MATHEMATICS

V. A. PLISS

ON THE PHENOMENON OF CONVERGENCE IN PERIODIC NONLINEAR SYSTEMS

(Presented by Academician V. I. Smirnov, 22 XII 1960)

Consider the system of differential equations

[
\frac{dx}{dt}=f(x,t),
\tag{1}
]

where (x={x^{(1)},\ldots,x^{(n)}}) and (f(x,t)={f^{(1)}(x,t),\ldots,f^{(n)}(x,t)}) are (n)-dimensional vectors. It is assumed with respect to (f(x,t)) that it is continuous, satisfies the condition of uniqueness of solutions of system (1) for all (x,t), and has period (\omega) in (t) for all (x): (f(x,t+\omega)=f(x,t)).

Let (\xi={\xi^{(1)},\ldots,\xi^{(n)}}) be an (n)-dimensional vector; put

[
|\xi|=\sum_{i=1}^{n}|\xi^{(i)}|.
]

Denote by (x(t,x_0,t_0)) the solution of system (1) with initial data (t_0,x_0).

We shall say that system (1) possesses the property of convergence if:

I. All solutions (x(t,x_0,t_0)) can be continued for all times (t\geq t_0).

II. System (1) has a unique (\omega)-periodic solution (x=\varphi(t)).

III. This solution is stable in the sense of Lyapunov.

IV. For any solution (x(t,x_0,t_0)) the relation

[
\lim_{t\to+\infty}|x(t,x_0,t_0)-\varphi(t)|=0
\tag{2}
]

holds.

Suppose that all solutions (x(t,x_0,t_0)) can be continued for (t\geq t_0). To the point (x_0) assign the point (x(\omega,x_0,0)). In this way we obtain a homeomorphic transformation (T) of the space (E_n) into itself.

We next establish conditions necessary and sufficient for the existence of convergence. These conditions are imposed on the transformation (T) and do not concern the behavior of the solutions for (0<t<\omega). In this they substantially generalize the conditions formulated in the paper ((^1)).

Theorem. In order that system (1) possess the property of convergence, it is necessary and sufficient that the following conditions be satisfied:

I. All solutions (x(t,x_0,t_0)) can be continued for (t\geq t_0).

II. There exists a closed, bounded set (F\subset E_n) which is mapped by the transformation (T) into itself.

III. There exists a continuous function (v(x,y)) of points (x) and (y) of the space (E_n) with the following properties: 1) (v(x,y)\geq 0) and (v(x,y)=0) if and only if (x=y); 2) for every (x\in F), (v(x,y)\to\infty) as (|y|\to\infty); 3) (v(T(x),T(y))\leq v(x,y)), and if (v(T(x),T(y))=v(x,y)), then (x=y).

We outline the proof.

Sufficiency. We shall prove that the transformation (T) has a fixed point. Consider the set

[
\Omega=\prod_{m=0}^{\infty} T^m(F).
]

It is not difficult to see that (\Omega) is closed and (T(\Omega)=\Omega). We shall show that (\Omega) is a point of the space (E_n). Suppose, on the contrary, that this is not so; then

[
\alpha=\sup_{x,y\in\Omega} v(x,y)>0.
\tag{3}
]

From the closedness of (\Omega) it follows that there exist (\bar{x}\in\Omega) and (\bar{y}\in\Omega) such that (v(\bar{x},\bar{y})=\alpha). From the equality (T(\Omega)=\Omega) it follows that (T^{-1}(\bar{x})\in\Omega) and (T^{-1}(\bar{y})\in\Omega); by virtue of property (3) of the function (v(x,y)) we then conclude that

[
v\bigl(T^{-1}(\bar{x}),T^{-1}(\bar{y})\bigr)>v(\bar{x},\bar{y})=\alpha.
\tag{4}
]

This contradicts relation (3). Consequently, the set (\Omega) is a point (\xi\in E_n). Obviously, (T(\xi)=\xi).

The function (x(t,\xi,0)) is an (\omega)-periodic solution of system (1). We shall show that it is unique. Indeed, if there existed a different (\omega)-periodic solution (x(t,\eta,0)), (\xi\ne\eta), then we would have (v(\xi,\eta)>v(T(\xi),T(\eta))=v(\xi,\eta)). The contradiction obtained proves the uniqueness of the solution (x(t,\xi,0)).

We shall show that the solution (x(t,\xi,0)) is Lyapunov stable. Let (\varepsilon>0). For this (\varepsilon), by the theorem on integral continuity, there is a (\delta_1>0) such that if (|\xi-y|\leqslant\delta_1), then

[
\bigl|x(t,y,0)-x(t,\xi,0)\bigr|>\varepsilon
\quad\text{for } 0\leqslant t\leqslant\omega.
\tag{5}
]

Put

[
\inf_{\delta_1\leqslant |\xi-y|\leqslant \varepsilon} v(\xi,y)=l.
]

By the continuity of (v) there is a (\delta>0) such that, when (|\xi-y|\leqslant\delta), (v(\xi,y)<l). Now take (y) such that (|\xi-y|\leqslant\delta). We shall prove that for all (n\geqslant0) one has

[
\bigl|\xi-T^n(y)\bigr|<\delta_1.
\tag{6}
]

We prove inequality (6) for (n=1). If (6) for (n=1) were not satisfied, then (|\xi-T(y)|\in[\delta_1,\varepsilon]), which follows from (5). But then (v(\xi,T(y))\geqslant l). By the same property 3) of the function (v), we have (v(\xi,T(y))\leqslant v(\xi,y)1), inequality (6) is established by induction. From relations (5) and (6), and from the periodicity of (f(x,t)), it follows that the solution (x(t,\xi,0)) is Lyapunov stable.

