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Qualitative Investigation of a Certain Second-Order Differential Equation in Control Theory
V. A. TABUEVA
A characteristic feature of systems with variable structure [1, 2] is their entry, beginning at a certain moment of time, into a sliding mode, which it is expedient to define as stable with respect to the origin of coordinates. In this case the representative point of the system of differential equations, from any point of the phase space, reaches a certain surface at some moment of time and, as time increases, does not leave it. The motion of the point along this surface occurs by virtue of the sliding equations, which, generally speaking, do not depend on the parameters of the controlled system; this in turn ensures robustness of the stability property of a system with variable structure with respect to changes in the parameters of the system within sufficiently broad limits.
A second-order system with variable structure can be described by the differential equation
\[ \ddot{x}+a\dot{x}+bx=-n\varphi(x), \tag{1} \]
where \(n=\operatorname{sign}x(Ax+\dot{x})\). Let the parameters of the system be such that \(a>0\), \(A>0\), and \(ab\) is of arbitrary sign. We assume the function \(\varphi(x)\) to be discontinuous for all \(x\ne 0\), piecewise differentiable, and satisfying, for all \(x\ne 0\), the inequality
\[ x\varphi(x)>0. \tag{2} \]
In particular, we assume that at \(x=0\) the function \(\varphi(x)\) may have a discontinuity of the first kind.
In the present paper the qualitative structure of the trajectories of equation (1) in the phase plane \(x,\dot{x}\) is studied under certain sufficiently general assumptions concerning the function \(\varphi(x)\); characteristic types of the region of attraction of the zero equilibrium state are revealed; criteria are established for the existence of a definite form of the region of attraction and of the corresponding pattern of arrangement of the trajectories of equation (1) in the phase plane.
We note that for a linear and a piecewise-linear function \(\varphi(x)\), equation (1) was considered earlier [3, 4, 5].
§ 1. Behavior of the System on the Switching Line \((R)\)
Equation (1) is equivalent to the system of differential equations
\[ \dot{x}=y,\quad \dot{y}=-ay-f(x), \tag{3} \]
where \(f(x)=bx+n\varphi(x)\), \(n=\operatorname{sign}x(Ax+y)\). For brevity, henceforth we shall
call system (3), for \(n=+1\), a \((+)\)-system, for \(n=-1\), a \((-)\)-system, and also put \(f_+(x)=bx+\varphi(x)\) and \(f_-=bx-\varphi(x)\).
The straight lines \(x=0\) and \(Ax+y=0\) are the switching lines of system (3), the lines of change of its structure. It is easy to see that the \(y\)-axis is “pierced” by its trajectories, i.e. the velocity vectors of the \((+)\)- and \((-)\)-systems at points of the \(y\)-axis are directed to the same side with respect to the axis itself.
The straight line \(Ax+y=0\), which henceforth we shall call the line \((R)\), depending on the properties of the function \(\varphi(x)\), may have both intervals of “piercing” and intervals of “sliding,” i.e. such intervals at whose points the velocity vectors of the \((+)\)- and \((-)\)-systems are directed into different half-planes with respect to the line \((R)\), and in such a way that the representative point of system (3) moves along these intervals toward the origin, by virtue of the differential equation [6] \(Ax+\dot{x}=0\).
Theorem 1. If for system (3) the inequality
\[ \frac{\varphi(x)}{x}>\left|A(a-A)-b\right| \tag{4} \]
is satisfied on the interval \(\varepsilon \leq x \leq \eta\), where \(\varepsilon\) and \(\eta\) are some numbers, then the corresponding segment of the line \((R)\) is a sliding segment of the system.
Proof. Let \(R(x,y)\equiv Ax+y=0\); then, by virtue of equations (3), we find
\[ \dot{R}=x\left[A(a-A)-b-n\frac{\varphi(x)}{x}\right]. \]
Taking into account the validity of inequality (4) for points with abscissas \(\varepsilon \leq x \leq \eta\), we obtain, for \(n=+1\), the equality \(\operatorname{sign}\dot{R}=-\operatorname{sign}x\), and for \(n=-1\), respectively, \(\operatorname{sign}\dot{R}=\operatorname{sign}x\), i.e. the velocity vectors of the \((+)\)- and \((-)\)-systems at the considered points of the line \((R)\) are directed to different sides with respect to the line, and moreover the velocity vector of the \((-)\)-system lies in the region of action of the \((+)\)-system and conversely. Therefore we may conclude that on the segment of the line \((R)\), for \(\varepsilon \leq x \leq \eta\), there is sliding. Obviously, the representative point of system (3) moves along such a segment, obeying the equation \(Ax+\dot{x}=0\), until condition (4) ceases to hold for the abscissa of a point of the line. Thereafter, as \(t\) increases, the motion of the representative point will be governed by the action of one of the \((+)\)- or \((-)\)-systems.
