UDC 517.919

CONSTRUCTION OF A SET OF SYSTEMS OF DIFFERENTIAL EQUATIONS HAVING PRESCRIBED INTEGRALS

R. G. MUKHARLYAMOV

In the work of N. P. Erugin [1] a method is given for constructing a set of systems of differential equations having a prescribed integral curve in the plane, and a detailed analysis of the constructed system is carried out. Making use of the nonuniqueness of the solution of the problem of constructing systems of differential equations from prescribed integrals, from the set obtained one can select systems satisfying additional conditions [2, 3]. Thus, in [4, 5] sets of systems of differential equations of stable motion along a prescribed curve in the plane are constructed. The construction of various special classes of systems of differential equations having prescribed integrals is also the subject of the works [6, 7]. Below a set of systems of differential equations having prescribed integral manifolds is constructed by the method proposed in [1], and the construction of systems from the condition of stability of these manifolds is determined.

§ 1. DEFINITION OF THE STRUCTURE OF THE SET OF SYSTEMS OF DIFFERENTIAL EQUATIONS

1. Let, in some domain \(G\) of the phase space \(X_n\) of the variables \(x_1,\ldots,x_n\), the equations

\[ \omega(x,t)=0, \tag{1} \]

where \(\omega\) is an \(s\)-dimensional and \(x\) an \(n\)-dimensional vector \((s \leq n)\), define “moving” hypersurfaces \(\Omega_1(t),\ldots,\Omega_s(t)\). Some of these hypersurfaces may have common parts \(M_{i_1 i_2 \ldots i_r}(t)\) \((i_1 < i_2 < \cdots < i_r \leq s)\). Assuming that for any \(t\) the functions \(\omega_1,\ldots,\omega_s\) are everywhere continuous in the domain \(G\) and possess continuous partial derivatives

\[ \frac{\partial \omega_i}{\partial t},\quad \frac{\partial \omega_i}{\partial x_j} \quad (i=1,\ldots,s;\ j=1,\ldots,n) \]

and that the rank of the functional matrix

\[ D=\frac{D(\omega_1,\ldots,\omega_s)}{D(x_1,\ldots,x_n)} \]

is equal to \(s\), it is required to construct a set of systems of differential equations

\[ \frac{dx}{dt}=P(x,t), \tag{2} \]

where \(P\) is an \(n\)-dimensional vector function, for which all the manifolds \(M_{i_1 i_2 \ldots i_r}(t)\) and hypersurfaces \(\Omega_{l_1}(t),\ldots,\Omega_{l_q}(t)\) \((l_1 \leq l_2 \leq \cdots \leq l_q \leq s)\) are integral.

For simplicity of reasoning, we shall assume that the common part \(M(t)=M_{12\ldots l}(t)\) is possessed only by the hypersurfaces \(\Omega_1(t), \Omega_2(t), \ldots, \Omega_l(t)\), and determine the structure of the set of systems of differential equations for which the manifold \(M(t)\) and each of the hypersurfaces \(\Omega_2(t), \ldots, \Omega_s(t)\) are integral. To this end we form the expressions

\[ \frac{\partial \omega_i}{\partial t} + \sum_{k=1}^{n} \frac{\partial \omega_i}{\partial x_k} P_k = \Phi_i \qquad (i=1,\ldots,s), \tag{3} \]

where \(\Phi_i\) are arbitrary continuous functions of the variables \(x_1,\ldots,x_n,t\).

To solve the problem posed, certain conditions must be imposed on these functions. Namely, since the \(n-l\)-dimensional “moving” manifold \(M(t)\) is to be an integral manifold for the system (2), the left-hand sides of the first \(l\) equalities (3) must vanish on this manifold. Consequently, the right-hand sides must be subject to the conditions that, for any \(t\),

\[ \left. \Phi_i(x,t) \right|_{\omega_1=\cdots=\omega_l=0} =0 \qquad (i=1,2,\ldots,l). \]

A function satisfying this condition will be denoted by

\[ \Phi_i = \Phi_i(\omega_1,\ldots,\omega_l; x_1,\ldots,x_n,t) = \Phi_i(\omega_1,\ldots,\omega_l; x,t) \]

\[ (i=1,\ldots,l). \]

At the remaining points of the domain \(G\), in particular at the points of the hypersurface \(\Omega_1(t)\) not belonging to the manifold \(M(t)\), this function may be arbitrary. In order that each of the hypersurfaces \(\Omega_2(t),\ldots,\Omega_s(t)\) can be integral for the system (2), the left-hand sides of all equalities (3), except the first, must vanish on the corresponding hypersurface. Consequently, for the right-hand sides, for any \(t\), the conditions must be satisfied

\[ \left. \Phi_i(x,t) \right|_{\omega_i=0} =0. \]

A function satisfying this condition will be denoted as follows:

\[ \Phi_i = \Phi_i(\omega_i; x_1,\ldots,x_n,t) = \Phi_i(\omega_i; x,t) \qquad (i=2,\ldots,s). \]

Thus, the functions \(\Phi_1,\ldots,\Phi_s\) must be chosen so that

\[ \begin{aligned} \Phi_1 &= \Phi_1(\omega_1,\ldots,\omega_l; x_1,\ldots,x_n,t),\\ \Phi_i &= \Phi_i(\omega_i; x_1,\ldots,x_n,t) \qquad (i=2,\ldots,s). \end{aligned} \tag{4} \]

