UDC 517.917

ON ONE METHOD OF STABILIZATION OF CONTROL SYSTEMS

E. A. BARBASHIN, N. G. YAROVOI

In this article an attempt is made to find a new method for stabilizing a control system. In the case of purely relay correction, stability under large initial conditions may fail to appear. If, in the correction, the first or higher powers of the controlled coordinate are used, then, by choosing a suitable control law, we can obtain the desired stability property of the system under arbitrary initial disturbances.

The present article is devoted to proving this assertion in the case when a quadratic function of the controlled coordinate is used in the correction law.

§ 1. STABILITY OF THE EQUILIBRIUM POSITION OF A DYNAMICAL SYSTEM OF ORDER \(n\)

Consider a differential equation of order \(n\) of the form

\[ x^{(n)} + a_1 x^{(n-1)} + \cdots + a_n x = -\alpha (Kx^2 + 1), \tag{1.1} \]

where \(a_1, \ldots, a_n\) are real numbers; \(K\) is a positive parameter; \(\alpha\) is determined by the relation

\[ \alpha = \operatorname{sign}(c_{n-1}x_1 + \cdots + x_n) \]

and satisfies the condition \(|\alpha| \leq 1\); \(x_1, x_2, \ldots, x_n\) are phase coordinates determined by the system of differential equations equivalent to equation (1.1):

\[ \begin{aligned} \dot{x}_1 &= x_2,\\ &\cdots\\ \dot{x}_{n-1} &= x_n,\\ \dot{x}_n &= -a_n x_1 - a_{n-1}x_2 - \cdots - a_1x_n - \alpha(Kx_1^2 + 1). \end{aligned} \tag{1.2} \]

The hyperplane

\[ w = c_{n-1}x_1 + c_{n-2}x_2 + \cdots + x_n = 0 \]

is called the switching hyperplane.

We shall find necessary and sufficient conditions for the existence of sliding at every point of the switching hyperplane.

Theorem 1. For sliding to exist on the entire switching hyperplane \(w=0\), it is necessary and sufficient that the conditions

\[ c_{i+1} = rc_i + a_{i+1}\quad (i=0, 1, \ldots, n-2), \tag{1.3} \]

\[ |f(r)| \leq 2\sqrt{K}, \tag{1.4} \]

be satisfied, where \(r = c_1 - a_1\) is a real parameter;

\[ f(r)=r^n+a_1r^{n-1}+\cdots+a_n=a_n+rc_{n-1}. \]

Proof of necessity. To derive conditions (1.3) and (1.4), we shall use the necessary and sufficient conditions for the existence of sliding (for example, [1], p. 424) on the hypersurface \(w=0\). These conditions have the form

\[ \lim_{w\to +0}\frac{dw}{dt}\leqslant 0,\qquad \lim_{w\to -0}\frac{dw}{dt}\geqslant 0. \tag{1.5} \]

The hyperplane \(w=0\) divides the phase space of the variables \(x_1,\ldots,x_n\) into two half-spaces in such a way that in the half-space \(w>0\) the system of differential equations (1.2) acts for \(\alpha=+1\), while in the half-space \(w\leqslant 0\) the system of equations acts for \(\alpha=-1\). Consequently, the conditions for the existence of sliding (1.5) in our case take the following form:

\[ (c_{n-1}-a_{n-1})x_2+\cdots+(c_1-a_1)x_n-a'_nx_1-(Kx_1^2+1)\leqslant 0, \]

\[ (c_{n-1}-a_{n-1})x_2+\cdots+(c_1-a_1)x_n-a_nx_1+(Kx_1^2+1)\geqslant 0. \tag{1.6} \]

In the phase space of the variables \(x_1,x_2,\ldots,x_n\), consider the following hypersurfaces:

\[ w_1=(c_{n-1}-a_{n-1})x_2+\cdots+(c_1-a_1)x_n-a'_nx_1-(Kx_1^2+1)=0, \]

\[ w_2=(c_{n-1}-a_{n-1})x_2+\cdots+(c_1-a_1)x_n-a_nx_1+(Kx_1^2+1)=0, \]

\[ w=c_{n-1}x_1+c_{n-2}x_2+\cdots+x_n=0. \]

It is not difficult to see that the hypersurfaces \(w_1=0\) and \(w_2=0\) are certain cylindrical hypersurfaces in the phase space of the variables \(x_1,x_2,\ldots,x_n\).