Corollary. If for system (3), on some interval \(\varepsilon \leq x \leq \eta\), the inequality
\[ \frac{\varphi(x)}{x}<\left|A(a-A)-b\right|, \tag{5} \]
is satisfied, then the corresponding segment of the line \((R)\) is a piercing segment of the system.
Thus, depending on the properties of the function \(\varphi(x)\) and on the values of the parameters of the system, the line \((R)\) may be either entirely a line of sliding or of piercing, or may consist of alternating segments of sliding and piercing. Since in the general case we do not assume the function \(\varphi(x)\) to be odd, let us consider the properties of the trajectories of system (3) depending on the character of the intersection of the line \((R)\) when \(x<0\). In the half-plane \(x>0\) such a study can be carried out analogously.
Suppose inequality (4) holds for all \(x<0\). Then the half-line \((R)\) \((x<0)\) is a half-line of sliding. It is easy to see that in this case, in the sector \(\Gamma\), defined by the inequalities \(Ax+y>0,\ x<0\), there necessarily exists exactly one trajectory of system (3) lying entirely in the sector, and such that, as time increases, the representative point of system (3) moves along this trajectory from any of its points to the origin, reaching it either in finite time or as \(t\to+\infty\). In what follows we shall call such a trajectory the stable separatrix of the origin.
For the analysis of the qualitative structure of system (3), an essential circumstance is the interval to which the origin belongs. If inequality (4) holds only on some interval of negative \(x\) adjacent to the origin, then the stable separatrix of the origin is located only partly in the sector \(\Gamma\) and necessarily intersects the line \((R)\) at a point situated to the left of the sliding segment. Finally, when the origin belongs to a sewing segment, then in this case in the sector \(\Gamma\) either no separatrices are incident to the origin at all, or there are two of them.
§ 2. STABILITY OF SYSTEM (3) AS A WHOLE
For system (3), the origin is a singular point of a specific type: owing to the presence of sliding, the representative point of system (3) can move toward the origin along the line \((R)\), arriving at it with finite velocity and in finite time. The property of variability of the structure of the system makes it possible to render it stable; therefore, in some cases the origin may be, for system (3), a stable equilibrium state.
Theorem 2. If the value of the parameter \(b\) and the properties of the function \(\varphi(x)\) are such that, for all \(x\ne0\), the inequality \(x f_{+}(x)>0\) is satisfied, then system (3) is stable as a whole with respect to the origin.
Proof. Consider the function
\[ v(x,y)=y^2+2\int_0^x f_{+}(x)\,dx. \tag{6} \]
Under the assumptions of the theorem it is positive definite for all \(x\ne0,\ y\ne0\). Its derivative, formed in view of equations (3), \(\dot v=-2ay^2-2(n-1)y\varphi(x)\), is, as shown in the article [7], a negative definite function everywhere as well. Indeed, for \(n=+1\) we obtain \(\dot v=-2ay^2\), while for \(n=-1\), \(\dot v=-2ay^2+4y\varphi(x)\), but in the domain \(x(Ax+y)<0\) we have \(y\varphi(x)<0\), since \(\operatorname{sign}\varphi(x)=\operatorname{sign}x\). The fulfillment of the conditions of the theorem of E. A. Barbashin and N. N. Krasovskii [8], which proves stability as a whole of the system, is then easily verified.
Let us note that Theorem 2 establishes stability as a whole of system (3) both in the case where the function \(\varphi(x)\) is continuous at \(x=0\), and in the case where the function \(\varphi(x)\) has a discontinuity of the first kind at \(x=0\). Varieties of the qualitative picture of the arrangement of the integral curves of system (3) are determined by the properties of the function \(\varphi(x)\) and by the character of the motion of the representative point along the line \((R)\) in the sense of the considerations set forth above.
