ON THE STABILITY OF MOTIONS OF A SYSTEM WITH RANDOM LAGS

E. A. LIDSKII

Motions described by linear differential equations with time lags are considered. The magnitude of the lag \(\eta(t)\) is a random Markov process. Sufficient conditions are determined for the asymptotic stability in probability of these motions. The indicated conditions restrict the rate of change of the mean value and the variance of the random function \(\eta(t)\). The method of solving the problem uses Lyapunov functions in its application to equations with time lags.

1. Consider the equation

\[ \frac{dx}{dt}=Ax(t)+Bx(t+\eta), \tag{1.1} \]

where \(x=\{x_1,\ldots,x_n\}\) is an \(n\)-dimensional vector of phase coordinates of the system; \(A=\|a_{ij}\|_1^n\), \(B=\|b_{ij}\|_1^n\) are constant square matrices; \(\eta(t)\) is a Markov random function.

We shall restrict ourselves to the case when \(\eta(t)\) is a purely discontinuous process [1, p. 136] and the quantity \(\eta\) can take values within the closed interval \(-h \le \eta \le 0\) \((h>0)\). A realization \(\eta(\omega,t)\) determines the lag \(\gamma(t)=\eta(\omega,t)\). We describe the process \(\eta(t)\) by means of the functions [1, p. 223] \(q(\alpha)\), \(q(\alpha,\beta)\), satisfying the conditions \(q(\alpha)\ge 0\), \(q(\alpha,\beta)\ge 0\), \(q(\alpha,-\infty)=0\), \(q(\alpha,\infty)=q(\alpha)\), \(q(\alpha,\beta)\) is a nondecreasing function of \(\beta\) and is continuous, essentially with respect to \(\alpha\), as a function of the parameter.

In this case

\[ P\,[\eta(t+\Delta t)=\alpha \mid \eta(t)=\alpha] = 1-\Delta t\,q(\alpha)+o(\Delta t), \]

\[ P\,[\eta(t+\Delta t)\le \beta \mid \eta(t)=\alpha] = \Delta t\,q(\alpha,\beta)+o(\Delta t), \]

where \(P[A\mid B]\) is conditional probability, and \(\Delta t\) is a small interval of time.

We shall consider two special cases of \(\eta(t)\):

1) \(q(\alpha,\beta)\) admits a density \(p(\alpha,\beta)\), continuous in \(\alpha\) and \(\beta\); then

\[ P\,[\eta(t+\Delta t)<\beta \mid \eta(t)=\alpha] = \Delta t\int_{-\infty}^{\beta} p(\alpha,\beta)\,d\beta+o(\Delta t); \tag{1.2} \]

2) \(\eta\) can take only a finite number of values

\[ \{\alpha_1,\ldots,\alpha_k\}, \]

\[ P\,[\eta(t+\Delta t)=\alpha_j \mid \eta(t)=\alpha_i] = p_{ij}\Delta t+o(\Delta t), \tag{1.3} \]

where \(p_{ij}\) is an entry of the given matrix \(\|p_{ij}\|_1^k\).

The realizations \(\eta(\omega,t)\) are step functions.

We shall assume that the realizations of the random function \(\eta(t)\) are continuous from the right.

The initial conditions at the instant \(t_0\), determining the motion (1.1) for \(t \geq t_0\), will be prescribed in the form of a segment of the trajectory \(x_0(t_0+\vartheta)\) \((-h \leq \vartheta \leq 0)\) and the value \(\eta(t_0)=\eta_0\).

These conditions determine the distribution of the random variables \(x(t)\), \(\eta(t)\) for \(t \geq t_0\), independently of \(x(\tau)\) for \(\tau < t-h\) and \(\eta(\tau)\) for \(\tau < t\).

In what follows, as an element of the realized trajectory we shall consider its segment \(x(\omega,t+\vartheta)\) for \(-h \leq \vartheta \leq 0\).