Now, when the conditions have been imposed on the right-hand sides of the equalities (3), let us determine the functions \(P_1,\ldots,P_n\). Assuming that

\[ \Delta^{(s)} = \left| \begin{array}{ccc} \dfrac{\partial \omega_1}{\partial x_1} & \ldots & \dfrac{\partial \omega_1}{\partial x_s} \\ \cdot & \cdot & \cdot \\ \dfrac{\partial \omega_s}{\partial x_1} & \ldots & \dfrac{\partial \omega_s}{\partial x_s} \end{array} \right| \ne 0, \]

and taking \(P_{s+1},\ldots,P_n\) to be arbitrary continuous functions of the variables \(x_1,\ldots,x_n\), we obtain from (3)

\[ P_j=\frac{1}{\Delta^{(s)}}\left[\sum_{i=1}^{s}\left(\Phi_i-\frac{\partial\omega_i}{\partial t}\right)\Delta^{(s)}_{ij}-\sum_{k=s+1}^{n}\Delta^{(s)jk}P_k\right]\quad (j=1,\ldots,s), \]

where \(\Delta^{(s)}_{ij}\) is the cofactor of the element \(\dfrac{\partial\omega_i}{\partial x_j}\) of the determinant \(\Delta^{(s)}\), and \(\Delta^{(s)jk}\) is the determinant obtained from \(\Delta^{(s)}\) by replacing its \(j\)-th column by the \(k\)-th column of the matrix \(D\). Thus, the desired set of systems of differential equations is obtained in the form

\[ \frac{dx_j}{dt}=\frac{1}{\Delta^{(s)}}\left[\sum_{i=1}^{s}\left(\Phi_i-\frac{\partial\omega_i}{\partial t}\right)\Delta^{(s)}_{ij}-\sum_{k=s+1}^{n}\Delta^{(s)jk}P_k\right]\quad (j=1,\ldots,s), \]

\[ \frac{dx_m}{dt}=P_m\quad (m=s+1,\ldots,n). \tag{5} \]

It contains \(s\) arbitrary continuous functions \(\Phi_1,\ldots,\Phi_s\), chosen according to (4), and \(n-s\) completely arbitrary continuous functions \(P_{s+1},\ldots,P_n\) of the variables \(x_1,\ldots,x_n,t\). If the given functions (1) do not depend explicitly on \(t\),

\[ \omega(x)=0, \tag{6} \]

then, choosing the functions \(\Phi_1,\ldots,\Phi_s,P_{s+1},\ldots,P_n\) also independent of \(t\),

\[ \Phi_1=\Phi_1(\omega_1,\ldots,\omega_s;x), \]

\[ \Phi_j=\Phi_j(\omega_j;x)\quad (j=2,\ldots,s), \]

\[ P_k=P_k(x)\quad (k=s+1,\ldots,n) \]

and taking into account that \(\dfrac{\partial\omega_i}{\partial t}=0\) \((i=1,\ldots,s)\), we obtain a system of the form

\[ \frac{dx_j}{dt}=\frac{1}{\Delta^{(s)}(x)} \left[\sum_{i=1}^{s}\Phi_i(x)\Delta^{(s)}_{ij}(x)- \sum_{k=s+1}^{n}\Delta^{(s)jk}(x)P_k(x)\right]\quad (j=1,\ldots,s), \tag{7} \]

\[ \frac{dx_m}{dt}=P_m(x)\quad (m=s+1,\ldots,n). \]

1. Suppose that all the hypersurfaces \(\Omega_1,\ldots,\Omega_s\), given respectively by equations (6), have a common part \(\Omega\). The set of systems of differential equations having an \((n-s)\)-dimensional integral manifold \(\Omega\) is written in the form (7), where the functions \(\Phi_i(x)\) are chosen as

\[ \Phi_i=\Phi_i(\omega,x)\quad (i=1,\ldots,s). \]

If at some point \(x'\) of the manifold \(\Omega\) all the functions \(P_{s+1}(x),\ldots,P_n(x)\) simultaneously vanish, and \(\Delta^{(s)}(x')\ne0\), then it is an equilibrium point for system (7).

If all determinants of order \(s\) of the matrix \(D\), except \(\Delta^{(s)}\), vanish on some set of points \(A\subset\Omega\), then on this set \(A\) the coordinates \(x_1,\ldots,x_s\) do not change, provided that at all points of the set \(A\) the functions \(P_{s+1}(x),\ldots,P_n(x)\) are continuous and bounded. This phenomenon can be avoided by choosing the functions \(P_{s+1}(x),\ldots,P_n(x)\) in the form

\[ P_k(x)=Q_k(x)\,\frac{1}{\prod_{j=1}^{s}\Delta^{(s)jk}}\quad (k=s+1,\ldots,n), \]

where \(Q_k(x)\) are bounded and nonzero everywhere on the set \(A\). But in this case the Lipschitz condition will not be satisfied, and the set \(A\) may consist of singular points. A singular point \(x'\) may also appear as a result of the fact that at the point \(x'\) the determinant \(\Delta^{(s)}\) vanishes. This singular point can be excluded by choosing the functions \(\Phi_i, P_k\) in the form of the products

\[ \Phi_i\Delta^{(s)},\quad P_k\Delta^{(s)}\quad (i=1,\ldots,s;\ k=s+1,\ldots,n). \]

If in some part \(B\) of the manifold \(\Omega\) one has

\[ \sum_{j=1}^{m}\left(\sum_{k=s+1}^{n}\Delta^{(s)jk}P_k\right)^2=0, \]

then in the domain \(B\) the coordinates \(x_1,\ldots,x_m\) do not change.