From conditions (1.6) it follows that, in the case of the existence of sliding, the hypersurface \(w_1=0\) must lie entirely in the half-space \(w>0\), and the hypersurface \(w_2=0\) must lie in the half-space \(w\leqslant 0\). Consequently, the hypersurfaces \(w_1=0\) and \(w_2=0\) can be tangent to the switching hyperplane along the generatrices of the cylindrical hypersurfaces \(w_1=0\) and \(w_2=0\), while remaining entirely in the half-spaces \(w>0\) and \(w\leqslant 0\), respectively.

Consider the vectors

\[ \operatorname{grad} w=\{c_{n-1},c_{n-2},\ldots,1\}, \]

\[ \operatorname{grad} w_1=\{-a'_n-2Kx_1,\ c_{n-1}-a_{n-1},\ldots,c_1-a_1\}, \]

\[ \operatorname{grad} w_2=\{-a'_n+2Kx_1,\ c_{n-1}-a_{n-1},\ldots,c_1-a_1\}. \]

Since the hypersurfaces \(w_1=0\) and \(w_2=0\) may be tangent to the switching hyperplane \(w=0\), it follows from this that the following conditions must be satisfiable:

\[ c_{n-1}-a_{n-1}=rc_{n-2}, \]

\[ c_{n-2}-a_{n-2}=rc_{n-3}, \]

\[ \cdots \]

\[ c_1-a_1=r. \]

These equalities can evidently be combined:

\[ c_{i+1}=rc_i+a_{i+1}\quad (i=0,1,\ldots,n-2). \tag{1.7} \]

Using conditions (1.7), conditions (1.5) can now be written in the form

\[ \left.\frac{dw}{dt}\right|_{\substack{\alpha=+1\\ w=0}} =-f(r)x_1-Kx_1^2-1\leqslant 0, \]

\[ \left.\frac{dw}{dt}\right|_{\substack{\alpha=-1\\ w=0}} =-f(r)x_1+Kx_1^2+1\geqslant 0, \tag{1.8} \]

and for conditions (1.8) to be satisfied it is evidently necessary and sufficient that the condition

\[ f^2(r)-4K\leqslant 0 \]

be satisfied, or the equivalent condition

\[ |f(r)|\leqslant 2\sqrt{K}. \]

The sufficiency of the conditions is obvious. The theorem is proved.

Let us proceed to the study of questions of stability of the equilibrium position of the system under consideration (1.2). Instead of the coordinate \(x_n\) of the phase space of the variables \(x_1,\ldots,x_n\), introduce a new coordinate

\[ w=c_{n-1}x_1+c_{n-2}x_2+\cdots+x_n . \]

Then the system of differential equations (1.2) is written in the following form:

\[ \dot{x}_i=x_{i+1}\quad (i=1,2,\ldots,n-2), \]

\[ \dot{x}_{n-1}=-\sum_{i=1}^{n-1}c_{n-i}x_i+w, \]

\[ \dot{w}=-f(r)x_1-\alpha(Kx_1^2+1). \tag{1.9} \]

In addition to the system (1.9), equivalent to system (1.2), consider the system of differential equations

\[ \dot{x}_i=x_{i+1}\quad (i=1,2,\ldots,n-2), \]

\[ \dot{x}_{n-1}=-\sum_{i=1}^{n-1}c_{n-i}x_i+w, \]

\[ \dot{w}=rw. \tag{1.10} \]

It is easy to see that the characteristic equation of the system of differential equations (1.10) is the equation

\[ \lambda^n+a_1\lambda^{n-1}+\cdots+a_n=f(r). \tag{1.11} \]

Theorem 2. Suppose the conditions for the existence of sliding (1.3) and (1.4) are satisfied.

