UDC 518.517.91/94

A. I. Kononenko, V. D. Milman

A Numerical Method for Finding Partially Stable Singular Points of Ordinary Differential Equations

(Presented by Academician L. V. Kantorovich, March 4, 1968)

1. In the note \((^1)\) an algorithm \(\mathfrak{A}=\mathfrak{A}(\{A_n\}_1^\infty;\ \{s_n\}_0^\infty;\ \{a_n\}_0^\infty)\) was constructed for control by the numerical Euler method, for finding an asymptotically stable singular point of a system of ordinary differential equations. The study of this algorithm was based on the converse of A. M. Lyapunov’s theorem obtained by I. G. Malkin \((^2)\). The present note is devoted to extending the algorithm \(\mathfrak{A}\) to the case of partial asymptotic stability. We note that partially stable systems arise, in particular, in passing from a stable differential equation of higher order to a system of first-order equations. The justification of convergence of the algorithm \(\mathfrak{A}\) under conditions of partial stability requires a corresponding converse of A. M. Lyapunov’s theorem. The results of V. V. Rumyantsev \((^3)\) and C. Corduneanu \((^4)\) available in this direction are insufficient for our purposes, since they concern partial stability in the small. In paragraph 2 of the present note, under some additional assumptions, we establish a converse of A. M. Lyapunov’s theorem on stability in the large for the case of partial stability. This, in particular, makes it possible to justify the convergence of the algorithm \(\mathfrak{A}\) for finding a saddle point of the Lagrange function \(\varphi(x,y)\) for a problem of concave programming (when the function \(\varphi(x,y)\) is strictly concave in \(x\) and linear in \(y\)).

2. Let \(E_m\) and \(E_{n-m}\) be Euclidean spaces. Consider the system of equations

\[ dx/dt=f(t,x,y),\qquad dy/dt=g(t,x,y), \tag{1} \]

where the vector \(x\) and the vector-function \(f(t,x,y)\) belong to \(E_m\), while the vector \(y\) and the vector-function \(g(t,x,y)\) belong to \(E_{n-m}\), and the vector-functions are twice continuously differentiable in some domain \(D=D_1\times E_{n-m}\) for all values \(t\ge 0\), where \(D_1\subset E_m\) and \(f(t,0,0)=g(t,0,0)=0\) for \(t\ge 0\). We shall denote the solution of system (1) with initial data \(x(t_0)=x_0,\ y(t_0)=y_0\) by \((x(t,t_0,x_0,y_0),\ y(t,t_0,x_0,y_0))\), assuming that it exists for all \(t\ge t_0\). We give the definitions of partial stability of the solution of system (1). The solution \(x=0,\ y=0\) is called \(x\)-stable if, for any \(\varepsilon>0\) and \(t_0\ge 0\), there exists a positive number \(\delta(\varepsilon,t_0)\) such that from the inequality \(\|x_0\|+\|y_0\|<\delta\) for \(t>t_0\) it follows that \(\|x(t,t_0,x_0,y_0)\|<\varepsilon\). If \(\delta(\varepsilon,t_0)\) does not depend on \(t_0\), then \(x\)-stability is called uniform (see \((^3)\)). Further, an \(x\)-stable solution \(x=0,\ y=0\) is called asymptotically \(x\)-stable (a. \(x\)-st.) in the large in the domain \(D\), if \(\lim_{t\to\infty} x(t,t_0,x_0,y_0)=0\) for all \((x_0,y_0)\in D\).

Asymptotic \(x\)-stability in the large in the domain \(D\) of the solution \(x=0,\ y=0\) of system (1) is called asymptotic \(x\)-stability in the large in the domain \(D\), uniformly with respect to \(t_0\), if \(\lim_{t\to\infty} x(t,t_0,x_0,y_0)=0\) uniformly with respect to \(t_0\ge 0\) for all \((x_0,y_0)\in D\).

Asymptotic \(x\)-stability in the large in the domain \(D\) of the solution \(x=0,\ y=0\) is called uniformly asymptotic \(x\)-stability (u.a. \(x\)-st.) in the large in the domain \(D\), if for any closed bounded domain \(\Omega\subset D\) the relation \(\lim_{t\to\infty} x(t,t_0,x_0,y_0)=0\) holds uniformly with respect to \(t_0\ge 0\) and \((x_0,y_0)\in\Omega\).

