Full Text
Brief Communications
UDC 517.917
On the Stability of Solutions of a Differential Equation with a Discontinuous Characteristic
R. M. Eidlinov
1. Consider the piecewise-linear differential equation
\[ x^{(n)}+a_n x^{(n-1)}+\ldots+a_1x=-Kx\operatorname{sign}x\sum_{i=1}^{n}c_i x^{(i-1)}, \]
which is equivalent to the system
\[ \frac{dx_i}{dt}=x_{i+1}\quad (i=1,2,\ldots,n-1), \]
\[ \frac{dx_n}{dt}=-\sum_{i=1}^{n}a_i x_i-Kx_1\operatorname{sign}x_1\sum_{i=1}^{n}c_i x_i. \tag{1} \]
Here \(a_i,\ c_i,\ K\) are constant parameters, with \(K>0\); \(n\) is the order of the system. The stability of solutions of equations of this kind was considered in [1, 2, 3]. In the phase space of the variables \(\{x_1,\ldots,x_n\}\), the switching hyperplanes
\[ x_1=0\quad\text{and}\quad \sum_{i=1}^{n}c_i x_i=0 \]
(the hyperplane \(S\)) are distinguished. In the case when the conditions for stable sliding on the hyperplane \(S\) [1, 2] are satisfied, the problem of overall stability of the solutions of system (1) reduces to the problem of the representative point reaching the hyperplane \(S\) for arbitrary initial conditions. In [3] this problem was solved for large values of \(K\). In the present work, under the assumption that the roots of the equation
\[ \sum_{i=1}^{n}c_i\lambda^{i-1}=0 \]
are real, necessary and sufficient conditions for reaching are obtained without restrictions on the magnitude of the parameter \(K\).
2. The phase space \(\{x_1,\ldots,x_n\}\) is divided by the switching hyperplanes into four regions:
\[ x_1>0;\qquad \sum_{i=1}^{n}c_i x_i>0 \quad \text{region } G_1, \]
\[ x_1<0;\qquad \sum_{i=1}^{n}c_i x_i>0 \quad \text{” } G_2, \]
\[ x_1<0;\qquad \sum_{i=1}^{n}c_i x_i<0 \quad \text{” } G_3, \]
\[ x_1>0;\qquad \sum_{i=1}^{n}c_i x_i<0 \quad \text{” } G_4. \]
In each of these regions, one of two systems of linear differential equations with constant coefficients is valid.
Let \(A_1\) be the matrix of coefficients of the system of differential equations (1) when
\[ \operatorname{sign} x_1 \sum_{i=1}^{n} c_i x_i > 0 \]
and \(A_2\) when
\[ \operatorname{sign} x_1 \sum_{i=1}^{n} c_i x_i < 0 . \]
We write these systems in vector form
\[ \frac{dx}{dt}=A_1x, \tag{2} \]
\[ \frac{dx}{dt}=A_2x. \tag{3} \]
With the notation introduced, we formulate the following theorem on the arrival of the representative point at the hyperplane \(S\).
If the roots of the characteristic equation
\[ \sum_{i=1}^{n} c_i \lambda^{i-1}=0 \]
are real and negative, then for the representative point to arrive at the hyperplane \(S\) it is necessary and sufficient that
\[ \mu_k<0 \quad (k=1,2,\ldots,m;\quad m\leq n), \]
where \(\mu_k\) are the real roots of the characteristic equation
\[ |\mu E-A_1|=0. \]
Moreover, the number of sign changes of the component \(x_1(t)\) before arrival at the hyperplane \(S\) does not exceed \(n\).
The necessity of the absence among the roots of the equation \(|\mu E-A_1|=0\) of real positive roots was shown in [1]. In a completely analogous way this can be done for zero roots.