If all roots of equation (1.11) have negative real parts, then the equilibrium position of system (1.9) is asymptotically stable in the large.

Proof. Since the characteristic equation (1.11) of the system of differential equations (1.10) has roots with negative real parts, there exists a definite positive

a quadratic form ([2], p. 62) \(V_1=V_1(x_1,x_2,\ldots,x_{n-1})\) such that its derivative, taken by virtue of the system of differential equations (1.10), has the form

\[ \frac{dV_1}{dt}=-x_1^2-x_2^2-\cdots-x_{n-1}^2 = W_1(x_1,x_2,\ldots,x_{n-1}). \]

Consider the quadratic form

\[ V=V_1+\frac{1}{2}\gamma w^2, \]

where \(\gamma>0\). Obviously, this form is positive definite. Let us find the time derivative, by virtue of the system of differential equations (1.9), of the function \(V(x_1,x_2,\ldots,x_{n-1},w)\):

\[ \frac{dV}{dt}=W_1+\gamma w[-f(r)x_1-\alpha(Kx_1^2+1)]. \]

But, by virtue of the conditions for the existence of sliding and of the switching law, we have

\[ \operatorname{sign} w=-\operatorname{sign}[-f(r)x_1-\alpha(Kx_1^2+1)]. \]

Consequently, there exists a positive definite quadratic form whose derivative, by virtue of system (19), equivalent to system (1.2), is negative. Hence the equilibrium position of system (1.2) is asymptotically stable in the large ([6], Theorem 12.2).

The theorem is proved.

In the case where the representative point \(M(t)\) reaches the switching hyperplane, the motion of this point will occur by virtue of the system of differential equations describing the sliding process. This system has the form ([3], p. 680)

\[ \dot{x}_i=x_{i+1}\quad (i=1,2,\ldots,n-2), \tag{1.12} \]

\[ \dot{x}_{n-1}=-\sum_{i=1}^{n-1} c_{n-i}x_i. \]

The question of stability of the system of differential equations (1.12) is solved by Theorem 3, analogous to a theorem of E. I. Gerashchenko [4].

Theorem 3. Let the conditions for the existence of sliding (1.3) and (1.4) be satisfied for the given value of \(r\). In order that the zero solution of system (1.12) be asymptotically stable, it is necessary and sufficient that all roots of equation (1.11), apart from the obvious one \(\lambda=r\), lie in the left half-plane of the complex variable.

The proof follows, first, from the fact that equation (1.11) splits into two equations: into \(\lambda=r\) and into the characteristic equation of the system of differential equations (1.12); second, system (1.12) is obtained from system (1.10) for \(w=0\).

§ 2. REACHING OF THE REPRESENTATIVE POINT OF THE PHASE SPACE ONTO THE SWITCHING SURFACE

To prove that the representative point of the phase space reaches the switching surface, we shall confine ourselves to consideration of a third-order system. In this case the system of differential equations (1.9) is rewritten as follows:

$$ \dot{x}_1=x_2, $$

$$ \dot{x}_2=-c_2x_1-c_1x_2+w, \tag{2.1} $$

$$ \dot{w}=-f(r)x_1-\alpha(Kx_1^2+1), $$

where \(c_1,\ c_2,\ 1\) are the coordinates of the normal vector to the switching plane, \(r=c_1-a_1\).

In the case where the representative point, moving according to the system of differential equations (2.1), reaches the switching plane \(w=0\) for \(0\leq t<\infty\), the ordinate \(w\) must change sign.