Theorem 1. In order that the solution \(x=0,\ y=0\) of system (1) be p.a. \(x\)-stable in the large in the domain \(D\), it is necessary and sufficient that there exist a function \(V(t,x,y)\) \(((x,y)\in D,\ t\geq 0)\) satisfying the following conditions:

1a) there exist continuous functions \(W_i(x)\) \((i=1,2,3)\) such that
\(W_1(x)\geq V(t,x,y)\geq W_2(x)\) for \((x,y)\in D,\ t\geq 0\), where \(W_1(0)=0\), \(W_2(x)>0\) for \(x\ne 0\), and \(W_2(x)\to\infty\) as \(x\to \partial D_1\);

1b)
\[ \frac{dV(t,x(t),y(t))}{dt} = \frac{\partial V}{\partial t}+(\operatorname{grad}_x V,f)+(\operatorname{grad}_y V,g) \leq -W_3(x)<0 \]
for \(x\ne 0\), \((x,y)\in D\) and \(t\geq 0\), for any solution \((x(t),y(t))\) of system (1);

1c) the function \(V(t,x,y)\) has continuous partial derivatives of the second order with respect to all arguments, uniformly bounded for \(t\geq 0\) and \(y\in E_{n-m}\).

In the direction of sufficiency, Theorem 1 is a trivial generalization of Lyapunov’s direct theorem on asymptotic stability. In proving necessity, I. G. Malkin’s method \((^2)\) of constructing the Lyapunov function is used. If p.a. \(x\)-stability is not assumed, then the indicated method of constructing the function no longer applies. However, the following holds.

Theorem 2. In order that the solution \(x=0,\ y=0\) be a. \(x\)-stable in the large in the domain \(D\), uniformly with respect to \(t_0\), it is necessary and sufficient that there exist a function \(V(t,x(t),y(t))\) \(((x,y)\in D,\ t\geq 0)\), where \(x(t)=x(t,t_0,x_0,y_0)\), \(y(t)=y(t,t_0,x_0,y_0)\), satisfying the following conditions:

2a) for \(x'\in D_1\) and \((x,y)\in D\) there exist continuous functions \(W_i(x',x,y)\) \((i=1,2,3)\), with \(W_i(x',x,y)>0\) for \(x'\ne 0\), such that
\[ W_1(x(t),x_0,y_0)\geq V(t,x(t),y(t))\geq W_2(x(t),x_0,y_0), \]
where \(W_1(0,x_0,y_0)=0\) and \(W_2(x,x_0,y_0)\to\infty\) as \(x\to\partial D_1\);

2b)
\[ \frac{dV(t,x(t),y(t))}{dt} = \frac{\partial V}{\partial t}+(\operatorname{grad}_x V,f)+(\operatorname{grad}_y V,g) \leq \]
\[ \leq -W_3(x,x_0,y_0)<0\quad (x\ne 0); \]

2c) the function \(V(t,x,y)\) has continuous partial derivatives of the second order with respect to all arguments, uniformly bounded for \(t\geq 0\).

1. We study in this section system (1) under the assumption that it has an a. \(x\)-stable in the large in the domain \(D\), uniformly with respect to \(t_0\), singular point \(x=0,\ y=0\). Let \(\Omega\) be an arbitrary closed domain from \(D\). We shall call the number \(\gamma(\Omega)\) a relaxation multiplier for the function \(V(t,x,y)\) in the domain \(\Omega\), if

\[ V(t,x,y)>V\bigl(t+\gamma(\Omega),\,x+\gamma(\Omega)f(t,x,y),\,y+\gamma(\Omega)g(t,x,y)\bigr) \]

for \(t\geq 0\) and \((x,y)\in\Omega\).

Lemma 1. For a function \(V(t,x,y)\) satisfying conditions 2a)—c), in any compact domain \(\Omega\subset D\) not containing the point \(x=0,\ y=0\), there exists a relaxation multiplier.

Let now, for an arbitrary initial value \((x_0,y_0)\in\Omega\), the sequences \(\{x_n\}_0^\infty,\ \{y_n\}_0^\infty\) be constructed by the Euler process

\[ x_{n+1}=x_n+\alpha_n f(t_n,x_n,y_n), \tag{2} \]
\[ y_{n+1}=y_n+\alpha_n g(t_n,x_n,y_n), \]

where \(\{\alpha_n\}_0^\infty\) is a sequence of positive numbers and
\[ t_n=\sum_{k=1}^{n-1}\alpha_k. \]

Then the following holds.