Example 1. Let \(\varphi(x)=Kx\), where \(K>0\) is a parameter. For \(K>-b\), the function \(f_{+}(x)=(b+K)x\) for all \(x\ne0\) satisfies the inequality of Theorem 2; therefore system (3) is stable as a whole with respect to the origin.
coordinates. The ratio of the numbers \(\dfrac{\varphi(x)}{x}=K\) and \(|A(a-A)-b|\) determines the varieties of the qualitative picture of the arrangement of the trajectories of this system. If \(K>\max[-b,\ |A(a-A)-b|]\), then the system has the arrangement of trajectories schematically represented in Fig. 1, \(a\). If, however, the relation \(-b<K<|A(a-A)-b|\) holds, then the trajectories are arranged as in Fig. 1, \(b\), respectively.
Fig. 1
The condition \(x f_{+}(x)>0\) for all \(x\ne0\) in Theorem 2 completely covers the case when \(b>0\), and partly the case when \(b<0\). Therefore, in what follows we shall consider system (3) only for \(b<0\), i.e., regarding the regulating system without correction as certainly unstable, we shall study the influence of the control, which changes the structure of the system, on its stability. In doing so, it is evidently meaningful to consider only such systems for which the inequality \(x f_{+}(x)>0\) holds at least in an arbitrarily small neighborhood of \(x=0\). Otherwise system (3) cannot have the origin as a stable equilibrium state.
§ 3. INVESTIGATION OF THE SHAPE OF THE REGION OF ATTRACTION
In the case when the function \(\varphi(x)\) is nonlinear and such that the function \(f_{+}(x)\) has zeros, system (3) is not stable in the large. The region of attraction of its zero equilibrium state is a part of the phase plane bounded, according to a well-known theorem of N. P. Erugin [9], by the separatrices of the system. Similar systems with discontinuous function \(f(x)\) were studied earlier, for example, in [10, 11].
Let the function \(f_{+}(x)\) satisfy the inequality \(x f_{+}(x)>0\) on the interval \((x_{2},x_{1})\), where \(x=x_{1}>0\) and \(x=x_{2}<0\) are the zeros of the function \(f_{+}(x)\) nearest to \(x=0\). We shall also suppose the function \(\varphi(x)\) to be such that the derivatives \(f_{+}'(x_{1})\), \(f_{+}'(x_{2})\) exist and are not equal to zero. Then, by virtue of our assumptions, we arrive at the conclusion that the inequalities \(f_{+}'(x_{1})<0\) and \(f_{+}'(x_{2})<0\) are valid; i.e., the points \((x_{1},0)\) and \((x_{2},0)\) are saddle-type singular points for system (3). Since, under the assumptions under consideration, system (3) has no limit cycles enclosing the origin, the qualitative structure of the system is determined only by the mutual arrangement of the separatrices of the singular points; at the same time, of course, one must take into account the character of the behavior of the system on the switching line \((R)\). The presence of sliding segments may lead to nonuniqueness of trajectories and, in particular, of the separatrices of system (3), when the trajectories, fill-
...covering a certain region of the phase plane, merge into a single trajectory [12].
Let us denote the separatrices of the \((+)\)-system according to Fig. 2, where, for example, \(s_1\) denotes the separatrix corresponding to the integral curve \(y=s_1(x)\). For the separatrices of system (3) we introduce the same notation, taking into account, however, that the trajectories of the systems coincide only as long as the representative point lies in the region determined by the inequality \(x(Ax+y)>0\). The separatrices of system (3) may be nonsmooth, “glued together” from the corresponding trajectories of the \((+)\)- and \((-)\)-systems, and may also contain rectilinear segments—sliding segments—and regions of nonuniqueness, as is shown, for example, in Fig. 2.
Fig. 2
We shall characterize the mutual arrangement of the separatrices by inequalities comparing the segments which they cut off either on the \(y\)-axis or on the line \((R)\).