To investigate the question of the representative point in the phase space of the variables \(x_1,\ x_2,\ w\) reaching the plane, we shall use the small-parameter method [5]. Introduce a small parameter \(\rho\) in (2.1) by the formula \(t=\rho\tau\) and make the change of variables of the form

$$ x_1=y_1,\qquad \rho x_2=y_2,\qquad \rho^2 w=y_3 \qquad \left(\rho=K^{-\frac13}\right). $$

Then system (2.1) is transformed into the system

$$ \frac{dy_1}{d\tau}=y_2, $$

$$ \frac{dy_2}{d\tau}=-\rho^2c_2y_1-\rho c_1y_2+y_3, $$

$$ \frac{dy_3}{d\tau}=-\rho^3 f(r)y_1-\alpha\rho^3(Ky_1^2+1)+\rho r y_3. $$

If \(K\to\infty\), then \(\rho\to0\). We obtain the simplified system of differential equations:

$$ \dot{y}_1=y_2,\qquad \dot{y}_2=y_3,\qquad \dot{y}_3=-\alpha y_1^2, \tag{2.2} $$

where

$$ \dot{y}_1=\frac{dy_1}{d\tau},\qquad \dot{y}_2=\frac{dy_2}{d\tau},\qquad \dot{y}_3=\frac{dy_3}{d\tau}. $$

We shall investigate the behavior of the trajectories of system (2.2) in the phase space of the variables \(y_1,\ y_2,\ y_3\).

Lemma 1. Let \(y_1(\tau),\ y_2(\tau),\ y_3(\tau)\) be a solution of system (2.2), determined at \(\tau=0\) by the initial point \(M_0(y_1^0,y_2^0,y_3^0)\), where \(y_3^0>0\). Then either there exists a number \(\tau_0\) \((0<\tau_0<\infty)\) such that \(y_3(\tau_0)=0\), or

$$ \lim y_1(\tau)=\lim y_2(\tau)=\lim y_3(\tau)=0 \quad \text{as } \tau\to\infty. $$

Proof. From the third equation of system (2.2) it follows that the quantity \(y_3(\tau)\) strictly decreases as \(\tau\) increases. If the function \(y_3(\tau)\) is discontinuable for \(\tau>0\), then there exists a number \(\tau_1\) such that \(\lim y_3(\tau)=-\infty\) as \(\tau\to\tau_1\). Hence the assertion of the lemma follows. If the function \(y_3(\tau)\) is defined and continuous for all positive values of \(\tau\), then suppose that \(\lim y_3(\tau)=L\) \((\tau\to\infty)\). The case \(L<0\) again leads us to the conclusion that the lemma is valid.

The assumption \(L>0\) leads to a contradiction, since from the first and second equations of system (2.2) it follows that \(y_2(\tau)\to\infty\) and \(y_1(\tau)\to\infty\) as \(\tau\to\infty\), and from the third equation it then follows that \(y_3(\tau)\to-\infty\). If

If \(L=0\), then we shall have \(y_3(\tau)>0\), whence it follows that \(y_2(\tau)\) increases. Suppose that \(\lim y_2(\tau)=L_1\) as \(\tau\to\infty\). If \(L\ne 0\), then from the first equation of system (2.2) it follows that \(\lim |y_1(\tau)|=\infty\), while from the third equation we again obtain \(\lim y_3(\tau)=-\infty\), which leads to a contradiction. Thus we have \(L=L_1=0\). But if \(L_1=0\), then the quantity \(y_2(\tau)\) will be negative and \(y_1(\tau)\) decreases. If \(\lim y_1(\tau)=L_2\) \((\tau\to\infty)\), then we again obtain a contradiction with the supposition that \(\lim y_3(\tau)=0\). For \(\alpha=-1\) and, consequently, \(y_3^0<0\), analogous arguments may be carried out.

Lemma 2. If the point \(M(\tau)\) moves from the position \(M_0\) along a trajectory of system (2.2), then it cannot remain in the domain \(|y_1|>\delta,\ y_3>0\) for more than \(y_3^0/\delta^2\) units of time.

