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On the Existence of an Interval of Stability of Motion According to the First Approximation
V. P. Rudakov
Let the unperturbed motion \(x \equiv 0\) and all possible perturbed motions \(x(t)\) be determined, respectively, by the systems
\[ \frac{dx}{dt}=P(t)x+X(t,x) \]
and
\[ \frac{dx}{dt}=P(t)x+X(t,x)+R(t,x). \tag{1} \]
It is assumed that in the domain
\[ T_1 \leq t \leq T_2,\qquad |x_i|\leq a \]
the matrix \(P=\|p_{ik}\|\) is a real and continuous function of time \(t\), while the vectors \(X=\{X_i\}\) and \(R=\{R_i\}\) are functions of time \(t\) and of the real vector \(x=\{x_i\}\) having the same properties \((i,k=1,\ldots,n)\). Moreover, \(X(t,0)=0\) and \(X(t,x)=O(x)\).
Definition (due to A. A. Lebedev [1]). The unperturbed motion is stable on the finite time interval \([t_0,t_1]\) \((T_1\leq t_0<t_1\leq T_2)\) if, for any sufficiently small number \(A>0\), there exists a number \(\eta(A)>0\) and a cycle \(V(t,x)=A\) such that on the given interval the following conditions are satisfied:
1) the diameter \(D(t)\) of the domain
\[ V(t,x)\leq A \tag{2} \]
does not exceed its initial value \(D(t_0)\);
2) for all initial values \(x(t_0)=x_0\) and continuously acting perturbations \(R\) satisfying the condition \(V(t_0,x_0)\leq A,\ |R_i|\leq \eta(A)\), the corresponding subsequent perturbations \(x(t)\) satisfy inequality (2).
In [2], sufficient conditions were indicated for the existence of an interval of stability according to the first approximation. They consist in the requirement that all roots of the characteristic equation \(\det(P(t_0)-\lambda E)=0\) have negative real part. Below these conditions are significantly extended.
Lemma 1. A function \(\Phi(t,x)\) satisfying the equation
\[ \frac{\partial \Phi}{\partial t} + \sum_{i=1}^{n} \left( \sum_{k=1}^{n} p_{ik}(t)x_k \right) \frac{\partial \Phi}{\partial x_i} =0 \tag{3} \]
and the initial condition
\[ \Phi(t_0,x)=\sum_{i=1}^{n} l_i x_i^2, \tag{4} \]
where \(l_i>0\), is a positive definite quadratic form with coefficients continuously differentiable on \([T_1,T_2]\).
To determine the coefficients of the form
\[ \Phi(t,x)=\sum_{i,k=1}^{n} b_{ik}(t)x_i x_k \quad (b_{ik}=b_{ki}) \tag{5} \]
we obtain the system
\[ b'_{ik}=-\sum_{j=1}^{n}\bigl(p_{ji}(t)b_{jk}+p_{jk}(t)b_{ji}\bigr) \quad (i,k=1,\ldots,n) \tag{6} \]
and the initial conditions \(b_{ii}(t_0)=l_i,\ b_{ik}(t_0)=0\ (i\ne k)\).
System (6) satisfies the condition of N. P. Erugin’s theorem ([3], Theorem 2.1) and, consequently, its solutions do not leave the class of coefficients of a sign-definite form, provided only that they belong to this class at the initial moment of time.
Let \(\gamma_m(t)\) be the smallest, at each moment of time, root of the equation \(\det(B-\gamma E)=0\), where \(B\) is the matrix of coefficients of the form (5).
Lemma 2. If \(p_{ss}(t_0)<0,\ l_s=1\) and \(l_i\gg 1\ (i\ne s)\), then there exists an interval \([t_0,t_0+\tau]\) \((\tau>0)\) on which the function \(\gamma_m(t)\) is continuously differentiable, and moreover \(\gamma'_m(t_0)>0\).
Lemma 3. If \(\gamma_m(t)\) is a positive continuously differentiable function, then for the existence of a sufficiently small number \(\varepsilon>0\) for which the inequality
\[ \exp \varepsilon (t_0-t) > \frac{\gamma_m(t_0)}{\gamma_m(t)} \tag{7} \]
is satisfied on some interval \([t_0,t_0+\tau]\), it is necessary and sufficient that \(\gamma'_m(t_0)>0\).
Theorem. If at least one of the elements of the main diagonal of the matrix \(P(t_0)\) is negative (for example, \(p_{ss}(t_0)\)), then there exists a nonzero interval \([t_0,t_0+\tau]\) on which the unperturbed motion is stable independently of the remaining elements of the matrix \(P(t_0)\), of the matrix \(P(t)-P(t_0)\), and of the vector \(X(t,x)\).
Proof. Consider the auxiliary function
\[ V(t,x)=\varphi^2(t)\Phi(t,x), \]
where
\[ \varphi(t)=\exp \frac{\varepsilon}{2}(t_0-t); \]
\(\varepsilon\) is a positive number for which inequality (7) is satisfied on the interval \([t_0,t_0+\tau]\).
Since the diameter of the region \(V\le A\) is equal to
\[ D(t)=D(t_0)\frac{\varphi(t_0)}{\varphi(t)} \sqrt{\frac{\gamma_m(t_0)}{\gamma_m(t)}}, \]
it follows from inequality (7) that \(D(t)\le D(t_0)\) for \(t\in[t_0,t_0+\tau]\).
The derivative \(dV/dt\), computed by virtue of system (1), taking into account that the form \(\Phi\) satisfies equation (3), has the form
\[ \frac{dV}{dt}=\frac{2\varphi'}{\varphi}V+ \sum_{i=1}^{n}(X_i+R_i)\frac{\partial V}{\partial x_i}. \]
On the surface \(V=A\), for any \(A\) the equality holds
\[ \left\{\frac{2\varphi'}{\varphi}V\right\}_{V=A}=-\varepsilon A. \]
By virtue of the relation of orders of smallness, for any sufficiently small number \(A\) the inequality
\[ \left\{\left|\sum_{i=1}^{n} X_i\frac{\partial V}{\partial x_i}\right|\right\}_{V=A}<\frac{\varepsilon A}{2}. \]
will be satisfied.
Since the derivatives \(\partial V/\partial x_i\) are bounded on the surface \(V=A\), for every \(A\) there exists a sufficiently small number \(\eta(A)\) such that, under the condition \(|R_i|<\eta(A)\), the inequality
\[ \left\{\left|\sum_{i=1}^{n} R_i\frac{\partial V}{\partial x_i}\right|\right\}_{V=A}<\frac{\varepsilon A}{2}. \]
will be satisfied.
It follows from what has been set forth that, for any sufficiently small numbers \(A\) and \(\eta(A)\), on the surface \(V=A\) the inequality
\[ \left\{\frac{dV}{dt}\right\}_{V=A}<0, \]
is valid; that is, all integral curves of system (1) that intersect the surface \(V=A\) intersect it from the outside inward.
Thus, on the interval \([t_0,t_0+\tau]\) all the conditions in the definition of stability are satisfied. The theorem is proved.
References
- Lebedev A. A. Proceedings of the Moscow Aviation Institute named after Ordzhonikidze, issue 50, 1955.
- Lebedev A. A. Proceedings of the Moscow Aviation Institute named after Ordzhonikidze, issue 112, 1959.
- Erugin N. P. Reports of the Academy of Sciences of the BSSR, 7, No. 9, 1963.
Received by the editors
September 5, 1964
Zhitomir Pedagogical Institute
named after I. Franko