We shall study the arrangement of the trajectories of system (3) by the method of Lyapunov functions. The function (6)
\[ v(x,y)=y^2+2\int_0^x f_+(x)\,dx \]
is positive definite and has a negative-definite derivative, formed by virtue of equations (3), in a neighborhood of the origin. The form of the curves \(v(x,y)=\mathrm{const}\) depends on the values of the integrals
\[ J_1=\int_0^{x_1} f_+(x)\,dx \]
and
\[ J_2=-\int_0^{x_2} f_+(x)\,dx . \]
Let us write the equation of the curve \(v(x,y)=\mathrm{const}\) passing through the point \((x_1,0)\) in the form \(y=\pm v_1(x)\), where
\[ v_1(x)=\left[2J_1-2\int_0^x f_+(x)\,dx\right]^{1/2}, \]
and the equation of the curve passing through the point \((x_2,0)\) in the form \(y=\pm v_2(x)\), where
\[ v_2(x)=\left[2J_2-2\int_0^x f_+(x)\,dx\right]^{1/2}. \]
From the properties of the function \(v(x,y)\) the validity of the following inequalities is obvious:
\[ \begin{gathered} v_1(x)<s_1(x),\quad -v_1(x)<m_1(x),\quad -v_1(x)>n_1(x),\\ v_2(x)>r_2(x),\quad -v_2(x)>n_2(x),\quad v_2(x)<s_2(x) \end{gathered} \tag{7} \]
for values of \(x\) which are common to the domains of existence of the radicals \(v_1(x)\) and \(v_2(x)\) and of the corresponding functions \(s_1(x), s_2(x), r_2(x), n_1(x), m_1(x), n_2(x)\).
Theorem 3. If, for system (3), the inequality \(x f_+(x)>0\) is satisfied on the interval \((x_2,x_1)\), where \(x=x_1>0\) and \(x=x_2<0\) are the zeros of the function \(f_+(x)\) closest to \(x=0\), then for the system there are three possible varieties of the form of the region of attraction of its zero equilibrium state, corresponding schematically to the qualitative pictures shown for the arrangement of the trajectories of the system—schemes I, II, III, Fig. 3.
The proof of the theorem can be carried out analogously to how this was done in [10]. At the same time, it can be shown without particular difficulty that the varieties of the qualitative portrait of system (3) depend on the mutual arrangement only of the separatrices \(r_2, s_1, n_2, m_1\) and, of course, on the character of the intersection of the straight line \((R)\) by the trajectories of the system.
Let the function \(\varphi(x)\) be such that \(J_1 < J_2\). The level lines of the function (6) are then arranged according to the inequality \(v_1(0) < v_2(0)\). Taking into account the relations (7), we see that the “lower” separatrices \(n_2\) and \(m_1\) are then arranged in the only possible way—according to the inequality \(n_2(0) < m_1(0)\). Moreover, the straight line \((R)\) is intersected by them in such a way that we have \(n_2(R) < m_1(R)\). Here and below, by \(n_2(R)\) and \(m_1(R)\) are denoted the points of intersection of the curves \(y = n_2(x)\) and \(y = m_1(x)\) with the straight line \((R)\), and the inequality means: the point \(n_2(R)\) lies on the straight line \((R)\) below the point \(m_1(R)\) (the point \(n_2(R)\) may also be absent). Obviously, the arguments concerning the arrangement of the lower separatrices in our case remain valid regardless of which segment of the straight line \((R)\) the points \(n_2(R)\) and \(m_1(R)\) belong to.
Fig. 3
Thus, if the points \(m_1(R)\) belong to a sliding segment, then the separatrix \(m_1\) of system (3) has a segment, and the representative point of system (3) moves along this segment toward the origin in such a way that the corresponding inequality (7) for the separatrix \(m_1\) is not violated. Further, a non-rough case is possible, generally speaking, when the point \(n_2(R)\) coincides with the left end of the sliding segment; in this case the separatrix \(n_2\), obviously, has a region of multivaluedness, but all these trajectories are located below the separatrix \(m_1\). For the “upper” separatrices in the case under consideration, when \(J_1 < J_2\), two varieties of their arrangement are possible—respectively, either according to the inequality \(s_1(R) < r_2(R)\), or \(s_1(R) > r_2(R)\). In order to avoid cumbersome arguments, we exclude the non-rough cases of the arrangement of the separatrices when \(s_1(R) = r_2(R)\); in particular, we shall include here also the case when the point \(r_2(R)\) belongs to the sliding segment of the straight line \((R)\), while the point \(s_1(R)\) is its right endpoint.