For a fixed function \(\eta(t)\), the solution of (1.1) is determined [2, p. 31] by the initial curve \(x_0(t_0+\vartheta)\). In the stochastic case, the initial conditions \(\{x_0(t_0+\vartheta),\eta_0\}\) generate a probability process, which may be visualized as a bundle of realizations of the motion (1.1)
\[ \{x(\omega,t)\}=\{x(x_0(t_0+\vartheta),\eta_0,t,\eta(\omega,t))\}, \]
corresponding to all possible \(\eta(\omega,t)\). In this case one may suppose that each realization \(\{x(\omega,t+\vartheta),\eta(\omega,t)\}\) satisfies the equation
\[ \lim_{\Delta t\to +0}\left(\frac{x_i(\omega,t+\Delta t)-x_i(\omega,t)}{\Delta t}\right) = \sum_{j=1}^{n} a_{ij}x_j(\omega,t) + \sum_{j=1}^{n} b_{ij}x_j(\omega,t+\eta(\omega,t)) \quad (i=1,\ldots,n). \]

The probability process \(x(t)\) generated by the conditions \(t=t_0\), \(\eta=\eta_0\), \(x_0(t_0+\vartheta)\), where \(x_0(t_0+\vartheta)\) is a prescribed function of \(\vartheta\), will be denoted by the symbol \(x(x_0(t_0+\vartheta),\eta_0,t)\), and the realizations of this process by the symbol \(x(x_0(t_0+\vartheta),\eta_0,t,\omega)\). Denote\(^1\)
\[ \left\|x(x_0(t_0+\vartheta_0),\eta_0,t+\vartheta,\omega)\right\| = \sup_{-h\leq \vartheta \leq 0} \left[ \sum_{n} x_i^2(x_0(t_0+\vartheta_0),\eta_0,t+\vartheta,\omega) \right]^{1/2}. \]

In the presence of a random lag \(\eta(t)\), one can give the following definition of stability and asymptotic stability in probability, generalizing Lyapunov’s definition [3] in the same direction as was done in [4] for systems described by ordinary differential equations with random parameters.

Definition 1.1. If for any two arbitrarily small numbers \(\varepsilon>0\), \(p>0\) one can indicate a number \(\delta>0\) such that for every motion (1.1) the inequality
\[ P\left[\left\|x(x_0(t_0+\vartheta_0),\eta_0,t+\vartheta,\omega)\right\|<\varepsilon\right] \tag{1.4} \]
holds for all \(t \geq t_0\), \(\left\|x_0(t_0+\vartheta)\right\|\leq \delta\) at \(t=t_0\), \(>1-p\), then the solution \(x=0\) of system (1.1) will be called stable in probability.

Definition 1.2. If, in addition to (1.4), for any \(\lambda>0\) and any \(\{x_0(t_0+\vartheta_0),\eta_0\}\) the condition
\[ \lim_{t\to\infty} P\left[ \left\|x(x_0(t_0+\vartheta_0),\eta_0,t+\vartheta,\omega)\right\|<\lambda \right] =1, \]
is fulfilled, then the solution \(x=0\) of system (1.1) will be called asymptotically stable in probability.

The motion of system (1.1) with constant lag \(\eta\equiv\eta_0\) will be denoted by the symbol \(x(x_0(t_0+\vartheta),\eta_0,t)^0\).

\(^1\) Here and below it is assumed that the variables \(\vartheta\) and \(\vartheta_0\) vary within the limits \(-h\leq \vartheta \leq 0\), \(-h\leq \vartheta_0 \leq 0\).

Suppose that for any fixed value \(\eta=\mu\) \((-h\leq \mu \leq 0)\) and initial fixed perturbation \(x_0(t_0+\vartheta_0)\), the motion \(x(x_0(t_0+\vartheta_0),\mu,t+\vartheta)^0\), where \(t\geq t_0\), uniformly with respect to \(\mu\) satisfies the inequality

\[ \left\|x\left(x_0(t_0+\vartheta_0),\mu,t+\vartheta\right)^0\right\| \leq B_1\left\|x_0(t_0+\vartheta_0)\right\|\exp[-a(t-t_0)], \tag{1.5} \]

where \(B_1\) and \(a\) do not depend on \(\mu\).

Condition (1.5) for a fixed constant \(\eta=\mu\) is equivalent to the asymptotic stability of motions of systems with delay in the sense of the definitions in [5, p. 155].