Lemma 2. Let \(\alpha_n\to 0\) \((n\to\infty)\). If the sequence \(\{(x_n,y_n)\}_0^\infty\subset\Omega\) defined by equalities (2) and
\[ \inf_{\,n;\ (x_0,y_0)\in D} W_1(x_n,x_0,y_0)>0, \]
then
\[ \sum_{n=0}^{\infty}\alpha_n<\infty \quad \text{(cf. \((^5)\)).} \]

Hence it follows

Theorem 3. If \(\{(x_n,y_n)\}_0^\infty \subset \Omega\), \(\lim\limits_{n\to\infty} a_n=0\), and
\[ \sum_{n=0}^{\infty} a_n=\infty, \]
then
\[ \lim_{n\to\infty} x_n=0. \]

Remark 1. In the case when the function \(V(t,x,y)\) satisfies conditions 1a)—c), Lemmas 1, 2 and Theorem 3 can be strengthened: instead of compactness of the domain \(\Omega\), one may require compactness of its projection \(\Omega_x\) onto \(E_m\).

1. We describe algorithms \(\mathfrak{R}_1\) and \(\mathfrak{R}_2\). Algorithm \(\mathfrak{R}_1\) differs from algorithm \(\mathfrak{R}\) (see (1)) in that the construction of the point \((x_{n+1}(j), y_{n+1}(j))\) depends on the fulfillment of the inequality
   \[ s_i \leq \sum_{k=1}^{n}\|x_k(j)-x_{k-1}(j)\|, \]
   and the transition to the next stage is carried out when the inequality \(\|x_{n+1}(j)\|>A_j\) is fulfilled, i.e., the conditions are checked with respect to the \(x\)-component.

Algorithm \(\mathfrak{R}_2\) differs from \(\mathfrak{R}_1\) only in the check of the condition for transition to the next stage, namely: the transition to the next stage is carried out when the inequality
\[ \|(x_{n+1}(j),y_{n+1}(j))\|>A_j \]
is fulfilled.

Theorem 4. Let
\[ \sup_{\substack{t\geq 0\\ y\in E_{n-m}}}\|f(t,x,y)\|<C(x)<\infty \]
and let the point \((0,0)\) be a p.a. \(x\)-stable singular point of system (1) in the whole space \(E_m\times E_{n-m}\). Then, for any initial point \((x_0,y_0)\), algorithm \(\mathfrak{R}_1\) stabilizes at some stage \(j_0\), and the sequence it generates,
\[ \{(x_n(j_0),y_n(j_0))\}_{n=0}^{\infty}, \]
is such that \(\lim\limits_{n\to\infty} x_n(j_0)=0\).

The proof is based on Theorem 1 and Remark 1.

Theorem 5. Let system (1) satisfy the conditions:

a) \(\sup\limits_{t\geq 0}\|f(t,x,y)\|<C(x,y)<\infty\);

b) for any \((x_0,y_0)\) there exists \(C_0(x_0,y_0)\) such that
\[ \|y(t,t_0,x_0,y_0)\|<C_0(x_0,y_0), \]
and let the point \((0,0)\) be an a. \(x\)-stable singular point, uniformly with respect to \(t_0\), in the whole space \(E_m\times E_{n-m}\). Then, for any initial point \((x_0,y_0)\), algorithm \(\mathfrak{R}_2\) stabilizes at some stage \(j_0\), and the sequence it generates,
\[ \{(x_n(j_0),y_n(j_0))\}_{n=0}^{\infty}, \]
is such that \(\lim\limits_{n\to\infty} x_n(j_0)=0\) and \(\{y_n(j_0)\}_{n=0}^{\infty}\) is contained in a bounded domain.

The proof uses Theorems 2, 3.

Corollary. Algorithm \(\mathfrak{R}_2\) is applicable to finding the value \(x^0\) corresponding to the saddle point \((x^0,y^0)\) of a functional \(\varphi(x,y)\), strictly concave in the variables \(x\) and convex in the variables \(y\), since the corresponding system for \(\varphi(x,y)\)
\[ dx/dt=\operatorname{grad}_x\varphi(x,y), \qquad dy/dt=-\operatorname{grad}_y\varphi(x,y) \]
satisfies the conditions of Theorem 5 (see (6)).

In conclusion, the authors gratefully note the constant attention of I. M. Glazman during the writing of this note.

Physical-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR

Institute of Chemical Physics
Academy of Sciences of the USSR

Received
26 II 1968

References

1. A. I. Kononenko, V. P. Milman, DAN, 167, No. 4 (1966).
2. I. G. Malkin, PMM, 18, no. 1–3, 129 (1954).
3. V. V. Rumyantsev, Vestn. Mosk. univ., ser. matem., mekh., astr., fiz., khim., no. 4 (1957).
4. C. Corduneanu, Rev. roumaine de mat. pures es appl., 9, No. 3, 229 (1964).
5. I. M. Glazman, DAN, 154, No. 5 (1964).
6. K. J. Arrow, L. Hurwicz, H. Uzawa, Studies in Linear and Nonlinear Programming, IL, 1962.