If the functions \(\Phi_1,\ldots,\Phi_m\) are identically equal to zero,

\[ \Phi_i\equiv 0\quad (i=1,\ldots,m), \]

then every manifold defined by the intersection of the hypersurfaces given by the equations

\[ \omega_i(x)+c_i=0\quad (i=1,\ldots,m), \]
\[ \omega_k(x)=0\quad (k=m+1,\ldots,s) \]

(provided that it exists) is an integral manifold for the system (7).

3. If in system (7) the functions \(\Phi_i\) are chosen as

\[ \Phi_i=\Phi_i(\omega_i;\,x)\quad (i=1,\ldots,s), \]

then each of the hypersurfaces \(\Omega_1,\ldots,\Omega_s\) will be integral for it.

In order that some function

\[ \omega_{s+1}(x)=0, \tag{8} \]

depending on the functions (6):

\[ \omega_{s+1}=\omega_{s+1}(\omega_1,\ldots,\omega_s), \tag{9} \]

also be an integral of the system (7), it is sufficient to require that the functions \(\Phi_1,\ldots,\Phi_s\) satisfy the additional condition

\[ \left.\sum_{i=1}^{s}\frac{\partial \omega_{s+1}}{\partial \omega_i}\Phi_i\right|_{\omega_{s+1}=0}=0. \tag{10} \]

Indeed, taking into account (9) and the equalities

\[ \frac{d\omega_i}{dt}=\Phi_i\quad (i=1,\ldots,s), \]

we shall have

\[ \frac{d\omega_{s+1}}{dt} =\sum_{i=1}^{s}\frac{\partial \omega_{s+1}}{\partial \omega_i}\frac{d\omega_i}{dt} =\sum_{i=1}^{s}\frac{\partial \omega_{s+1}}{\partial \omega_i}\Phi_i, \]

i.e., under condition (10), the function (8) is an integral of system (7).

In the case

\[ \sum_{i=1}^{s}\frac{\partial \omega_{s+1}}{\partial \omega_i}\Phi_i \equiv 0 \]

the function

\[ \omega_{s+1}(x)=c, \]

where \(c\) is an arbitrary constant, is a first integral of system (7).

1. Let now the matrix \(D\) be such that

\[ \Delta^{(s)}\ne 0,\quad \Delta^{(s)jk}\equiv 0\quad (j=1,\ldots,s;\ k=s+1,\ldots,n). \]

This can occur, for example, when the collection of functions \(\omega_1,\ldots,\omega_s\) depends only on \(s\) coordinates \(x_1,\ldots,x_s\). Then, choosing the functions \(\Phi_i\) also to depend only on \(x_1,\ldots,x_s\), we obtain the desired system, which will consist of two parts:

\[ \frac{dx_j}{dt}=\frac{1}{\Delta^{(s)}}\sum_{i=1}^{s}\Delta_{ij}^{(s)}\Phi_i(\omega_i;x)\quad (j=1,\ldots,s), \tag{11} \]

\[ \frac{dx_k}{dt}=P_k(x)\quad (k=s+1,\ldots,n). \tag{11'} \]

The first part does not depend on the second, which is completely arbitrary. A system of the form (11) is also obtained in the case \(s=n\), i.e., a system whose integrals are the functions

\[ \omega_i(x)=0\quad (i=1,\ldots,n), \tag{12} \]

can be written in the form

\[ \frac{dx_j}{dt}=\sum_{i=1}^{n}\Delta_{ij}^{(n)}\Phi_i(\omega_i;x)\quad (j=1,\ldots,n). \tag{13} \]

Here the factor \(\frac{1}{\Delta^{(n)}}\) is included in the functions \(\Phi_i\) \((i=1,\ldots,n)\).

Now system (13) may be regarded as a system of general form, independently of the fulfillment of the condition \(\Delta^{(n)}\ne 0\).

In order that on the hypersurfaces \(\Omega_1,\Omega_2,\ldots,\Omega_n\), given by equations (12), there be no equilibrium points or singular points, it is necessary that

\[ \left.\sum_{\substack{i=1\\ i\ne j}}^{n}\left(\Delta_{ij}^{(n)}\Phi_i\right)^2\right|_{\omega_j=0}\ne 0. \]

1. Suppose we have a system (13) for which the hypersurfaces \(\Omega_1,\ldots,\Omega_n\), given by equations (12), are integral. Let us take one more hypersurface \(\Omega_{n+1}\), given by the equation

\[ \omega_{n+1}(x)=0 \tag{14} \]

and located between the hypersurfaces (12), and let us investigate how the functions \(\Phi_i(\omega_i;x)\) \((i=1,\ldots,n)\) must be chosen so that \(\Omega_{n+1}\) is also an in-

integral hypersurface for system (13). If (14) is an integral hypersurface, then it must satisfy

\[ \sum_{j=1}^{n}\sum_{i=1}^{n}\frac{\partial \omega_{n+1}}{\partial x_j}\Delta_{ij}^{(n)}\Phi_i(\omega_i;\,x)=\Phi_{n+1}(\omega_{n+1};\,x). \tag{15} \]