The fulfillment of condition (1.5) alone, however, does not ensure asymptotic stability in probability of the motions of the stochastic system (1.1). We pose the problem: to indicate additional restrictions on the properties of the random process \(\eta(t)\) which, together with (1.5), are sufficient to ensure the stability of solutions of (1.1) in the sense of definitions (1.1)—1.2).

2. Let us make some preliminary remarks.

Suppose that, for fixed \(\eta=\mu\), condition (1.5) is satisfied. Then one can construct a positive definite functional \(V^0(x(\vartheta),\mu)\), satisfying the estimate

\[ \lim_{\Delta t\to +0}\sup\left(\frac{\Delta V^0}{\Delta t}\right) < -\|x(\vartheta)\|. \tag{2.1} \]

The functional \(V^0\) can be constructed in the form [5, p. 192]

\[ V^0(x_0(\vartheta_0),\mu) = 2\int_0^T \left\|x\left(x_0(\vartheta_0),\mu,\xi+\vartheta\right)^0\right\|\,d\xi + \sup\left(\left\|x\left(x_0(\vartheta_0),\mu,\right.\right.\right. \]

\[ \left.\left.\left. \xi+\vartheta\right)^0\right\| \ \text{for } 0\leq \xi \leq T\right), \quad \text{where } T=\frac{1}{a}\ln 2B. \tag{2.2} \]

Remark 2.1. On the left-hand side of inequality (2.1) stands the right upper derivative number of the function \(v_0(t)\), obtained by substituting into the functional \(V^0(x(\vartheta),\mu)\) the element of the trajectory \(x(x_0(\vartheta),\mu,t+\vartheta)^0\) (for \(t\geq 0\)) of the motion of system (1.1) with fixed \(\eta=\mu\).

Let us now consider some functional \(V(x(\vartheta),\eta)\). If in the expression \(V(x(\vartheta),\eta)\) one substitutes, instead of \(x(\vartheta)\) and \(\eta\), \(\eta(\eta_0,t)\) and \(x(x_0(\vartheta),\eta_0,t+\vartheta)\), then as a result one obtains a random function \(v(t)\). Denote by the symbol

\[ \begin{aligned} \lim_{\Delta t\to +0}\sup\left(\frac{\Delta M\{V\}}{\Delta t}\right) &= \lim_{\Delta t\to +0}\sup \frac{M\{v(\Delta t)-v(0)\}}{\Delta t} = \lim_{\Delta t\to +0}\sup \frac{1}{\Delta t}\times \\ &\quad \times \Big[ M\{V(x(x_0(\vartheta),\eta(0),\Delta t+\vartheta),\eta(\eta(0),\Delta t)) \mid x_0(\vartheta),\eta(0)\} \\ &\qquad - V(x_0(\vartheta),\eta(0)) \Big] \end{aligned} \tag{2.3} \]

\[ (\eta(0)=\eta_0), \]

where \(M\{A\mid B\}\) is conditional mathematical expectation.

One may assert that if for the stochastic equation (1.1) there exists a functional \(V(x(\vartheta),\eta)\), positive definite [5, p. 171] for all \(\eta\in[-h,0]\) and admitting an infinitesimal upper limit uniformly in \(\eta\), and if the quantity

\[ \lim_{\Delta t\to +0}\sup\left(\frac{\Delta M\{V\}}{\Delta t}\right) \]

is negative definite, then the solution \(x=0\) of equation (1.1) is asymptotically stable in probability.

The proof of the assertion just made is constructed according to the same plan as in the analogous theorem [4] for stochastic equations without delay, with the introduction of certain changes [5, p. 172] caused by the transition from ordinary equations to equations with delays. This method of proof is applicable here, since in the space of trajectory elements \(x(t+\vartheta)\) \((-h \leq \vartheta \leq 0)\) and \(\eta(t)\) the process \(\{x(t), \eta(t)\}\) has the Markov property.

