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ON THE STABILITY OF THE ZERO SOLUTION OF A CLASS OF DIFFERENTIAL EQUATIONS OF ORDER \(n\)
B. N. SKACHKOV
Let us consider the question of the stability of the zero solution of a differential equation of the following form:
\[ x^{(n)}+p_1x^{(n-1)}+\cdots+p_{n-1}x'+p_nx=0, \tag{1} \]
where \(p_1,\ldots,p_n\) are continuous real functions of the arguments \(t,x,x',\ldots,x^{(n-1)}\), satisfying, for certain constants \(l_i^-\), \(l_i^+\), the conditions
\[ l_i^- \leq p_i \leq l_i^+ \qquad (i=1,2,\ldots,n). \tag{2} \]
In the article, sufficient conditions are given for the stability of the zero solution of equation (1), which become necessary and sufficient for equation (1) when the \(p_i\) are constant. An analogous problem for \(n=2,3,4\) and \(p_1,p_2,p_3=\mathrm{const}\) was studied earlier in [1—3]. In the present article, the results obtained for the general case of equation (1) in [4] are improved.
For the further investigation, let us replace equation (1) by an equivalent system of differential equations, putting
\[ x^{(s)}=y_{s+1}\quad (s=0,1,\ldots,n-1). \tag{3} \]
Then
\[ \left\{ \begin{aligned} \dot y_s&=y_{s+1}\quad (s=1,2,\ldots,n-1),\\ \dot y_n&=-\sum_{k=1}^{n}p_{n-k+1}y_k . \end{aligned} \right. \tag{4} \]
We transform system (4) to another form. For this purpose let us specify the numbers
\[ \varkappa_1,\varkappa_2,\ldots,\varkappa_n, \tag{5} \]
the first \(2l\) of which are complex, pairwise conjugate, and the remaining ones real,
\[ \varkappa_{2j-1}=\lambda_j+i\mu_j,\quad \varkappa_{2j}=\lambda_j-i\mu_j,\quad \mu_j>0\quad (j=1,2,\ldots,l), \]
\[ \varkappa_k=\lambda_k\quad (k=2l+1,\ldots,n,\quad 0\leq 2l\leq n), \tag{6} \]
and put
\[ y_{s+1}=\sum_{k=1}^{n}\alpha_{sk}z_k\quad (s=0,1,\ldots,n-1), \tag{7} \]
where
\[ \alpha_{sk}= \sum_{\substack{i_1+i_2+\cdots+i_k=s-k+1\\ 0\leq i_1,i_2,\ldots,i_k\leq s-k+1}} \varkappa_1^{i_1}\varkappa_2^{i_2}\cdots\varkappa_k^{i_k} \quad (k\leq s+1),\qquad \alpha_{sk}=0\quad (k\geq s+2). \tag{8} \]
\[ (s=0,1,\ldots,n+1,\quad k=1,2,\ldots,n). \]
From formula (7) it follows that
\[ z_{s+1}=\sum_{k=1}^{n}\alpha_{sk}^{-1}y_k \quad (s=0,1,\ldots,n-1), \tag{9} \]
where
\[ \alpha_{sk}^{-1}=(-1)^{s-k+1} \sum_{i_1,i_2,\ldots,i_{s-k+1}}\chi_{i_1}\chi_{i_2}\cdots \chi_{i_{s-k+1}} \quad (k\le s),\quad (s=0,1,\ldots,n), \]
\[ \alpha_{s,s+1}^{-1}=1,\quad \alpha_{sk}^{-1}=0\ (k>s+1)\quad (k=1,2,\ldots,n). \tag{10} \]
The index \(i_1,i_2,\ldots,i_{s-k+1}\) denotes all possible combinations of the numbers \(1,2,\ldots,s\) taken \(s-k+1\) at a time.
The relations
\[ \alpha_{s+1,k}=\alpha_{s,k-1}+\chi_k\alpha_{sk}\quad (\alpha_{s0}=0), \]
\[ \alpha_{s+1,k}^{-1}=\alpha_{s,k-1}^{-1}-\chi_{s+1}\alpha_{sk}^{-1}\quad (\alpha_{s0}^{-1}=0) \tag{11} \]
are evident.