Suppose that the point \(s_1(R)\) is located on the straight line \((R)\) above the point \(r_2(R)\). Then, regardless of which segment it belongs to,
point \(r_2(R)\), the separatrix \(s_1\) intersects the \(y\)-axis higher than the separatrix \(r_2\). The trajectories of system (3) are arranged according to scheme I, Fig. 3, when the domain of attraction of the zero equilibrium state is unbounded in the directions both along the semiaxis \(+y\) and along the semiaxis \(-y\), bounded by the separatrices \(s_1, s_2, n_2, n_1\).
Now suppose that \(s_1(R)<r_2(R)\). At the point \(s_1(R)\) the line is pierced by the separatrix \(s_1\); therefore, even in the case when \(r_2(R)\) is a point of the sliding segment, the separatrix \(r_2\) is situated no lower than the separatrix \(s_1\), so that the relation \(r_2(0)\geq s_1(0)\) holds, i.e., system (3) has the arrangement of trajectories according to scheme II, Fig. 3, when the domain of attraction is unbounded only in the direction \(-y\) and is bounded by the separatrices \(s_1\) and \(n_1\).
Let us next consider system (3) when the inequality \(J_1>J_2\) is satisfied; this uniquely determines the arrangement of the separatrices \(s_1\) and \(r_2\) according to the inequality \(r_2(0)<s_1(0)\). In this case the lower separatrices can be arranged relative to one another again in only two different ways (we exclude the nonrough case of coincidence of separatrices). Then, if the inequality \(n_2(R)<m_1(R)\) is satisfied, we have scheme I for the form of the domain of attraction of the zero equilibrium state of the system; if \(n_2(R)>m_1(R)\), we obtain scheme III for the arrangement of the trajectories of the system, when the domain of attraction is a strip extending infinitely in the direction of the semiaxis \(+y\) and bounded by the separatrices \(s_2, n_2\).
From the enumeration carried out of the possible arrangement of the separatrices \(s_1, r_2, m_1\), and \(n_2\), it is clear that the possible varieties of the form of the domain of attraction of system (3) are exhausted by the rough cases considered, shown schematically in Fig. 3.
In conclusion we note that, obviously, satisfaction of the inequality \(J_1>J_2\) is a sufficient condition for the presence of one of the possible schemes of arrangement of the trajectories of system (3), I or III, and for the absence of the arrangement of trajectories according to scheme II. Satisfaction of the inequality \(J_1<J_2\) is a sufficient condition for the existence of either scheme I or scheme II, and, correspondingly, for the absence of scheme III. Equality of the integrals \(J_1=J_2\) uniquely determines the arrangement of the trajectories of the system according to the qualitative portrait shown schematically in Fig. 3, I.
Example 2. Let the function \(\varphi(x)\) be given in the form
\[ \varphi(x)= \begin{cases} H_1, & \text{for } x>\dfrac{H_1}{K},\\[6pt] Kx, & \text{for } -\dfrac{H_2}{K}\leq x\leq \dfrac{H_1}{K},\\[6pt] -H_2, & \text{for } x<-\dfrac{H_2}{K}, \end{cases} \qquad H_1\ne H_2. \]
For \(b>0\) system (3) in the present case has the property \(x f_+(x)>0\) for all \(x\ne0\); therefore, according to Theorem 2, it is stable in the large for arbitrary positive values of \(H_1, H_2, K\). The values of these parameters influence the character of the trajectories near the origin, and the magnitude of the sliding interval of the line \((R)\). Obviously, for \(K<|A(a-A)-b|\) the system has no sliding, and for \(K>|A(a-A)-b|\) sliding occurs on the interval \(-H_2<x|A(a-A)-b|<H_1\).
For \(b<0\) and \(K>-b\) system (3) still has a stable equilibrium state at the origin; its domain of attraction has one of the three forms indicated in Fig. 3, depending on the relation between the quantities \(H_1\) and \(H_2\).