We prove sufficiency. Consider the motion of the representative point \(M(t)\) in phase space. Let \(M(0)\subset G_1\). Then the system of differential equations (2) acts. We study the solution \(x_1(t)\) of this system. If the limit of the function \(x_1(t)\) exists, then, according to the conditions of the theorem, we have
\[ \lim_{t\to\infty} x_1(t)=0. \]
From the analysis of the general solution of the system of differential equations (2) it is not difficult to see that in this case
\[ \lim_{t\to\infty} \frac{d^i x_1(t)}{dt^i} = \lim_{t\to\infty} x_{i+1}(t) =0 \quad (i=1,2,\ldots,n-1). \]
As a result, for the case under consideration we have: the representative point either leaves the region \(G_1\) in finite time, or asymptotically approaches the origin of the phase space inside the region \(G_1\).
If the function \(x_1(t)\) has no definite limit, then there exist partial limits, among which one can distinguish the greatest and the least [4]. In the present case, for the function \(x_1(t)\) as the solution of the system of differential equations (2), restricted by the condition of the theorem, we have
\[ \overline{\lim}_{t\to\infty} x_1(t)=A,\quad 0<A<+\infty, \]
\[ \underline{\lim}_{t\to\infty} x_1(t)=B,\quad 0>B>-\infty . \]
From this it follows immediately that in a finite time \(t_1\) the representative point leaves the region \(G_1\), either through the hyperplane \(S\) (and then the theorem is proved), or through the hyperplane \(x_1=0\), entering the region \(G_2\).
Let \(M(t_1)\subset G_2\). Then the system of differential equations (3) acts. We analyze the solution \(x_1(t)\) of this system. Suppose that the function \(x_1(t)\) has a limit. Consider 3 cases:
1)
\[ \lim_{t\to\infty} x_1(t)=0; \]
2)
\[ \lim_{t\to\infty} x_1(t)=\pm\infty; \]
3)
\[
\lim_{t\to\infty} x_1(t)=A,\qquad A=\text{const}.
\]
1) Analogously to what was described above, in this case we have
\[ \lim_{t\to\infty}\frac{d^i x_1(t)}{dt^i} = \lim_{t\to\infty}x_{i+1}(t)=0 \qquad (i=1,2,\ldots,n-1), \]
and the representative point either leaves the domain \(G_2\) in finite time, or asymptotically tends to the origin of the phase space while remaining in the domain \(G_2\).
2) From an analysis of the general solution of the system of differential equations (3) and of the discontinuity property of the function \(x_1(t)\) and its derivatives, we obtain the result
\[ \operatorname{sign}\lim_{t\to\infty}\frac{d^i x_1(t)}{dt^i} = \operatorname{sign}\lim_{t\to\infty}x_{i+1}(t) = \operatorname{sign}\lim_{t\to\infty}x_1(t), \]
\[ (i=1,2,\ldots,k-1;\ k\leq n), \]
\[ \lim_{t\to\infty}x_{i+1}(t)=0 \qquad (i=k,k+1,\ldots,n). \]
If \(\lim_{t\to\infty}x_1(t)=+\infty\), then the function \(x_1(t)\) changes sign in finite time. If, however,
\[ \lim_{t\to\infty}x_1(t)=-\infty, \]
then, taking into account that \(c_i>0\), we obtain
\[ \lim_{t\to\infty}\sum_{i=1}^{n} c_i x_i(t)=-\infty \]
and, consequently, the representative point reaches the hyperplane \(S\) in finite time.
3) Investigating the general solution of the system of differential equations (3), it is not difficult to obtain for this case
\[ \lim_{t\to\infty}x_{i+1}(t)=0 \qquad (i=1,2,\ldots,n-1). \]
Taking \(A>0\) and \(A<0\), and reasoning analogously to how this was done in item 2, it is not difficult to obtain the result: the representative point leaves the domain \(G_2\) in finite time.
Finally, suppose that the function \(x_1(t)\) has no definite limit. Let us again consider the partial limits of the function \(x_1(t)\), among which we distinguish the greatest and the least. If
\[ \operatorname{sign}\overline{\lim}_{t\to\infty}x_1(t) = -\operatorname{sign}\underline{\lim}_{t\to\infty}x_1(t), \]
then we again obtain the previous conclusion about the behavior of the representative point in the domain \(G_2\). However, the greatest and least limits of the function \(x_1(t)\) may have the same signs in the case when the characteristic equation \(|\lambda E-A_{12}|=0\), besides roots with negative real part, has one zero root and several purely imaginary roots. It can be shown that in this case the functions \(x_{i+1}(t)\) \((i=1,2,\ldots,n-1)\) and
\[ \sum_{i=2}^{n} c_i x_i(t) \]
are sign-alternating for sufficiently large \(t\) [5].
