k=1,\ldots,m;\ r_1+\cdots+r_m=n)\); the remaining (in general rectangular) blocks of \(A(t)\) consist of zeros; in particular, the \(A_k(t)\) may coincide with the Jordan blocks \(L_{k r}[\lambda_r(t)]\) (see 1, p. 29) of the matrix \(A(t)\).
The vector function \(f(t,x)=\{f_1(t,x),\ldots,f_m(t,x)\}\) is defined and continuous for \(t>\theta\) and all vectors (columns) \(x=\{x_1,\ldots,x_m\}\) from the \(n\)-dimensional vector space \(R^{(n)}\) over the field of real numbers (respectively, \(x_k\in R^{(r_k)}\) is an \(r_k\)-dimensional vector (column); \(k=1,\ldots,m\)), where it satisfies the conditions of uniqueness and continuability of solutions of system (1) and the growth restrictions on the channel couplings
\[ \|f_i(t,x)\|\le b_{i1}\|x_1\|+\cdots+b_{im}\|x_m\|, \tag{2} \]
where the constants \(b_{ij}\ge 0\) \((i,j=1,\ldots,m)\).
Let \(H_i(t,\tau)\) denote the Cauchy–Green matrix of the canonical fundamental system of solutions of the system of equations
\[ \dot{x}_i=A_i(t)x_i\quad (i=1,\ldots,m) \tag{3} \]
and suppose that there exist numbers \(C_i>0\) such that
\[ \int_{\tau}^{t}\|H_i(s,\tau)\|^2\,ds<C_i \quad \text{for } t\ge \tau>\theta \quad (i=1,\ldots,m), \tag{4} \]
and, moreover, the matrices of M. Roseau (see [2], p. 204)
* Reported on October 1, 1964, at the seminar of V. V. Nemytskii at Moscow State University.
\[ R_i(t,\tau)\equiv \int_\tau^t A_i(s)H_i(s,\tau)\,ds \to \{0\} \quad \text{as } t-\tau \to 0 \tag{5} \]
uniformly with respect to \(t\ge \tau>\theta\).
Lemma 1. Each function
\[ V_i(t,x_i)\equiv \int_t^{+\infty} \|H_i(\tau,t)x_i\|^2\,d\tau \quad (i=1,\ldots,m) \tag{6} \]
under conditions (4), (5) is defined and continuous for \(t>\theta\) and \(x_i\in R^{(r_i)}\), where it admits an infinitely small upper bound.
Proof. Applying the Cauchy–Schwarz inequality to the right-hand side of expression (6) and taking (4) into account, we obtain
\[ V_i(t,x_i)<C_i x_i^2 \quad (i=1,\ldots,m) \tag{7} \]
for \(t>\theta\) and \(x_i\in R^{(r_i)}\) \((x_i\ne \vec 0)\).
The last inequality and the continuity of the matrix \(H_i(t,\tau)\) for \(t\ge \tau>\theta\) \((i=1,\ldots,m)\) prove the assertion of Lemma 1.
Lemma 2. Each function \(V_i(t,x_i)\) (6) is positive definite for \(t>\theta\) and \(x_i\in R^{(r_i)}\) \((i=1,\ldots,m)\).
Proof. The Cauchy–Green matrix \(H_i(t,\tau)\) satisfies the well-known matrix Cauchy–Hamilton equation
\[ \frac{\partial H_i(t,t_0)}{\partial t} = A_i(t)H_i(t,t_0) \quad (t\ge t_0>\theta;\ i=1,\ldots,m). \]
Integrating both sides of this equation over the interval from \(\tau\) to \(t\) \((t\ge \tau\ge t_0>\theta)\), we obtain
\[ H_i(t,t_0)-H_i(\tau,t_0) = \int_\tau^t A_i(s)H_i(s,t_0)\,ds. \]
If we take into account that
\[ H_i(t,\tau)\equiv H_i(t,t_0)H_i(t_0,\tau) \quad \text{and} \quad H_i^{-1}(t,\tau)\equiv H_i(\tau,t), \]
then from the expression obtained above and from (5) there follow the identities
\[ R_i(t,\tau)\equiv \{H_i(t,t_0)-H_i(\tau,t_0)\}H_i(t_0,\tau)\equiv H_i(t,\tau)-E_{r_i}, \]
valid for \(t\ge \tau\ge t_0>\theta\), where \(E_{r_i}\) is the identity matrix of order \(r_i\) \((i=1,\ldots,m)\).
Hence the integrand in (6) takes the following form:
\[ \|H_i(t,\tau)x_i\|^2 = x_i^2+2(R_i x_i,x_i)+\|R_i(t,\tau)x_i\|^2 \tag{8} \]
for \(t\ge \tau>\theta\) \((i=1,\ldots,m)\).