Indeed, from the last equation of system (2.2) we have
\[ y_3(\tau)< y_3^0-\delta^2\tau, \]
where \(\tau\) is counted from the instant of time starting from which the inequality \(y_1^2>\delta^2\) is satisfied. If \(\tau_1>y_3^0/\delta^2\), then \(y_3(\tau_1)\) will already be a negative quantity. Thus, during the interval of time \(0\le \tau\le \tau_1\), the point \(M(\tau)\) falls either on the plane \(y_3=0\) or into the domain \(|y_1|<\delta\).

Lemma 3. If the conditions \(|y_1^0|<\delta,\ |y_2^0|<\delta,\ 0<y_3^0<\delta\) are satisfied, where \(y_1^0,\ y_2^0,\ y_3^0\) are the coordinates of the initial point \(M_0\), then the solution of system (2.2) satisfies, in the domain \(y_3>0\), the inequalities
\[ 1)\ -2\delta<y_1(\tau)<5\delta,\quad 2)\ -\delta<y_2(\tau)<4\delta,\quad 3)\ 0<y_3(\tau)<\delta . \]

Proof. From the last equation of system (2.2) there follows directly the validity of the third of the inequalities to be proved. To prove inequality 2), suppose that \(y_3^0>0\). Since \(y_1(\tau)\) and \(y_2(\tau)\) can only increase, denote by \(\tau_0\) the instant of time when \(y_1(\tau_0)=\delta\), and by \(\tau_0'\) the instant of time when \(y_2(\tau_0')=\delta\). Denote by \(\tau_1\) the instant of encounter of the point \(M(\tau)\) with the plane \(y_3=0\).

By virtue of Lemma 2, for \(\tau_1=\infty\) we have that \(|y_1(\tau)|<\delta\) for all \(\tau>0\); if \(\tau_0'=\infty\), then inequality 2) will hold. Two cases are possible. In the first case suppose that \(\tau_0\le \tau_0'\). From the second equation of system (2.2) it follows, for \(\tau>\tau_0'\), that
\[ y_2(\tau)-y_2(\tau_0')\le y_3^0(\tau-\tau_0'), \]
but since, by Lemma 2, \(\tau-\tau_0'<\tau_1-\tau_0'<1\), we obtain \(y_2(\tau)<2\delta\) for all \(\tau\) in the interval \(\tau_0'\le \tau\le \tau_1\). In the second case suppose that \(\tau_0'<\tau_0\), and estimate the difference \(\tau_0-\tau_0'\). Then from the second equation of system (2.2) we obtain
\[ \delta<y_2(\tau)<y_3^0(\tau-\tau_0')+\delta \quad \text{for } \tau>\tau_0'. \tag{2.3} \]
The left-hand side of inequality (2.3) gives the estimate
\[ \delta(\tau_0-\tau_0')+y_1(\tau_0')<y_1(\tau_0), \]
whence it follows that \(\tau_0-\tau_0'<2\). The right-hand side of inequality (2.3) gives the estimate \(y_2(\tau)<\delta(\tau-\tau_0+\tau_0-\tau_0')+\delta<4\delta\), since \(\tau-\tau_0<1\) according to Lemma 2.

Consider the case when \(y_2^0<0\). If \(y_2(\tau)\) does not change sign, then inequality 2) is fulfilled. If \(y_2(\tau)\) changes sign at the instant of time \(\tau_3\), then two cases are possible.

In the case \(|y_1(\tau_3)| < \delta\) for \(\tau > \tau_3\), the point \(M(\tau)\) falls under the conditions considered above. If, however, \(y_1(\tau_3) > -\delta\), then, by Lemma 2, the point \(M(\tau)\) cannot remain outside the strip \(|y_1| < \delta\) for more than one unit of time; during this time \(y_2(\tau)\) can increase by no more than \(\delta\), and therefore the point \(M(\tau)\) again enters the region \(|y_1| < \delta,\ |y_2| < \delta\) and, consequently, will satisfy estimate 2).