If the functional \(V(x(\vartheta), \eta)\) is continuous in \(\eta\) and satisfies a Lipschitz condition in \(x(\vartheta)\), then the quantity (2.3) satisfies the inequality

\[ \lim_{\Delta t \to +0}\sup \left( \frac{\Delta M\{V\}}{\Delta t} \right) \leq \lim_{\Delta t \to +0}\sup \left( \frac{\Delta V}{\Delta t} \right)_{\eta_0} + \int_{-h}^{0} [V(x(\vartheta), \beta)-V(x(\vartheta), \eta_0)]\, d_\beta q(\eta_0,\beta). \tag{2.4} \]

Here
\[ \lim_{\Delta t \to +0}\sup \left( \frac{\Delta V}{\Delta t} \right)_{\eta_0} \]
is computed for fixed \(\eta=\eta_0\). Verification of inequality (2.4) is carried out in the same way as was done in [6] when computing the quantity corresponding to (2.3).

In the special cases (1.2) and (1.3), the integral in the right-hand side of (2.4) is replaced by one of the following expressions:

\[ 1).\quad \int_{-h}^{0} [V(x(\vartheta),\beta)-V(x(\vartheta),\eta_0)]\,p(\eta_0,\beta)\,d\beta; \]

\[ 2).\quad \sum_{l\ne s}^{k} p_{sl} [V(x(\vartheta),a_l)-V(x(\vartheta),a_s)], \]

where \(\eta_0=\eta_s\).

1. We shall now indicate those restrictions on the function \(\eta(t)\) under which condition (1.5) ensures asymptotic stability in probability of system (1.1).

Theorem 3.1. If, for motions (1.1), for every fixed \(\eta=\mu \in [-h,0]\) condition (1.5) is satisfied, then one can indicate a constant \(Q>0\) such that, when the inequality

\[ \frac{dM\{|\eta(\tau)-\eta_0|\}}{dt}<Q \tag{3.1} \]

is fulfilled, the motion \(x=0\) of system (1.1) is asymptotically stable in probability.

Remark 3.1. The symbol \(dM\{|\eta(\tau)-\eta_0|\}/dt\) denotes the derivative of the quantity \(M\{|\eta(\tau)-\eta_0| \mid \eta(0)=\eta_0\}\), computed as \(\tau \to +0\).

Remark 3.2. In the special cases (1.2) and (1.3), a sufficient condition for the fulfillment of (3.1) is the fulfillment of the inequalities, for case 1),

\[ p(\alpha,\beta)< \frac{Q_1}{|\beta-\alpha|} \quad (Q_1=\mathrm{const};\ \beta,\alpha\in[-h,0]); \]

for case 2),

\[ \sum_{l\ne s}^{k} p_{sl}|a_l-a_s|<Q_2 \quad (Q_2=\mathrm{const}), \]

where \(Q_1,Q_2\) are sufficiently small constants.

Proof. We shall use in the proof the stability criterion for the motion \(x=0\) of system (1.1), formulated in Section 2. We choose as \(V(x(\vartheta),\eta)\) the functional (2.2), setting
\[ V(x(\vartheta),\eta)=V^0(x(\vartheta),\eta). \]
Then, by virtue of (2.1) and (2.4), we obtain
\[ \lim_{\Delta t\to +0}\sup\left(\frac{\Delta M\{V\}}{\Delta t}\right) \leq -\|x_0(\vartheta)\|+ \int_{-h}^{0}\bigl[V(x_0(\vartheta),\beta)- V(x_0(\vartheta),\eta_0)\bigr]\,d_\beta q(\eta_0,\beta). \tag{3.2} \]

For ordinary equations with lags, in which \(\eta\) plays the role of a fixed parameter, the following property holds.

The difference of two solutions \(x(x_0(\vartheta_0),\eta,t+\vartheta)^0\), corresponding to two different lags \(\eta=\beta\) and \(\eta=\eta_0\), satisfies the inequality
\[ \|x(x_0(\vartheta_0),\beta,t+\vartheta)^0 - x(x_0(\vartheta_0),\eta_0,t+\vartheta)^0\| \leq L_1\|x_0(\vartheta_0)\|\,|\beta-\eta_0| \tag{3.3} \]
\[ (L_1>0,\ t\geq 0). \]

The proof of (3.3) is carried out by considering the corresponding integral inequalities, just as in the case of dependence on the initial condition \(x_0(\vartheta_0)\) [5, p. 174].