§ 4. SUFFICIENT CRITERIA FOR THE EXISTENCE OF I, II, III SCHEMES OF ARRANGEMENT OF THE TRAJECTORIES OF SYSTEM (3)
If system (3), satisfying the requirements of § 3, can be integrated to the end, then by directly checking the mutual arrangement of the separatrices one can establish the form of the qualitative picture of the arrangement of its trajectories. In the case where the function \(\varphi(x)\) is nonlinear, of interest are sufficient conditions for the existence of one or another pattern of arrangement of the trajectories of the system. The derivation of such criteria is connected with finding estimates for the separatrices of system (3); moreover, the criteria are the more effective the more accurately the estimate of the separatrices \(n_2, m_1, s_1, r_2\) is found.
As curves forming the so-called “forks” for the separatrices, one can use noncontact curves passing through the points \((x_1,0)\), \((x_2,0)\). In particular, for system (3), curves of the family (6) \(v(x,y)=\mathrm{const}\) are noncontact on certain intervals of the \(x\)-axis. In addition, as a Lyapunov function for system (3) one may use the function employed in [7],
\[ \omega(x,y)=y^2+2\int_0^x [bx+n\varphi(x)]\,dx, \tag{8} \]
where \(n=\operatorname{sign} x(Ax+y)\). Its derivative, formed by virtue of the equations of system (3), is written in the form \(\dot{\omega}=-2ay^2\), which also shows that the curves \(\omega(x,y)=\mathrm{const}\) are noncontact on some intervals of the \(x\)-axis. The level lines of the function (8) are composed of the lines \(v(x,y)=\mathrm{const}\) in that part of the phase plane where the inequality \(x(Ax+y)>0\) is satisfied, and of the curves
\[ y^2+2\int_0^x f_-(x)\,dx=C \]
in the sector \(x(Ax+y)<0\).
Theorem 4. If for system (3), satisfying the conditions of § 3, with \(J_1<J_2\) the inequality
\[ (A^2+b)X^2<J_1+J_2, \tag{9} \]
is satisfied, where \(x=X\) is the negative root of the equation
\[ J_2-J_1=2\int_0^x \varphi(x)\,dx, \tag{10} \]
then we have the arrangement of its trajectories corresponding to scheme I, Fig. 3.
Proof. Consider the curves \(y=v_2(x)\) and \(y=\omega_1(x)\), where
\[ v_2(x)=\left[2\int_x^{x_2} f_+(x)\,dx\right]^{\frac12} \quad\text{and}\quad \omega_1(x)=\left[2\int_x^{x_1} f(x)\,dx\right]^{\frac12}. \]
These curves, when the inequality \(J_1<J_2\) is fulfilled, necessarily intersect at the point \(N(X,Y)\), whose abscissa is the negative root of equation (10); moreover, from the properties of the function \(\varphi(x)\) the uniqueness of such a point is evident. Inequality (9), equivalent to the inequality \(AX+Y>0\), determines the position of the point \(N\) above the straight line \((R)\), and thereby the validity of the relation \(s_1(R)>r_2(R)\). Both curves under consideration are positive level lines of the Lyapunov functions (6) and (8), and therefore are noncontact and are intersected by the trajectories of the system in such a way that the inequalities \(s_1(x)>\omega_1(x)\) hold at least for \(X\le x<x_1\) and \(r_2(x)<v_2(x)\) at least for \(x_2<x\le X\), i.e., in the present case the separatrices \(r_2\) and \(s_1\) can be arranged in a unique way
according to the inequality \(s_1(0)>r_2(0)\), determining, for \(J_1<J_2\), the disposition of the trajectories of the system according to scheme I, Fig. 3.
In conclusion we note that, for \(J_1<J_2\), it is evident that for the existence of a disposition of trajectories of type scheme I it is sufficient in general that there be no intersection of the curve \(y=\omega_1(x)\) with the line \((R)\), i.e., it is sufficient that there be no negative root of the equation
\[ A^2x^2+2\int_0^x f_-(x)\,dx=2J_1\qquad (x<0) \]
or that inequality (4) hold for \(x_0<x<0\), where \(x=x_0\) is the abscissa of the point \(r_2(R)\). Analogous conditions for the existence of the picture of the disposition of the trajectories of system (3), corresponding to scheme I, Fig. 3, for \(J_1>J_2\), can easily be obtained.