For the new variables we have
\[ \dot z_s=\chi_s z_s+z_{s+1}\quad (s=1,2,\ldots,n-1), \]
\[ \dot z_n=-\sum_{k=1}^{n}P_k z_k+\chi_n z_n, \tag{12} \]
where
\[ P_k=\sum_{s=0}^{n}p_s\alpha_{n-s,k}\quad (p_0=1),\quad (k=1,2,\ldots,n). \tag{13} \]
From the properties (6) of the numbers (5) it follows that in formulas (8) and (10) the numbers
\[ \alpha_{s,2j},\ \alpha_{sk}\quad (s=0,1,\ldots,n;\ j=1,2,\ldots,l;\ k=2l+1,\ldots,n) \]
and
\[ \alpha_{2j,k}^{-1},\ \alpha_{sk}^{-1}\quad (k=1,2,\ldots,n;\ j=0,1,\ldots,l-1;\ s=2l+1,\ldots,n) \tag{14} \]
are real. Hence, also from formulas (11), it follows that
\[ \operatorname{Im}\alpha_{s,2j-1}=\mu_j\alpha_{s,2j},\quad \operatorname{Im}\alpha_{2j-1,k}^{-1}=-\mu_j\alpha_{2j-2,k}^{-1} \]
\[ (s=0,1,\ldots,n;\ k=1,2,\ldots,n;\ j=1,2,\ldots,l). \]
From the last formulas and the reality of the numbers (14) and of the variables \(y_k\), it follows that in expression (9) \(z_{2j-1}, z_s\) \((j=1,2,\ldots,l;\ s=2l+1,\ldots,n)\) are real, \(\operatorname{Im} z_{2j}=-\mu_j z_{2j-1}\) \((j=1,2,\ldots,l)\), while in expression (13)
\[ P_{2j}=-\frac{1}{\mu_j}\operatorname{Im}P_{2j-1}\quad (j=1,2,\ldots,l). \]
Therefore, putting
\[ x_{2j-1}=-\mu_j z_{2j-1},\quad x_{2j}=i\mu_j z_{2j-1}+z_{2j}\quad (j=1,2,\ldots,l), \]
\[ x_s=z_s\quad (s=2l+1,\ldots,n), \tag{15} \]
we arrive at the real system of equations
\[ 2l<n \begin{cases} \dot{x}_{2j-1}=\lambda_j x_{2j-1}-\mu_j x_{2j},\\[2mm] \dot{x}_{2j}=\mu_j x_{2j-1}+\lambda_j x_{2j}-\dfrac{1}{\mu_{j+1}}x_{2j+1} \quad (j=1,2,\ldots,l)\;(\mu_{l+1}=-1),\\[2mm] \dot{x}_s=\lambda_s x_s+x_{s+1}\quad (s=2l+1,\ldots,n),\\[2mm] \dot{x}_n=\displaystyle\sum_{j=1}^{l}\dfrac{1}{\mu_j} \bigl(\operatorname{Re}P_{2j-1}x_{2j-1}-\operatorname{Im}P_{2j-1}x_{2j}\bigr) -\displaystyle\sum_{k=2l+1}^{n}P_kx_k+\lambda_nx_n; \end{cases} \tag{16} \]
\[ 2l=n \begin{cases} \dot{x}_{2j-1}=\lambda_j x_{2j-1}-\mu_j x_{2j},\\[2mm] \dot{x}_{2j}=\mu_j x_{2j-1}+\lambda_j x_{2j}-\dfrac{1}{\mu_{j+1}}x_{2j+1} \quad (j=1,2,\ldots,l-1),\\[2mm] \dot{x}_{n-1}=\lambda_l x_{n-1}-\mu_l x_n,\\[2mm] \dot{x}_n=\displaystyle\sum_{j=1}^{l}\dfrac{1}{\mu_j} \bigl(\operatorname{Re}P_{2j-1}x_{2j-1}-\operatorname{Im}P_{2j-1}x_{2j}\bigr) +\mu_l x_{n-1}+\lambda_l x_n. \end{cases} \]
The transformations (3), (7), (15) are such that the problems of stability of the trivial solutions of equation (1) and of the systems (4), (12), (16) are equivalent.
We perform one more change of variables, putting in system (16)
\[ x_{2j-1}=r_j\cos\varphi_j,\qquad x_{2j}=r_j\sin\varphi_j \quad (j=1,2,\ldots,l). \tag{17} \]
For the new variables we obtain
\[ 2l<n \begin{cases} \dot{r}_j=\lambda_j r_j-\dfrac{1}{\mu_{j+1}}r_{j+1}\cos\varphi_{j+1}\sin\varphi_j \quad (j=1,2,\ldots,l-1),\\[2mm] \dot{r}_l=\lambda_l r_l+x_{2l+1}\sin\varphi_l,\\[2mm] \dot{x}_s=\lambda_s x_s+x_{s+1}\quad (s=2l+1,\ldots,n-1),\\[2mm] \dot{x}_n=\displaystyle\sum_{j=1}^{n}\dfrac{1}{\mu_j}r_j \bigl(\operatorname{Re}P_{2j-1}\cos\varphi_j-\operatorname{Im}P_{2j-1}\sin\varphi_j\bigr) -\displaystyle\sum_{k=2l+1}^{n}P_kx_k+\lambda_nx_n; \end{cases} \]
\[ 2l=n \begin{cases} \dot{r}_j=\lambda_j r_j-\dfrac{1}{\mu_{j+1}}r_{j+1}\cos\varphi_{j+1}\sin\varphi_j \quad (j=1,2,\ldots,l-1),\\[2mm] \dot{r}_l=\displaystyle\sum_{j=1}^{l}\dfrac{1}{\mu_j}r_j \bigl(\operatorname{Re}P_{2j-1}\cos\varphi_j-\operatorname{Im}P_{2j-1}\sin\varphi_j\bigr) +\lambda_l r_l. \end{cases} \tag{18} \]
The transformation (17) is such that the problems of stability of the solutions
\(x_1=x_2=\cdots=x_n=0\) of system (16) and
\(r_1=r_2=\cdots=r_l=x_{2l+1}=\cdots=x_n=0\) of system (18) are equivalent. Therefore we shall not write down the equations for \(\varphi_j\) \((j=1,2,\ldots,l)\), remembering, however, that \(\varphi_j\) in equations (18) are certain functions of \(t\).