Suppose \(x_1(t)<0\) for \(t_1<t\leq\infty\). Then \(c_1x_1(t)<0\) on the same half-interval of \(t\). For sufficiently large \(t\), the function
\[ c_1x_1(t)+\sum_{i=2}^{n}c_i x_i(t) \]
will either be sign-alternating (and then the representative point reaches the hyperplane \(S\)), or will have constant sign, with
\[ \operatorname{sign}\sum_{i=1}^{n}c_i x_i(t)=\operatorname{sign}x_1(t)<0, \]
i.e., in the latter case as well the representative point reaches the hyperplane \(S\). Finally, if the partial limits of the function \(x_1(t)\) are positive, then the representative point leaves the domain \(G_2\) in finite time through the hyperplane \(x_1=0\).
From the analysis of the motion of the representative point in the half-space
\[ \sum_{i=1}^{n}c_i x_i>0 \]
it follows that it remains to prove the validity of the theorem only in the case of multiple intersections by the representative point of the hyperplane \(x_1=0\). To prove the validity of the theorem in this case as well, let us consider the function \(x_1(t)\), taking into account that this function is sign-alternating, and that the time intervals on which the sign of the function remains unchanged are finite.
It follows from system (1) that on the hyperplane \(x_1=0\) the phase-velocity vector has no discontinuity; hence the function \(x_1(t)\) and \(n\) of its derivatives \(\bigl(x_{i+1}(t)\) for \(i=1,2,\ldots,n-1\bigr)\) are continuous. We now make use of one variant of Rolle’s generalized theorem [6]. According to the conditions of this theorem, take a finite interval of time on which the function \(x_1(t)\) vanishes at \(n\) points \((0<t<t_2)\). Then, under the condition that the roots of the polynomial
\[
\sum_{i=1}^{n} c_i \lambda^{i-1}=0
\]
are real, within the interval so chosen the following equality holds:
\[
\sum_{i=1}^{n} c_i \frac{d^{\,i-1} x_1(t_3)}{dt^{\,i-1}}
=
\sum_{i=1}^{n} c_i x_i(t_3)=0,
\qquad 0<t_3<t_2,
\]
i.e., the representative point falls onto the hyperplane \(S\), having previously crossed the hyperplane \(x_1=0\) at most \(n\) times.
Analogous arguments may be carried out for the half-space
\[
\sum_{i=1}^{n} c_i x_i<0.
\]
Thus the theorem is proved.
The author thanks E. A. Barbashin for guidance in carrying out this work.
References
-
S. V. Emel’yanov, V. A. Taran. Izv. AN SSSR, OTN, Energetika i avtomatika, No. 3, 1962.
-
E. A. Barbashin, V. A. Tabueva. Avtomatika i telemekhanika, 23, No. 10, 1962.
-
S. V. Emel’yanov, V. I. Utkin. Izv. AN SSSR, OTN, Tekhnicheskaya kibernetika, No. 2, 1964.
-
G. M. Fikhtengol’ts. A Course of Differential and Integral Calculus, Vol. 1. Gostekhizdat, 1948, p. 162.
-
N. I. Akhiezer. Lectures on Approximation Theory. Moscow, Gostekhizdat, 1947, pp. 37–38.
-
G. Pólya, G. Szegő. Problems and Theorems in Analysis, Part II, 2nd ed. Moscow, State Publishing House of Technical-Theoretical Literature, 1956, Section V, Chapter 1, § 8.
-
E. I. Gerashchenko. Izv. AN SSSR, OTN, Tekhnicheskaya kibernetika, No. 4, 1963.
Received by the editors
December 2, 1965
Ural Polytechnic Institute
named after S. M. Kirov