Otherwise, (15) can be written as

\[ \sum_{i=1}^{n}\nabla^{(n)in+1}\Phi_i(\omega_i;\,x)=\Phi_{n+1}(\omega_{n+1};\,x), \tag{16} \]

where \(\nabla^{(n)in+1}\) is obtained by replacing the \(i\)-th row of the determinant \(\Delta^{(n)}\) by the row

\[ \frac{\partial \omega_{n+1}}{\partial x_1},\,\ldots,\,\frac{\partial \omega_{n+1}}{\partial x_n}. \]

We shall assume that the functions \(\omega_1,\ldots,\omega_n\) are single-valued and that \(\Delta^{(n)}\ne 0\) in the domain \(G\). Under these conditions, from the system of equations

\[ \omega_i=\omega_i(x_1,\ldots,x_n)\qquad (i=1,\ldots,n) \]

one can find

\[ x_i=x_i(\omega_1,\ldots,\omega_n)\qquad (i=1,\ldots,n) \]

and, substituting them into (14), regard \(\omega_{n+1}\) as a function of \(\omega_1,\ldots,\omega_n\). The determinant \(\nabla^{(n)in+1}\) is transformed as follows:

\[ \nabla^{(n)in+1} = \sum_{j=1}^{n}\frac{\partial \omega_{n+1}}{\partial \omega_j}\nabla^{(n)ij} = \frac{\partial \omega_{n+1}}{\partial \omega_i}\Delta^{(n)}, \]

and condition (16) takes the form

\[ \Delta^{(n)}\sum_{i=1}^{n}\frac{\partial \omega_{n+1}}{\partial \omega_i}\Phi_i(\omega_i;\,x)=\Phi_{n+1}(\omega_{n+1};\,x). \]

Since \(\Delta^{(n)}\ne 0\), including it among the functions \(\Phi_1,\ldots,\Phi_n\), the last equality can be rewritten in the form

\[ \sum_{i=1}^{n}\frac{\partial \omega_{n+1}}{\partial \omega_i}\Phi_i(\omega_i;\,x)=\Phi_{n+1}(\omega_{n+1};\,x). \]

Hence we obtain the condition: in order that the hypersurface \(\Omega_{n+1}\) be integral for system (13), the functions \(\Phi_1(\omega_1;\,x), \Phi_2(\omega_2;\,x),\ldots,\Phi_n(\omega_n;\,x)\) must be chosen so that

\[ \sum_{i=1}^{n}\frac{\partial \omega_{n+1}}{\partial \omega_i} \Phi_i\bigl(\omega_i;\,x_1(\omega_1,\ldots,\omega_n),\,\ldots,\,x_n(\omega_1,\ldots,\omega_n)\bigr)=0 \]

when \(\omega_{n+1}(\omega_1,\ldots,\omega_n)=0\).

1. Let now the functions (12) be dependent and the rank of the matrix \(D\) equal to \(n-1\). The set of systems for which the functions (12) are integrals will be assumed constructed in the form (13). It can be shown that in this case any of the functions

\[ \omega_i(x)+c_i=0\qquad (i=1,\ldots,n), \tag{17} \]

where \(C_i\) are arbitrary constants, is a first integral of the system

\[ \frac{dx_j}{dt}=\sum_{i=1}^{n}\Delta_{ij}^{(n)}\Phi_i(x)\qquad (j=1,\ldots,n), \tag{18} \]

and no conditions are imposed on the functions \(\Phi_i\). Indeed, differentiating any of the functions (17), by virtue of system (18) we obtain

\[ \sum_{j=1}^{n}\frac{\partial \omega_s}{\partial x_j} \sum_{i=1}^{n}\Delta_{ij}^{(n)}\Phi_i = \sum_{i=1}^{n}\nabla^{(n)is}\Phi_i=0, \]

since for \(s\ne i\) the determinant \(\nabla^{(n)is}\) has two identical rows, while for \(s=i\), \(\nabla^{(n)ii}=\Delta^{(n)}=0\) by virtue of the dependence of the functions \(\omega_1,\ldots,\omega_n\).

Remark. In [1], for an \(n=2\) system whose integrals are the functions \(\omega_1(x_1,x_2)=0\), \(\omega_2(x_1,x_2)=0\), a system was obtained in the form (18) under the condition

\[ \Phi_i=\Phi_i(\omega_i;\ x_1,x_2)\qquad (i=1,2). \]

Let us see what conditions must be imposed in this case on the functions \(\Phi_1,\ldots,\Phi_n\) so that the function \(\omega_{n+1}(x)=0\) is also an integral of system (13).