We shall prove that, for arbitrary (x_0,t_0),

[
\lim_{t\to+\infty}\bigl|x(t,x_0,t_0)-x(t,\xi,0)\bigr|=0.
\tag{7}
]

Since all solutions (x(t,x_0,t_0)) are continuable for (t\geqslant t_0), to prove (7) it is sufficient to establish the relation

[
\lim_{k\to\infty} T^k(x_0)=\xi.
\tag{8}
]

From property 2) of the function (v) it follows that the set

[
G{v(\xi,x)\leqslant v(\xi,x_0)}
]

is bounded. Denote by (S) the set

[
\prod_{m=0}^{\infty} T^m(G).
]

Just as in the proof of the existence of the fixed point (\xi), we establish that (S) is a fixed point of the transformation (T). From the uniqueness of (\xi) it follows that (S=\xi). Clearly, all limit points of the sequence (T^k(x_0)) are contained in (S). Hence (8) follows. The sufficiency of the conditions of the theorem is established.

Necessity. Put

[
\rho(t,x_0,y_0,t_0)=\sum_{k=1}^{n}\left[x^{(k)}(t,x_0,t_0)-x^{(k)}(t,y_0,t_0)\right]^2 .
]

Form the function

[
w(x,y,t)=\int_t^{\infty}G(\rho(\tau,x,y,t))\,d\tau ,
\tag{9}
]

where (G(z)) is some function defined for (z\geq 0), positive for (z>0), and vanishing for (z=0).

Following the ideas of Massera ((^2)), the function (G) can be chosen so that the integral on the right-hand side of (9) will converge uniformly for all (|x|\leq A,\ |y|\leq A,\ 0\leq t\leq A), where (A>0) is a constant. Thus the function (w(x,y,t)) is defined and continuous for all (x,y) and for (t\geq 0).

Let us show that (w(x,y,t)) is (\omega)-periodic in the argument (t). We have

[
w(x,y,t+\omega)=\int_{t+\omega}^{\infty}G(\rho(\tau,x,y,t+\omega))\,d\tau
=\int_{\infty}^{t}G(\rho(\tau+\omega,x,y,t+\omega))\,d\tau .
\tag{10}
]

But from the (\omega)-periodicity of (f(x,t)) it is not difficult to derive the relation

[
\rho=(\tau+\omega,x,y,t+\omega)=\rho(\tau,x,y,t),
]

and therefore from (10) we obtain

[
w(x,y,t+\omega)=\int_t^{\infty}G(\rho(\tau,x,y,t))\,d\tau=w(x,y,t).
\tag{11}
]

We now show that along solutions of system (1) (w(x,y,t)) decreases. We have

[
\rho(\tau,x(t,x_0,t_0),x(t,y_0,t_0),t)=\rho(\tau,x_0,y_0,t_0).
\tag{12}
]

Substituting (12) into (8) and differentiating with respect to (t), we obtain

[
\frac{d}{dt}w(x(t,x_0,t_0),x(t,y_0,t_0),t)=-G(\rho(t,x_0,y_0,t_0)).
]

Hence, and from the property of the function (G(z)), it follows that if (x_0\ne y_0), then for all (t) one has

[
\frac{d}{dt}w(x(t,x_0,t_0),x(t,y_0,t_0),t)<0.
\tag{13}
]

Thus, along solutions of system (1), (w) decreases.

The function (G) can be chosen so that for any (x) satisfying the condition (|x|\leq B), where (B>0) is arbitrary but fixed, one has

[
w(x,y,0)\to\infty \quad \text{as } |y|\to\infty .
\tag{14}
]

Now put (v(x,y)=w(x,y,0)). It is not difficult to see that then all the conditions of the theorem will be fulfilled.

As an example, consider the system

[
\frac{dx_k}{dt}=\sum_{\nu=1}^{n}b_{k\nu}x_\nu+h_k f(\sigma)+p_k(t),\qquad
\sigma=\sum_{i=1}^{n}\alpha_i x_i
\quad (k=1,\ldots,n).
\tag{15}
]

With respect to (f(\sigma)), it is assumed that it is differentiable for all (\sigma) and that (f'(\sigma) > 0). The functions (p_k(t)) are assumed to be continuous for all (t) and (p_k(t+\omega)=p_k(t)). Suppose that system (15) has the following canonical form (3):

[
\frac{dz_k}{dt}=\lambda_k z_k+f(y)+q_k(t),\qquad
y=\sum_{i=1}^{n}\gamma_i z_i
\quad (k=1,\ldots,n),
\tag{16}
]

where all (\lambda_k) have negative real parts and among them there are (s) real ones and (\frac12(n-s)) pairs of complex conjugates. Suppose further that the system of quadratic equations

[
-2a_k\sum_{i=1}^{n}\frac{a_i}{\lambda_i+\lambda_k}+\gamma_k=0
\quad (k=1,\ldots,n)
\tag{17}
]

has (s) real roots and (\frac12(n-s)) pairs of complex conjugate roots. Under fulfillment of these conditions, system (15) possesses the property of convergence.

Leningrad State University
named after A. A. Zhdanov

Received
25 XI 1960

CITED LITERATURE

¹ V. I. Zubov, Vestn. LGU, No. 1 (1960).
² J. L. Massera, Ann. of Math., 50, No. 3 (1949).
³ A. I. Lur’e, Some Nonlinear Problems in the Theory of Automatic Control, 1951.