From inequality (3.3) and the form of expression (2.2) of the functional \(V^0(x_0(\vartheta),\mu)\), for the parameter values \(\mu=\beta\), \(\mu=\eta_0\), there follows the validity of the inequality
\[ |V(x_0(\vartheta),\beta)-V(x_0(\vartheta),\eta_0)| \leq L\|x_0(\vartheta)\|\,|\beta-\eta_0| \tag{3.4} \]
\[ (L>0). \]

Taking (3.2), (3.3), and (3.4) into account, we obtain
\[ \lim_{\Delta t\to +0}\sup\left(\frac{\Delta M\{V\}}{\Delta t}\right) \leq \|x_0(\vartheta)\| \left[ -1+ L\int_{-h}^{0}|\beta-\eta_0|\,d_\beta q(\eta_0,\beta) \right]. \]

We note that
\[ \int_{-h}^{0}|\beta-\eta_0|\,d_\beta q(\eta_0,\beta) = \lim_{\Delta t\to +0} \frac{M\{|\eta(\Delta t)-\eta_0|\}}{\Delta t} = \frac{dM\{|\eta(\tau)-\eta_0|\}}{dt}. \]

A sufficient condition for asymptotic stability in probability is the definiteness and negativity of the quantity (2.3). If in condition (3.1) one chooses \(Q<1/L\), then (2.3) will be negative definite. Therefore, for \(Q<1/L\), the constructed functional \(V\) satisfies the requirements of the stability criterion and, consequently, the motion \(x=0\) of system (1.1) is asymptotically stable in probability.

The theorem is proved.

Let us note in conclusion that the additional condition (3.1) of Theorem 3.1, which together with (1.5) ensures asymptotic stability in probability of the motions (1.1), has the meaning of a restriction on the rate of change of the mean value and variance of the random function \(\eta(t)\). Thus, just as in the case of ordinary linear stochastic equations [7], asymptotic stability in probability is ensured if the probability of large changes of the random parameter \(\eta\) is small.

Remark 3.3. The result obtained can be generalized to motions described by nonstationary equations of the form (1.1), and also to the case when the matrices \(A\) and \(B\) in (1.1) are functions of a random parameter \(\eta\). Here the investigation is carried out taking into account the results of [7].

Remark 3.4. Condition (1.5) is equivalent to [5, p. 195] the fulfillment of the inequality

\[ \operatorname{Re}\lambda_i < -\rho \;(\rho=\operatorname{const}>0) \tag{3.5} \]

for the roots \(\lambda_i\) of the characteristic equation
\[ \left|a_{ij}+b_{ij}e^{\lambda\mu}-\lambda\delta_{ij}\right|_1^n=0 \]
of system (1.1) for every \(\eta=\mu\) \((-h\leq \mu\leq 0)\). Therefore the following assertion is valid:

if (3.5) is satisfied, then one can specify such a constant \(Q>0\) that, when inequality (3.1) is fulfilled, the solution \(x=0\) of system (1.1) is asymptotically stable in probability.

Remark 3.5. While the present article was being prepared, the author became aware of a paper by G. A. Agasandyan, in which systems with random lags are also considered. The stability problem considered by us differs from the problems studied in the indicated paper.

REFERENCES

1. Doob J. Stochastic Processes. IL, Moscow, 1956.
2. Myshkis A. D. Linear Differential Equations with Retarded Argument. Gostekhizdat, 1951.
3. Lyapunov A. M. The General Problem of the Stability of Motion. Gosenergoizdat, 1956.
4. Kac I. Ya., Krasovskii N. N. PMM, vol. XXIV, No. 5, 1960.
5. Krasovskii N. N. Some Problems in the Theory of Stability of Motion. Fizmatgiz, 1959.
6. Krasovskii N. N., Lidskii E. A. Avtomatika i telemekhanika, vol. XXII, No. 10, 1961.
7. Lidskii E. A. SMZh, vol. IV, No. 5, 1963.

Received by the editors
September 5, 1964

Ural Polytechnic Institute
named after S. M. Kirov