Form the expression

\[ \sum_{j=1}^{n}\frac{\partial \omega_{n+1}}{\partial x_j}\frac{dx_j}{dt} = \sum_{i,j=1}^{n}\frac{\partial \omega_{n+1}}{\partial x_j}\Delta_{ij}^{(n)}\Phi_i = \sum_{i=1}^{n}\nabla^{(n)i\,n+1}\Phi_i. \tag{19} \]

Assuming that \(\omega_n=\omega_n(\omega_1,\ldots,\omega_{n-1})\), after transforming expression (19) we obtain

\[ \sum_{j=1}^{n}\frac{\partial \omega_{n+1}}{\partial x_j}\frac{dx_j}{dt} = \nabla^{(n)n\,n+1} \left[ \Phi_n-\sum_{i=1}^{n-1}\frac{\partial \omega_{n+1}}{\partial \omega_i}\Phi_i \right]. \tag{20} \]

Hence we obtain that if the function \(\omega_{n+1}\) depends on \(\omega_1,\ldots,\omega_{n-1}\), namely \(\omega_{n+1}=\omega_{n+1}(\omega_1,\ldots,\omega_{n-1})\), then \(\nabla^{(n)n\,n+1}=0\), and expression (20) is identically zero. Consequently, the function

\[ \omega_{n+1}(x)-c=0 \tag{21} \]

for any constant \(c\) is a first integral of system (13). If, however, the functions \(\omega_1(x),\ldots,\omega_{n-1}(x),\omega_{n+1}(x)\) are independent, then in order that the function \(\omega_{n+1}(x)=0\) be an integral of system (18), the condition

\[ \left. \Phi_n(x)-\sum_{i=1}^{n-1}\frac{\partial \omega_{n+1}}{\partial \omega_i}\Phi_i(x) \right|_{\omega_{n+1}(x)=0} =0 \tag{22} \]

must be fulfilled.

In the case where expression (22) is identically equal to zero, the function (21) is a first integral.

1. Let us construct a set of systems of differential equations that have the hypersurfaces \(\Omega_1,\ldots,\Omega_{n-1},\Omega_n\), specified respectively by the equations

\[ \begin{gathered} \omega_1(x)=0,\\ \cdots\\ \omega_{n-1}(x)=0,\\ \omega_n(x)=\omega_1(x)-c=0, \end{gathered} \tag{23} \]

where \(c\) is a fixed constant, as integrals. The functions \(\omega_1,\ldots,\omega_{n-1}\) are regarded as independent. The problem can be solved in two ways. The desired set of systems may be constructed analogously to (18). It will have the form

\[ \frac{dx_j}{dt}=\Delta^{(n)}_{ij}\Phi_1(x)+\Delta^{(n)}_{nj}\Phi_2(x) \qquad (j=1,\ldots,n), \tag{24} \]

since \(\Delta^{(n)}_{ij}\equiv 0\) \((i=2,\ldots,n-1;\ j=1,\ldots,n)\). Then any of the functions

\[ \omega_i(x)+c_i=0 \qquad (i=1,\ldots,n-1), \]

where \(c_i\) are arbitrary constants, is a first integral of system (24) in the entire domain \(G\), where the functions \(\Phi_1,\Phi_2\) are defined.

One may propose yet another method. Let us introduce into consideration a function
\(\omega=\omega(\omega_1,\ldots,\omega_{n-1})\), which vanishes when \(\omega_1=0\) and \(\omega_1=c\), for example,

\[ \omega=\omega_1(\omega_1-c)\varphi(\omega_2,\ldots,\omega_{n-1}), \]

and construct a set of systems of differential equations having the integrals

\[ \begin{aligned} \omega(x)&=0,\\ \omega_2(x)&=0,\\ &\ldots\\ \omega_{n-1}(x)&=0 \end{aligned} \tag{25} \]

in the form (7). The system obtained will have as an integral any of the functions (23). If some hypersurfaces along which \(\omega(x)\) vanishes have a common part, then it consists either of points where the uniqueness condition is violated, or of points in which at least part of the coordinates remains unchanged.

Let some hypersurfaces \(\Omega_1,\ldots,\Omega_l\) among \(\Omega_1,\ldots,\Omega_s\), given by equations (6), have a common part \(M=M_{12\ldots l}\), and let system (7), where

\[ \Phi_i=\Phi_i(\omega_i;x) \qquad (i=1,\ldots,s), \]

satisfy the conditions of the theorem on uniqueness of solutions on the manifold \(M\). If the initial point \(x^0=x(t_0)\) belongs to the manifold \(M\), then under its subsequent motion by virtue of system (7) the point \(x=x(t)\) will not leave any of the hypersurfaces \(\Omega_1,\ldots,\Omega_l\), and hence will remain on the manifold \(M\). If, however, \(x^0\) does not belong to the manifold \(M\), but is located on one of the hypersurfaces \(\Omega_1,\ldots,\Omega_l\), then the point \(x(t)\), moving along the corresponding hypersurface, will not reach the manifold \(M\) for any finite value of \(t\).