By virtue of (5), for every positive number \(\varepsilon>0\) such that \(2\varepsilon+\varepsilon^2<1/2\), there exists \(\delta=\delta(\varepsilon)>0\) such that
\[ \|R_i(t,\tau)\|<\varepsilon \quad \text{for } t-\tau<\delta \quad (t\ge \tau>\theta;\ i=1,\ldots,m). \tag{9} \]
Applying the Cauchy–Schwarz inequality to the last two terms of the right-hand side of (8), under conditions (9), we obtain
\[ \|H_i(t,\tau)x_i\|^2>\frac12 x_i^2 \quad \text{for } t-\tau<\delta \quad (t\ge \tau>\theta;\ i=1,\ldots,m). \]
Hence, taking into account expression (6), we obtain the required inequalities
\[ V_i(t,x_i)>\int_t^{t+\delta}\|H_i(s,t)x_i\|^2\,ds>\frac{\delta}{2}x_i^2(t)>0,\quad x_i\ne \vec 0;\quad i=1,\ldots,m), \]
which prove the assertion of Lemma 2.
Lemma 3. The function \(V_i[t,x_i(t)]\) from (6) decreases monotonically for \(t\ge t_0>0\), if \(x_i(t)\) is a solution of system (3) starting at the point \(x_i(t_0)\ne \vec 0\).
Proof. The derivative by virtue of system (3) is
\[ \dot V_i[t,x_i(t)]\equiv -x_i^2(t)+2\int_t^{+\infty}\left\{H_i(\tau,t)x_i(t)\right\}\cdot \left\{\frac{\partial H_i(\tau,t)}{\partial t}x_i(t)+ H_i(\tau,t)\dot x_i(t)\right\}\,d\tau \]
\[ \equiv -x_i^2(t)<0\quad \text{for } t\ge t_0>0 \text{ and } x_i(t_0)\ne \vec 0 \]
\[ (i=1,\ldots,m), \]
since the well-known identities hold
\[ \frac{\partial H_i(\tau,t)}{\partial t}\equiv -H_i(\tau,t)A_i(t) \quad\text{and}\quad \dot x_i(t)=A_i(t)x_i(t), \]
and \(x_i(t)\) cannot become the zero vector in a finite interval of time \(t\).
Theorem 1. Under conditions (4), (5), each system (3) \((i=1,\ldots,m)\) has a uniformly asymptotically stable (in the sense of A. M. Lyapunov) zero-solution.
The proof follows from the three lemmas given above and the known theorem of K. P. Persidskii [3].
Corollary. On the set \(M\) of all defined continuous and bounded for \(t>0\) square matrices \(A_i(t)\) of the \(r_i\)-th order \((i=1,\ldots,m;\ r_1+\cdots+r_m=n)\), conditions (4), (5) are equivalent in essence to the numbers of L. V. Khokhlova [4]
\[ K_i\equiv\left\{\operatorname*{borne\,sup}_{t>\tau>0}\int_\tau^t\|H_i(t,s)\|\,ds\right\}^{-1}>0 \quad (i=1,\ldots,m). \tag{10} \]
In Theorem 1 of the cited work it is proved that (10) is a necessary and sufficient condition for each system (3), on the above-mentioned set \(M\) of matrices \(A_i(t)\), to have a uniformly asymptotically stable zero-solution.
Consequently, it is enough to prove that conditions (4), (5) follow from inequalities (10).
For this it is sufficient to use the fact, known from the same work [4], that under conditions (10) there exist positive numbers \(\alpha_i\) and \(\beta_i\) such that
\[ \|H_i(t,\tau)\|<\alpha_i \quad\text{and}\quad \int_\tau^t\|H_i(s,\tau)\|\,ds<\beta_i, \quad \text{for } t>\tau>0\quad (i=1,\ldots,m). \tag{11} \]
Therefore, under conditions (10) one may put in (4)
\[ C_i=\alpha_i\beta_i\quad (i=1,\ldots,m). \]
In the same case condition (5) is also satisfied, since
\[ \|R_i(t,\tau)\|< \max_{t\geq s\geq \tau}\|A_i(s)\|\int_\tau^t \|H_i(s,\tau)\|\,ds < A_i\alpha_i(t-\tau)\to 0 \]
as \(t-\tau\to 0\) uniformly with respect to \(t>\tau>\theta\) (here
\(A_i\equiv \operatorname{borne}\sup_{s>\theta}|A_i(s)|;\ i=1,\ldots,m\)).
Remark 1. Condition (4) does not in general follow from condition (5), as is shown by the following example.