Introduce the notation:
\[ \overline{P}_s=\max_{t,x,x',\ldots,x^{(n-1)}}|P_s|\quad (s=1,2,\ldots,n); \]
\[ Q = \begin{cases} -\displaystyle\min_{t,x,x',\ldots,x^{(n-1)}} P_n, & \text{for } 2l<n,\\[6pt] \displaystyle\max_{t,x,x',\ldots,x^{(n-1)}} \left[(\operatorname{Re} P_{2l-1}\cos\varphi_l-\operatorname{Im} P_{2l-1}\sin\varphi_l)\sin\varphi_l\right], & \text{for } 2l=n. \end{cases} \]
Theorem 1. The zero solution of system (18), and hence also of equation (1), is stable in the large if the numbers (5), satisfying the properties (6), can be chosen so that the inequalities
\[ \lambda_s<0\quad (s=1,2,\ldots,l,2l+1,\ldots,n), \]
\[ \sum_{j=1}^{l}(-1)^j \overline{P}_{2j-1}\gamma_{j-1} + \sum_{k=2l+1}^{n-1}(-1)^{k-l}\overline{P}_k\gamma_k + (-1)^{n-l}Q\gamma_{n-1} + \]
\[ +(-1)^{n-l}\gamma_n>0 \quad \text{for } 2l<n, \tag{19} \]
\[ \lambda_s<0\quad (s=1,2,\ldots,l), \]
\[ \sum_{j=1}^{l-1}(-1)^j\overline{P}_{2j-1}\gamma_{j-1} + (-1)^l Q\gamma_{l-1} + (-1)^l\gamma_l>0 \quad \text{for } 2l=n, \]
where
\[ \gamma_0=1,\qquad \gamma_j=\prod_{m=1}^{j}\lambda_m\mu_m \quad (j=1,2,\ldots,l), \]
\[ \gamma_{2l}=\prod_{m=1}^{l}\lambda_m\mu_m,\qquad \gamma_k=\prod_{m=1}^{l}\lambda_m\mu_m \prod_{m=2l+1}^{k}\lambda_m \quad (k=2l+1,\ldots,n). \]
Proof. According to Theorem I [5], system (18) will be stable in the large if, for the matrix \(A\) obtained from the matrix of coefficients of the right-hand sides of system (18) by replacing the diagonal elements by their maxima and the remaining elements by the maxima of their moduli, there is a vector \(a>0\) such that
\[ A'a<0, \tag{20} \]
where the prime denotes transposition.
Taking into account that
\[ \max\left|\operatorname{Re}P_{2j-1}\cos\varphi_j-\operatorname{Im}P_{2j-1}\sin\varphi_j\right| \leq \overline{P}_{2j-1} \quad (j=1,2,\ldots,l), \]
we obtain
\[ A= \left\| \begin{array}{cccccccccc} \lambda_1 & \dfrac{1}{\mu_2} & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & 0 \\ 0 & \lambda_2 & \dfrac{1}{\mu_3} & \cdot & \cdot & \cdot & \cdot & \cdot & 0 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \lambda_l & 1 & 0 & \cdot & 0 \\ \cdot & \cdot & \cdot & \cdot & 0 & \lambda_{2l+1} & 1 & \cdot & 0 \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \dfrac{1}{\mu_1}\overline{P}_1 & \dfrac{1}{\mu_2}\overline{P}_3 & \dfrac{1}{\mu_3}\overline{P}_5 & \cdot & \dfrac{1}{\mu_l}\overline{P}_{2l-1} & P_{2l+1} & P_{2l+2} & \cdot & Q+\lambda_n \end{array} \right\|. \tag{21} \]
For the case \(2l=n\), the changes in the matrix (21) are obvious.
According to Theorem II [6], inequality (20) has a solution if, for the successive principal minors of the matrix (21) \(A_i\) \((i=1,2,\ldots,n-l)\), the inequalities
\[ (-1)^i A_i > 0 \]
are satisfied.
The inequalities (19) are precisely these latter inequalities in expanded form. The theorem is proved.
The quantities \(Q\) and \(\overline{P}_s\) \((s=1,2,\ldots,n)\) are determined by means of the inequalities (2).
References
- Starzhinskii V. M. PMM, XVI, issue 3, 1952.
- Starzhinskii V. M. PMM, XVI, issue 4, 1952.
- Starzhinskii V. M. PMM, XIX, issue 4, 1955.
- Skachkov B. N. Vestnik LGU, No. 19, 1962.
- Skachkov B. N. Vestnik LGU, No. 19, 1960.
- Skachkov B. N. Vestnik LGU, No. 7, 1960.
Received by the editors
November 17, 1964
Leningrad Branch
of the V. A. Steklov
Institute of Mathematics