Finally, let us show the validity of inequality 1). If the point \(M(\tau)\) enters the region \(y_1 > \delta\), then it will not return to the strip \(|y_1| < \delta\) and must reach the plane \(y_3 = 0\). But in this case, from the first equation of system (2.2) it follows that

\[ y_1(\tau) - y_1(\tau_0) < y_2(\tau)(\tau - \tau_0) \quad \text{for } \tau_0 \leq \tau \leq \tau_1 . \]

From Lemma 2 it follows that \(\tau - \tau_0 < 1\), and from inequality 2) we have \(0 < y_2(\tau) < 4\delta\). Thus, \(-\delta < y_1(\tau) < 5\delta\) for \(\tau > \tau_0\).

If the point \(M(\tau)\) enters the region \(y_1(\tau) < -\delta\) at the time \(\tau_4\), then either it reaches the plane \(y_3 = 0\) without returning to the strip \(|y_1| < \delta\), or it returns to this strip. In the first case we obtain \(y_1(\tau) > -2\delta\). In the second, the situation considered earlier is obtained.

The lemma is proved.

For the transition from system (2.2) to the system of differential equations (2.1), we shall need the following general result.

Along with the equation

\[ \dot{x} = X(x,t) \tag{2.4} \]

consider the equation

\[ \dot{y} = X(y,t) + R(y,t), \tag{2.5} \]

where \(x,\ y\) are \(n\)-dimensional vectors; \(X(x,t),\ R(x,t)\) are vector functions. Suppose that in some region of phase space, on the time interval \(t_0 \leq t \leq t_0 + T\), the conditions

\[ \|X(x,t) - X(y,t)\| \leq L\|x-y\| \tag{2.6} \]

and

\[ \|R(y,t)\| \leq M\|y\| \tag{2.7} \]

are satisfied.

Suppose further that, on the time interval under consideration, the solution \(x(t)\) of system (2.4) satisfies the inequality

\[ \|x(t)\| \leq \varepsilon . \tag{2.8} \]

Lemma 4. Suppose that conditions (2.6), (2.7), and (2.8) are satisfied on the time interval \(t_0 \leq t \leq t_0 + T\). Then the estimate

\[ \|y(t)-x(t)\| \leq \frac{M\varepsilon}{M+L} \left[\exp (M+L)(t-t_0) - 1\right], \tag{2.9} \]

holds, where \(x(t)\) and \(y(t)\) are the corresponding solutions of equations (2.4) and (2.5), determined by identical initial conditions.

Proof. Starting from the inequality

\[ \|y(t)-x(t)\| \leq \int_{t_0}^{t} \left[\|X(y,t)-X(x,t)\|+\|R(y,t)\|\right]\,dt, \]

we obtain

\[ \|y(t)-x(t)\| \leq \int_{t_0}^{t} \left[L\|y(t)-x(t)\|+M\|y\|\right]\,dt . \]

Since

\[ \|y(t)\|\leqslant \|y(t)-x(t)\|+\|x(t)\| \quad \text{and} \quad \|x(t)\|\leqslant \varepsilon, \]

we obtain

\[ \|y(t)-x(t)\|\leqslant \int_{t_0}^{t} [M\varepsilon+(M+L)\|y(t)-x(t)\|]\,dt . \]

Inequality (2.9) follows directly from Lemma 1.1 ([6], Chapter 1). Along with system (2.2), consider the system

\[ \frac{dy_1}{d\tau}=y_2,\qquad \frac{dy_2}{d\tau}=y_3+\rho\varphi_1(y_1,y_2),\qquad \frac{dy_3}{d\tau}=-\alpha y_1^2+\rho F_1(y_1,y_2,y_3,\tau). \tag{2.10} \]

Let the functions \(\varphi_1(y_1,y_2)\) and \(F_1(y_1,y_2,y_3,\tau)\) in the domain \(|y_1|<\infty,\ |y_2|<\infty,\ y_3>0,\ 0\leqslant \tau\) satisfy the conditions

\[ |\varphi_1(y_1,y_2)|<L_1(|y_1|+|y_2|), \]

\[ |F_1(y_1,y_2,y_3,\tau)|<L_2(|y_1|+|y_2|+|y_3|), \tag{2.11} \]

where \(L_1,\ L_2\) are certain constants, with \(L_1>0,\ L_2>0\).