Example 1. Construct a set of systems of differential equations whose integral manifold is the line of intersection of the sphere

\[ \omega_1(x,y,z)=x^2+y^2+z^2-R^2=0,\qquad R\ne 0 \tag{26} \]

and the elliptic paraboloid

\[ \omega_2(x,y,z)=\frac{x^2}{a^2}+\frac{y^2}{b^2}-z=0,\qquad 0\ne a\ne b\ne 0. \tag{27} \]

Form the matrix

\[ D= \begin{pmatrix} 2x & 2y & 2z\\ \dfrac{2x}{a^2} & \dfrac{2y}{b^2} & -1 \end{pmatrix}. \]

Put

\[ \Delta^{(2)}= \begin{vmatrix} 2y & 2z\\[2pt] \dfrac{2y}{b^2} & -1 \end{vmatrix} =-2y\left(1+\frac{2z}{b^2}\right). \]

By virtue of the assumptions concerning \(R,a,b\), \(\Delta^{(2)}\) does not vanish identically. According to (7), the desired set of systems is obtained in the form

\[ \begin{aligned} \frac{dx}{dt}&=P,\\ \frac{dy}{dt}&=\frac{1}{\Delta^{(2)}}\left[-\Phi_1-2z\Phi_2+2x\left(1+\frac{2z}{a^2}\right)P\right],\\ \frac{dz}{dt}&=\frac{2y}{\Delta^{(2)}}\left[\Phi_2-\frac{1}{b^2}\Phi_1-2x\left(\frac{1}{a^2}-\frac{1}{b^2}\right)P\right], \end{aligned} \tag{28} \]

where \(P=P(x,y,z)\), \(\Phi_1=\Phi_1(\omega_1,\omega_2,x,y,z)\), \(\Phi_2=\Phi_2(\omega_1,\omega_2,x,y,z)\).

For \(\Phi_1=\Phi_2=0\), \(P=-y\left(1+\dfrac{2z}{b^2}\right)\), one obtains the system

\[ \frac{dx}{dt}=-y\left(1+\frac{2}{b^2}z\right), \]

\[ \frac{dy}{dt}=x\left(1+\frac{2}{a^2}z\right), \]

\[ \frac{dz}{dt}=2\left(\frac{1}{a^2}-\frac{1}{b^2}\right)xy, \]

constructed in [7].

§ 2. CONSTRUCTION OF A SET OF SYSTEMS OF DIFFERENTIAL EQUATIONS HAVING A PRESCRIBED STABLE INTEGRAL MANIFOLD

Let the manifold \(\Omega\) be formed by the intersection of the hypersurfaces \(\Omega_1,\ldots,\Omega_s\), given respectively by the equations

\[ \omega_i(x)=0 \qquad (i=1,\ldots,s;\ s<n). \]

We shall assume that \(\Omega\) is a smooth, bounded manifold, i.e., the rank of the matrix \(D\) is everywhere on \(\Omega\) equal to \(s\), and there exists a ball of finite radius inside which the manifold \(\Omega\) lies. When these conditions are fulfilled, the set of points constituting \(\Omega\) is compact in the space \(X_n\) (\(X_n\) is \(n\)-dimensional Euclidean space).

Considering the distance \(\rho(y,\Omega)\) from a point \(y\in X_n\) to the manifold \(\Omega\) in the usual sense,

\[ \rho(y,\Omega)=\inf_{x\in\Omega}\rho(y,x) \]

and understanding by an \(\varepsilon\)-neighborhood of the manifold \(\Omega\) the set \(\Omega_\varepsilon\subset X_n\) for any point \(y\) of which the inequality \(\rho(y,\Omega)\leqslant\varepsilon\) holds, we introduce the notions of a stable and an unstable manifold.

Definition 1. The manifold \(\Omega\) is called stable if, for any \(\varepsilon>0\), one can find such an \(\eta>0\) (\(\eta\leqslant\varepsilon\)) that from \(x(t_0)=x^0\in\Omega_\eta\) it follows that \(x(t)=x\in\Omega_\varepsilon\) for any \(t>t_0\). Otherwise, the manifold \(\Omega\) is unstable.

Definition 2. The manifold \(\Omega\) is called asymptotically stable if it is stable and

\[ \lim_{t\to\infty}\rho(x(t),\Omega)=0. \]

Let us determine how the functions \(\Phi_i, P_m\) \((i=1,\ldots,s;\ m=s+1,\ldots,n)\), entering into the right-hand sides of system (7), should be chosen so that the manifold \(\Omega\) is stable (asymptotically) or unstable. In doing so we shall assume that the functions \(\Phi_i, P_m\) are chosen so as not to contradict the conditions for existence and uniqueness of solutions of system (7) in the domain \(\Omega_h\) \((h>\varepsilon)\).

Introduce into consideration a function \(V=V(\omega,x)\), single-valued, continuous, possessing continuous partial derivatives \(\dfrac{\partial V}{\partial \omega_i}, \dfrac{\partial V}{\partial x_j}\) \((i=1,\ldots,s;\ j=1,\ldots,n)\), and satisfying the conditions

\[ V(0,x)=0,\qquad \frac{\partial V(0,x)}{\partial x_j}=0\quad (j=1,\ldots,n) \tag{29} \]

in the domain \(\Omega_h\). Its derivative \(V'=\dfrac{dV(\omega,x)}{dt}\) by virtue of (3) and (7) is determined by the relation

\[ \begin{aligned} V'&=\sum_{i=1}^{s}\frac{\partial V}{\partial \omega_i}\frac{d\omega_i}{dt} +\sum_{j=1}^{n}\frac{\partial V}{\partial x_j}\frac{dx_j}{dt} \\ &=\sum_{i=1}^{s}\frac{\partial V}{\partial \omega_i}\Phi_i +\sum_{j=1}^{s}\frac{\partial V}{\partial x_j}\frac{1}{\Delta^{(s)}} \left[ \sum_{i=1}^{s}\Phi_i\Delta_{ij}^{(s)} -\sum_{k=s+1}^{n}\Delta^{(s)jk}P_k \right] \\ &\quad +\sum_{j=s+1}^{n}\frac{\partial V}{\partial x_j}P_j . \end{aligned} \tag{30} \]

From (29) and (30) it follows that \(V'(0,x)=0\).