In some cases, as contactless curves one may use the curves of the family
\[ (y+ax)^2+2\int_0^x f_+(x)\,dx=C, \]
considered in detail in [12]. In particular, it is easy to verify the validity, for example, of the following inequalities:
\[ s_1(x)>-ax+\left[\,2J_1+a^2x_1^2-2\int_0^x f_+(x)\,dx\,\right]^{\frac12} \]
at least for \(0\le x<x_1\), and also
\[ n_2(x)<-ax-\left[\,2J_2+a^2x_2^2-2\int_0^x f_+(x)\,dx\,\right]^{\frac12} \]
at least for \(x_2<x\le 0\). Then, analogously to the proof of the corresponding theorems of [10], one can prove the validity of the criteria formulated in the following theorem 5. The fact that the point \(r_2(R)\) may belong to the sliding segment obviously does not affect the results.
Theorem 5. For the existence of scheme I of the disposition of the trajectories of system (3), satisfying the assumptions of § 3, it is sufficient that one of the inequalities hold:
for \(J_1<J_2\), the inequalities
\[ 2(J_2-J_1)<a^2x_1^2 \]
\[ \text{or}\qquad [\,2J_1+a^2x_1^2\,]^{\frac12}> \max_{x_2<x<0}\left[-\frac{f_+(x)}{a}\right]; \]
for \(J_1>J_2\), the inequalities
\[ 2(J_1-J_2)<a^2x_2^2 \]
\[ \text{or}\qquad -[\,2J_2+a^2x_2^2\,]^{\frac12}< \min_{0<x<x_1}\left[-\frac{f_+(x)}{a}\right]. \]
Moreover, applying, for example, the method of successive approximations, one can also obtain certain curves estimating the separatrices of system (3). Obviously, using the arguments given in [12], it is easy to verify the validity of the following estimates:
\[ s_1(x)<a(x_1-x)+\omega_1(x)\equiv y_1(x) \tag{11} \]
for \(\tilde x\le x<x_1\), where \(\tilde x\) is the abscissa of the point of intersection of the curve \(y=y_1(x)\) with the line \((R)\), and also
\[ r_2(x)>a(x_2-x)+v_2(x)\equiv y_2(x) \tag{12} \]
for those values of \(x\) for which \(y_2(x)>0\). Analogous estimates can also be written for the separatrices \(n_2\) and \(m_1\).
Theorem 6. If, for system (3) satisfying the assumptions of § 3, with \(J_1<J_2\) the inequality
\[ a(x_1--x_2)<\sqrt{2J_1}-\sqrt{2J_2}, \tag{13} \]
holds, then the system has an arrangement of trajectories corresponding to scheme II, Fig. 3.
The proof of the theorem can be carried out analogously to how this was done earlier in [10], using inequalities (11), (12). We note that here a nonrough case is possible, when the point \(r_2(R)\) belongs to the sliding segment, and the point \(s_1(R)\) to the right end of this segment. Then the separatrix \(s_1\), which together with the separatrix \(n_1\) bounds the domain of attraction, has a region of multivaluedness, but the domain of attraction still has the form schematically indicated in Fig. 3, II.
By similar arguments one can justify an analogous theorem establishing the existence of a domain of attraction of the zero equilibrium state of system (3) of the form indicated schematically in Fig. 3, III.
Let \(y_1(R)\) and \(y_2(R)\) be, respectively, the points of intersection of the line \((R)\) with the curves \(y=y_1(x)\) and \(y=y_2(x)\), where the functions \(y_1(x)\) and \(y_2(x)\) are defined by relations (11) and (12). Then the validity of the following proposition is easily verified.
Theorem 7. If, for system (3) satisfying the conditions of § 3, with \(J_1<J_2\) the point \(y_1(R)\) lies below the point \(y_2(R)\), then the trajectories of system (3) are arranged in such a way that the domain of attraction has the form schematically indicated in Fig. 3, II.
In an analogous form one can formulate a criterion for the existence of an arrangement of trajectories of system (3), for \(J_1>J_2\), corresponding to the scheme in Fig. 3, III.
We note that, in the arguments carried out in § 4, the aim was mainly only to indicate methods for obtaining criteria, as well as ways of increasing their effectiveness.