Lemma 5. Every solution of system (2.10), determined by the initial point \(M_0(y_1^0,y_2^0,y_3^0)\) for \(\tau=0\), lying in the domain \(|y_1|<\delta,\ |y_2|<\delta,\ 0<y_3<\delta\), satisfies in the half-space \(y_3>0\) the inequalities

\[ |y_1|<5\delta+A_1\rho\delta,\qquad |y_2|<4\delta+A_2\rho\delta,\qquad y_3<\delta+A_3\rho\delta . \]

The validity of Lemma 5 follows directly from Lemma 4. Indeed, from Lemma 3 it follows (as is seen from the proof of Lemma 3) that the point \(M(\tau)\), moving according to system (2.2), remains outside the domain \(|y_1|<\delta,\ |y_2|<\delta,\ y_3>0\) for no more than three units of time; moreover, it does not leave the domain

\[ |y_1|<5\delta,\qquad |y_2|<4\delta,\qquad 0<y_3<\delta . \]

Hence it follows that the norm of the vector \(P(0,\rho\varphi_1,\rho F_1)\), characterizing the deviation of systems (2.2) and (2.10), can, taking (2.11) into account, be estimated as follows: \(\|P\|\leqslant N_1\rho\|F\|\), where \(F\) denotes the vector with projections \(y_1,y_2,y_3\). In addition, according to Lemma 3, we have \(\|F\|\leqslant N_2\delta\). Thus, setting \(M=N_1\rho\) and \(\varepsilon=N_2\delta\), we can apply the estimate of Lemma 4 and obtain an estimate for the deviations of the solutions of systems (2.2) and (2.10). The required result will follow from this estimate.

Theorem 4. Let \(c_1>0,\ c_2>0\), and, in addition, let the conditions for the existence of sliding be satisfied. Then there exists a positive number \(K_0\) such that for \(K>K_0\) the equilibrium position of system (2.1) will be asymptotically stable for arbitrary initial disturbances.

Proof. From Lemmas 1 and 5 it follows that, for sufficiently large \(K\), any point of the phase space either reaches the switching plane \(c_2x_1+c_1x_2+x_3=0\) in finite time and will continue its motion according to the differential equations

\[ \dot{x}_1=x_2,\qquad \dot{x}_2=-c_2x_1-c_1x_2, \]

whose zero solution is asymptotically stable for arbitrary initial disturbances, or it reaches the origin. Indeed, in the sec-

case the solution of system (2.2), tending to zero as $\tau \to \infty$, is surrounded by a tube of sufficiently small radius $\varepsilon_1$. Then if the point of the phase space leaves the tube, it will find itself under the conditions of the case considered above and, consequently, will fall onto the switching plane. If, however, the representative point of the phase space remains inside the tube, then it will fall onto the switching plane in some neighborhood of the origin.

References

1. Dolgolenko Yu. V. Proceedings of the 2nd All-Union Conference on the Theory of Automatic Control. Academy of Sciences of the USSR, 1, 1955, pp. 421—438.

2. Malkin I. G. Theory of Stability of Motion. Moscow—Leningrad, Gostekhizdat, 1952.

3. Aizerman M. A., Gantmakher F. R. Proceedings of the 1st International Congress IFAC. Academy of Sciences of the USSR, 1961, pp. 679—688.

4. Gerashchenko E. I. Izv. AN SSSR, Technical Cybernetics, No. 4, 157—163, 1963.

5. Barbashin E. A., Tabueva V. A. Applied Mathematics and Mechanics, 27, 4, 664—671, 1963.

6. Barbashin E. A. Introduction to Stability Theory. “Nauka,” 1967.

Received by the editors
January 3, 1967

Institute of Mathematics, Academy of Sciences of the BSSR,
Ural State University named after A. M. Gorky