Let us give the following definitions.

Definition 3. The function \(V\) is called sign-definite with respect to \(\Omega\) if it takes values of only one definite sign for any \(x\in\Omega_h'\) and vanishes only at points of the manifold \(\Omega\).

Definition 4. The function \(V\) is called sign-constant positive (negative) with respect to \(\Omega\) if it is nonnegative (nonpositive) in \(\Omega_h\).

Definition 5. The function \(V\) is called sign-changing with respect to \(\Omega\) if it can take both positive and negative values in \(\Omega_h\).

Now we can formulate and prove theorems that make it possible to judge the stability of the manifold \(\Omega\).

Theorem 1. The manifold \(\Omega\) is stable if one can find a positive-definite with respect to \(\Omega\) function \(V\), whose derivative \(V'\) is a sign-constant negative with respect to \(\Omega\) function.

Since the set of points constituting the manifold \(\Omega\) is compact, for it there exists a finite \(\varepsilon\)-net, however small \(\varepsilon>0\) may be \((\varepsilon<h)\). As an \(\varepsilon\)-net for the manifold \(\Omega\) we take its \(\varepsilon\)-neighborhood \(\Omega_\varepsilon\), bounded by a finite number of pieces \(\Gamma_{\varepsilon j}\) of the hypersurfaces \(\Omega_{\varepsilon j}\) \((j=1,\ldots\)

..., \(N\)), having no common points with \(\Omega\). Denote by \(\overline{\Gamma}_{\varepsilon j}\) the closure of the set of points constituting \(\Gamma_{\varepsilon j}\). As a closed and bounded set in \(X_n\), \(\overline{\Gamma}_{\varepsilon j}\) is compact, and the function \(V\) has on it a positive lower bound \(l_j>0\).

Let

\[ l=\min\{l_1,\ldots,l_N\}. \]

Consider an arbitrary solution \(x=x(t)\) of system (7), whose initial value \(x^0=x(t_0)\) belongs to some \(\eta\)-neighborhood \(\Omega_\eta\) of the manifold \(\Omega\), \((\eta<\varepsilon)\). We shall assume the number \(\eta\) to be so small that

\[ V(\omega(x^0),x^0)<l. \]

Since the function \(V\) does not increase in the domain \(\Omega_\varepsilon\), the solution \(x=x(t)\) of system (7), satisfying the initial condition \(x^0=x(t_0)\), cannot for any \(t>t_0\) reach the boundary of \(\Omega_\varepsilon\); consequently, it will remain in the \(\varepsilon\)-neighborhood of \(\Omega\).

Theorem 2. The manifold \(\Omega\) is asymptotically stable if there exists a function \(V\), positive definite with respect to \(\Omega\), whose derivative \(V'\) is negative definite with respect to \(\Omega\).

By Theorem 1 the manifold \(\Omega\) is stable. Consequently, there exists such an \(\eta(\varepsilon)\) that, if \(x^0\in\Omega_\eta\), then \(x\in\Omega_\varepsilon\) for every \(t>t_0\). Since \(V'\) is everywhere negative in \(\Omega_\varepsilon\) and vanishes only on the manifold \(\Omega\), then, by uniqueness of the solution of system (7), it cannot vanish for any finite \(t\) along the solution \(x(t)\). Then the function \(V\) decreases monotonically.

Suppose that \(V(\omega,x)>\alpha>0\). Then \(\rho(x(t),\Omega)>a>0\), i.e., there exists such an \(\Omega_a\) into which the point \(x(t)\) cannot fall. The set \(\Omega_{h-a}\) of points \(x\) for which \(a\leqslant\rho(x,\Omega)\leqslant h\) is closed and bounded, i.e., is compact, and the function \(V'\) attains on it the exact upper bound \(-b\). From the equality

\[ V(t)-V(t_0)=\int_{t_0}^{t} V'(t)\,dt \]

we have

\[ V(t)\leqslant V(t_0)-b(t-t_0) \]

for \(t>t_0\). Since for large \(t\) this contradicts the assumption that \(V\) is positive definite with respect to \(\Omega\), it follows that \(V(t)\to0\) as \(t\to\infty\). Consequently, \(\rho(x(t),\Omega)\to0\) as \(t\to\infty\), i.e., the manifold \(\Omega\) is asymptotically stable.

Theorem 3. The manifold \(\Omega\) is unstable if one can choose a function \(V\), not necessarily sign-definite with respect to \(\Omega\), which tends to zero together with \(\rho(x(t),\Omega)\), and whose derivative \(V'\) is sign-definite with respect to \(\Omega\), and if for arbitrarily small \(\eta\) the function \(V\) can assume in \(\Omega_\eta\) the sign of \(V'\).

We shall assume that \(V'\) is positive definite on the compact set \(\Omega_h\), containing the manifold \(\Omega\). Then \(V(t)>V(t_0)\) for every \(t>t_0\). According to the condition of the theorem, for any \(\eta>0\) one can choose such an \(x^0\) that \(x^0\in\Omega_\eta\) and \(V(\omega(x^0),x^0)>0\).