If \(A(t)=-1/2t\) and \(\theta=0\), then \(H(t,\tau)=+\sqrt{\tau/t}\) and
\[ \int_\tau^t A(s)H(s,\tau)\,ds=(t-\tau)/\{1+H(\tau,t)\}\to 0 \]
as \(t-\tau\to 0\) uniformly with respect to \(t\geq \tau>0\); however, in the special case when \(t=e\tau\) (const. \(e=2.7182\ldots\), the base of natural logarithms),
\[ \int_\tau^t H^2(s,\tau)\,ds=\tau\to +\infty . \]
Remark 2. If the matrix \(A(t)\equiv \{a_{ij}(t)\}\) \((i,j=1,\ldots,n)\) from (1) is triangular (i.e., \(a_{ij}(t)\equiv 0\) for \(t>\theta\) and \(i<j\), in particular, for \(m=n\)), then conditions (4), (5) are satisfied a fortiori if there exist positive numbers \(a_{ij}\) and \(G_i\) \((i,j=1,\ldots,n,\ i\geq j)\) such that \(|a_{ij}(t)|\leq a_{ij}\) and
\[ \int_\tau^t \left\{\exp\int_s^t a_{ii}(\sigma)\,d\sigma\right\}\,ds\leq G_i \]
for \(t\geq \tau>0\) \((i,j=1,\ldots,n;\ i\geq j)\).
Let \(p_1,\ldots,p_m\) be arbitrary positive numbers, and let \(E\equiv \{e_{ij}\}\) be a constant square symmetric matrix of order \(m\) with elements
\[ e_{ij}= \begin{cases} (1-2c_i b_{ii})p_i, & \text{for } i=j,\\ -(p_i c_i b_{ij}+p_j c_j b_{ji}), & \text{for } i\ne j, \end{cases} \tag{12} \]
where \(b_{ij}\) and \(c_i\) are the constants from (2) and (4), respectively \((i,j=1,\ldots,m)\).
Theorem 2. If there exist positive numbers \(p_1,\ldots,p_m\) such that the matrix \(E\) with elements (12) is positive definite, then, under conditions (2), (4), and (5), the zero-solution of system (1) is uniformly asymptotically stable in the large.1
Proof. We shall prove that, under the assumptions of Theorem 2, the bundle
\[ V(t,x)\equiv p_1V_1(t,x_1)+\cdots+p_mV_m(t,x_m) \]
is an A. M. Lyapunov function for system (1) [6]. Obviously, for this it is enough to prove the negative definiteness of the derivative, along system (1),
\[ \dot V(t,x)\equiv \sum_{i=1}^m p_i\left\{-x_i^2+2\int_t^{+\infty} [H_i(\tau,t)x_i,\ H_i(\tau,t)f_i(t,x)]\,d\tau\right\}. \]
Applying the Cauchy–Schwarz inequality to the right-hand side of the last expression, by virtue of (2), (4), and the positive definiteness of the matrix \(E\) (see [5], p. 22), we obtain
\[ \dot V(t,x)\leq -\sum_{i=1}^m p_i\left\{-x_i^2+2c_i|x_i|\,\|f_i(t,x)\|\right\} \leq -\sum_{i,j=1}^m e_{ij}\|x_i\|\,\|x_j\|<0, \]
provided only that \(x(t_0)\ne \vec 0\), which was required to be proved.
Example. Consider the system of ordinary differential equations
\[ \left. \begin{aligned} \ddot{x}+a_1(t)\dot{x}+a_0(t)x&=\varphi(t,x,y),\\ \dot{y}+b_0(t)y&=\psi(t,x,y) \end{aligned} \right\}, \tag{13} \]
which describes a process of automatic control similar to that given in Example 2 of [7].
The functions \(a_0(t)\), \(a_1(t)\), and \(b_0(t)\), defined, continuous, and bounded for \(t>\theta\), satisfy the following conditions.
There exist positive numbers \(\alpha,\beta,\alpha_2\), and \(\beta_2\) such that
\[ a_1^2(t)\geq \alpha^2+4a_0(t),\qquad \beta a_1(t)\leq \beta^2+a_0(t)\quad \text{for } t>\theta \tag{14} \]
and
\[ H_2(t,\tau)\equiv \exp\left\{-\int_\tau^t b_0(s)\,ds\right\}<\alpha_2, \qquad \int_\tau^t H_2(t,s)\,ds<\beta_2 \quad \text{for } t\geq \tau>\theta . \tag{15} \]
The functions \(\varphi(t,x,y)\) and \(\psi(t,x,y)\), defined and continuous for \(t>\theta\) and all real \(x\) and \(y\), satisfy the conditions for uniqueness and continuability of solutions of system (13) and the growth restrictions on the channel connections (2):
\[ \begin{aligned} |\varphi(t,x,y)|&\leq b_{11}|x|+b_{12}|y|,\\ |\psi(t,x,y)|&\leq b_{21}|x|+b_{22}|y| \end{aligned} \qquad \text{for } t>\theta, \tag{16} \]
where additional restrictions will subsequently be imposed on the constants \(b_{ij}\geq 0\) \((i,j=1,2)\), ensuring uniform asymptotic stability in the large of the zero solution of system (13).