§ 5. ESTIMATE OF THE DOMAIN OF ATTRACTION
After the form of the qualitative picture of the arrangement of trajectories has been established, it makes sense to estimate the domain of attraction in the case when the function \(\varphi(x)\) is nonlinear and satisfies the requirements of § 3. This can be done by knowing curves that estimate the separatrices.
Let, for system (3), the domain of attraction have the form indicated in Fig. 3, I, i.e., it is bounded by the separatrices \(s_1,s_2,n_1,n_2\). Then, as an approximate estimate of the domain of attraction, one can take the domain bounded by the curves that are the corresponding estimates of these separatrices. Thus, the separatrix \(s_1\) can be estimated from below by the curve
\[ y=-ax+\left[a^2x_1^2+2\int_x^{x_1} f_+(x)\,dx\right]^{\frac12} \]
for \(0\le x<x_1\), extending it for \(\tilde{x}_0\le x<0\) by the corresponding curve of the family \(\omega(x,y)=c\), i.e., by the curve
\[ y=\left[a^{2}x_{1}^{2}+2J_{1}-2\int_{0}^{x} f_{-}(x)\,dx\right]^{\frac12}, \]
where \(x=\tilde{x}_{0}\) is the abscissa of the point of intersection of this curve with the line \((R)\); further, for \(x<\tilde{x}_{0}\) the curve can be continued by a curve of the family \(v(x,y)=c\), passing through the point \((\tilde{x}_{0},-A\tilde{x}_{0})\). The separatrix \(s_{2}\) should be replaced by its upper estimate, i.e., for example, by the curve
\[ y=a(x_{2}-x)+\left[2\int_{x}^{x_{2}} f_{+}(x)\,dx\right]^{\frac12} \]
for \(x<x_{2}\). The region bounded by these curves in the half-plane \(y>0\) is a certain estimate of the domain of attraction in our case. In an analogous way, an estimate of the domain of attraction in the half-plane \(y<0\) can be obtained.
Next suppose that system (3) has a domain of attraction of the form schematically shown in Fig. 3, II. In this case the boundary of the domain of attraction consists of the separatrices \(s_{1}\) and \(n_{1}\). To estimate the domain of attraction, one should take a lower estimate of the separatrix \(s_{1}\). As such a curve one may take, for example, the curve \(y=\omega_{1}(x)\) for \(X\le x<x_{1}\) and continue it by the corresponding curve of the family (8). The separatrix \(n_{1}\) can be estimated from below, for example, by the curve
\[ y=a(x_{1}-x)-\left[2\int_{x}^{x_{1}} f_{+}(x)\,dx\right]^{\frac12} \quad \text{for } x>x_{1}. \]
Entirely analogous considerations may be used to estimate the domain of attraction of type III, Fig. 3.
References
- Taran V. A. Survey. Automation and Remote Control, 25, No. 1, 1964.
- Barbashin E. A. UMN, vol. XIX, issue 6 (120), 1964.
- Bermant M. A., Bakakin A. V. and Ezerov V. B. Automation and Remote Control, 25, No. 7, 1964.
- Barbashin E. A., Pechorina I. N. and Eidinov R. M. Automation and Remote Control, 24, No. 1, 1963.
- Emelyanov S. V. and Fedotova A. I. Automation and Remote Control, 23, No. 10, 1962.
- Barbashin E. A., Tabueva V. A. Automation and Remote Control, vol. XXIII, No. 10, 1962.
- Barbashin E. A., Gerashchenko E. I. Differential Equations, vol. I, No. 1, 1965.
- Barbashin E. A., Krasovskii N. N. DAN SSSR, 86, No. 3, 1952.
- Erugin N. P. PMM, vol. XV, issue 2, 1951.
- Tabueva V. A. On the question of the form of the domain of attraction of the zero solution of a certain second-order differential equation. Matematicheskii sbornik, 47 (89), issue 2, 1959.
- Tabueva V. A. Izv. vuzov, Mathematics, No. 2 (3), 1958.
- Tabueva V. A. Siberian Mathematical Journal, 4, No. 2, 1963.
- Tabueva V. A. Izv. vuzov, Mathematics, No. 2 (15), 1960.
Received by the editors
May 8, 1965
V. A. Steklov Mathematical Institute.
Sverdlovsk Branch