Let \(\mu=\inf \rho(x,\Omega)\) for all \(x\in\Omega_\eta\) such that \(V\geqslant V_0\). Since \(\eta\) is arbitrarily small, we may assume \(\eta<h\). The function \(V'\) on the compact set \(\Omega_{h-\mu}\) has the exact lower bound \(l\). Consequently,

\[ V(t)=V(t_0)+\int_{t_0}^{t} V'(t)\,dt \geqslant V(t_0)+l(t-t_0), \]

i.e., \(V\) increases without bound, whence it follows that \(x(t)\), for some \(t\), will leave the set \(\Omega_h\).

Remark. Choose the functions \(\Phi_i\) not depending explicitly on \(x_1,\ldots,x_n\):

\[ \Phi_i=\Phi_i(\omega)\quad (i=1,\ldots,s), \]

assuming here that the right-hand sides of system (7) satisfy the conditions of the existence and uniqueness theorem for solutions in the domain \(\Omega_h\). Then the question of stability of the manifold \(\Omega\) reduces to the investigation of the stability of the trivial solution of the system

\[ \frac{d\omega_i}{dt}=\Phi_i(\omega)\quad (i=1,\ldots,s). \]

Example 2. Let us return to Example 1. Choose the functions \(\Phi_1,\Phi_2,P\) entering into the right-hand sides of system (28) so that the motion along the line of intersection of the sphere (26) and the elliptic paraboloid (27) is asymptotically stable. To this end let

\[ \Phi_1=-2(\alpha x^2+\beta y^2)z\,\omega_1, \tag{31} \]

\[ \Phi_2=-(\gamma x^2+\delta y^2)\omega_2, \tag{32} \]

\[ P=-\frac{a^2xz(\alpha\omega_1+\gamma\omega_2)}{a^2+2z} +a^2y^m(b^2+2z)f(x,y,z), \]

where \(f(x,y,z)\) is an arbitrary function and the constants \(\alpha,\beta,\gamma,\delta,m\) satisfy the conditions: \(\alpha>0\), \(\beta>0\), \(\gamma>0\), \(\delta>0\), \(m>1\). System (28) is written in the form

\[ \frac{dx}{dt} =-\frac{a^2xz(\alpha\omega_1+\gamma\omega_2)}{a^2+2z} +a^2y^m(b^2+2z)f(x,y,z), \]

\[ \frac{dy}{dt} =-\frac{b^2yz(\beta\omega_1+\delta\omega_2)}{b^2+2z} -b^2xy^{m-1}(a^2+2z)f(x,y,z), \tag{33} \]

\[ \frac{dz}{dt} =\frac{x^2(a^2\gamma\omega_2-2\alpha z\omega_1)}{a^2+2z} +\frac{y^2(b^2\delta\omega_2-2\beta z\omega_1)}{b^2+2z} +2(b^2-a^2)xy^m f(x,y,z), \]

where in place of \(\omega_1\) and \(\omega_2\) their expressions in terms of \(x,y,z\) must be substituted:

\[ \omega_1=x^2+y^2+z^2-R^2,\qquad \omega_2=\frac{x^2}{a^2}+\frac{y^2}{b^2}-z, \]

and \(\alpha,\beta,\gamma,\delta,m\) and \(f(x,y,z)\) must be chosen so that, on the integral line and in some neighborhood of it, system (33) has no singular points. Putting \(V=\omega_1^2+\omega_2^2\), taking (31), (32) into account, we obtain

\[ V'=2\omega_1\Phi_1+2\omega_2\Phi_2 =-4z(\alpha x^2+\beta y^2)\omega_1^2 -2(\gamma x^2+\delta y^2)\omega_2^2. \]

By virtue of the conditions \(R\ne 0\), \(a\ne 0\), \(b\ne 0\), there exists a domain

\[ z>\varepsilon_1>0,\qquad x^2+y^2>\varepsilon_2>0, \]

where \(V'\) is negative definite; whence it follows that system (33) will describe an asymptotically stable motion along the line of intersection of a sphere and an elliptic paraboloid.

References

1. N. P. Erugin, PMM, vol. XVII, no. 6, 1952.
2. A. S. Galiullin, Some Problems of Stability of Programmed Motion. Kazan, Tatknigoizdat, 1960.
3. A. S. Galiullin, On problems of dynamic programming. Transactions of UDN, vol. V. Theoretical Mechanics, issue II. Moscow, 1964.
4. I. A. Mukhametzyanov, Construction of a set of systems of differential equations of stable motion according to a prescribed program. Transactions of UDN, vol. I. Theoretical Mechanics, issue I. Moscow, 1963.
5. I. A. Mukhametzyanov, Some problems of stability of motion arising in the construction of the control law for an object under programmed regulation. Transactions of UDN, vol. V. Theoretical Mechanics, issue II. Moscow, 1964.
6. M. I. Almukhamedov, On the construction of a differential equation whose limit cycles are prescribed curves. Izv. vuzov, Mathematics, no. 1, 1965.
7. M. B. Ignat’ev, Holonomic Automatic Systems. Moscow–Leningrad, Publishing House of the Academy of Sciences of the USSR, 1963.

Received by the editors
31 July 1965

Peoples’ Friendship University
named after Patrice Lumumba