The homeomorphism \(\{x_1\equiv x,\ x_2\equiv \dot{x},\ x_3\equiv y\}\) brings system (13) to the general form (1), where the matrices are
\[ A_1(t)\equiv \begin{pmatrix} 0 & 1\\ -a_0(t) & -a_1(t) \end{pmatrix} \quad \text{and} \quad A_2(t)\equiv \{-b_0(t)\}\quad (m=2), \]
and the functions are
\[ f_1(t,x)\equiv \{0,\varphi(t,x_1,x_3)\} \quad \text{and} \quad f_2(t,x)\equiv \{\psi(t,x_1,x_3)\}. \]
Under conditions (14) there exist numbers \(\rho>\delta>\gamma\) such that \(\delta-\gamma=\alpha\) and \(\gamma\geq \beta\), and, in addition, the equation
\[ \lambda^2+a_1(t)\lambda+a_0(t)=0, \]
defined for all \(t>\theta\), has roots \(\lambda_1(t)\) and \(\lambda_2(t)\) satisfying the inequalities
\[ -\rho\leq \lambda_1(t)\leq -\delta<-\gamma\leq \lambda_2(t)\leq -\beta \quad \text{for } t>\theta . \tag{17} \]
K. Olech [8] proved that, under conditions (17), the homogeneous equation
\[ \ddot{z}+a_1(t)\dot{z}+a_0(t)z=0 \]
has two linearly independent solutions
\[ \omega_i=\exp\left\{\int_\tau^t \mu_i(s)\,ds\right\} \quad \text{for } t\geq \tau>\theta \quad (i=1,2), \tag{18} \]
where certain functions \(\mu_1(t)\) and \(\mu_2(t)\), continuous for \(t>\theta\), satisfy the same inequalities
\[ -\rho\leq \mu_1(t)\leq -\delta<-\gamma\leq \mu_2(t)\leq -\beta . \tag{19} \]
Olech’s result (18) makes it possible to determine directly the Cauchy–Green matrix
\[ H_1(t;\tau)\equiv \frac{1}{\mu(\tau)} \left\{ \begin{array}{cc} \mu_2(\tau)\omega_1-\mu_1(\tau)\omega_2, & \omega_2-\omega_1 \\[2mm] \mu_1(t)\mu_2(\tau)\omega_1-\mu_2(t)\mu_1(\tau)\omega_2, & \mu_2(t)\omega_2-\mu_1(t)\omega_1 \end{array} \right\} \tag{20} \]
for \(t\geq \tau>\theta\) of the system \(x'=A_1(t)x'\), where \(\mu(\tau)\equiv \mu_2(\tau)-\mu_1(\tau)\), and the vector \(x'\equiv \{x_1(t),x_2(t)\}\).
From expression (20) and inequalities (19), the values \(\alpha_1=2l/\alpha\) and \(\beta_1=2l/\alpha\beta\) from (11) are easily computed (where \(l=\max(1,\rho,\rho\gamma,\rho\beta/\delta)\)), and hence also the number \(C_1=\alpha_1\beta_1=4l^2/\alpha^2\beta\) from (4) (corresponding to the matrix (20)).
In our case the matrix \(E\equiv\{e_{ij}\}\) with elements \(e_{ij}\) \((i,j=1,2)\) from (12) takes the form
\[ E\equiv \left\{ \begin{array}{cc} (1-2c_1b_{11})p_1, & -(p_1c_1b_{12}+p_2c_2b_{21}) \\[1mm] -(p_1c_1b_{12}+p_2c_2b_{21}), & (1-2c_2b_{22})p_2 \end{array} \right\}, \]
where \(c_2=\alpha_2\beta_2\) from (15).
The necessary condition for positive definiteness of the matrix \(E\) is determined by the well-known Sylvester criterion:
\[ 1-2c_1b_{11}>0,\qquad \det E>0. \tag{21} \]
In particular, conditions (21) are satisfied when \(b_{11}=b_{22}=0\) and \(p_1=p_2=1/2\), provided only that \(c_1b_{12}+c_2b_{21}<1\).
In our notation the last inequality takes the form
\[ (4l^2/\alpha^2\beta)b_{12}+\alpha_2\beta_2b_{21}<1; \]
it determines a sufficient condition for uniform asymptotic stability in the large of the zero solution of system (13) under conditions (14)—(16), where \(b_{11}=b_{22}=0\) (we note that \(b_{12}\) characterizes the maximum effectiveness of the control element of the nonautonomous controlled object, and \(b_{21}\) the maximum transfer ratio with respect to the displacement of the regulator with time adaptation).
References
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Received by the editors
16 December 1964
Peoples’ Friendship University
named after Patrice Lumumba
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On the last term see [5], p. 7. ↩