UDC 519.271

CYBERNETICS AND CONTROL THEORY

V. V. KALASHNIKOV

ON THE PROBLEM OF STUDYING THE STABILITY OF FUNCTIONING OF COMPLEX SYSTEMS

(Presented by Academician V. M. Glushkov, 16 II 1967)

Among the problems relating to the study of the general properties of individual classes of complex systems, an essential place is occupied by the study of the stability of their functioning. In what follows it will be assumed that the state of a complex system is described by a set of generalized coordinates \(z\), and the process of its functioning is completely determined by the change of these coordinates in time, i.e., by functions \(z(t,\alpha)\), \(t \in T\), which are, generally speaking, realizations of some random process \(Z(t,\alpha)\). Here \(T\) is the totality of the time instants under consideration, and \(\alpha\) is an element of some abstract set \(\Lambda\), which in what follows we shall call the set of parameters. A one-parameter family of functionals \(F_\tau\) is given on the realizations \(z(t,\alpha)\) for any \(\alpha \in \Lambda\), i.e. \(F_\tau = F_\tau\{z(t,\alpha), t \leqslant \tau; t,\tau \in T\}\). For fixed \(\alpha\) and fixed realization \(z(t,\alpha)\), the functional \(F_\tau\) is a function of time \(\tau\), \(\tau \in T\). In the space \(\Omega\) of all possible real functions with domain \(T\), select a class of sets \(\mathcal{B}_\mu\), consisting of all sets \(B\) satisfying some property \(\mu\). Denote by \(B(t)\) the range of values at the instant \(t\) of all functions belonging to the set \(B\). Let \(y(t)\) be some real function with domain \(T\). We shall agree to distinguish types of inclusions \(y(t) \in B\), determined by some parameter \(d\). To each \(d\) there corresponds, generally speaking, some subset \(T_d \subset T\). For definiteness, by the inclusion \(y(t) \in B\) we shall understand that the function \(y(t)\), \(t \in T\), is an element of \(B\). In this case there is no need to define the set \(T_0\). By the inclusion \(y(t) \in B\) we shall understand that the value of the function \(y(t)\) is an element of \(B(t)\); \(T_1 = T\). In principle, other types of inclusions are possible as well, but for the purposes of the present note these two types suffice. In the set \(\Lambda\) we select a class of subsets \(\mathcal{A}_\lambda\), consisting of all sets \(A\) satisfying the property \(\lambda\). We shall agree to distinguish two numbers: \(a\) and \(a-0\). In this case the inequality \(b > a-0\) is understood as \(b \geqslant a\). The definition given below indicates the connection between the various previously introduced definitions of stability, although it is of methodological rather than practical interest.

Definition 1. A complex system is \((\lambda,\mu)\)-stable with respect to \((\varepsilon_0,F,d)\), \(-0 \leqslant \varepsilon_0 \leqslant 1\), if for any \(\varepsilon > \varepsilon_0\) and any set \(B \in \mathcal{B}_\mu\) there exists a set \(A \in \mathcal{A}_\lambda\) such that, for any \(\alpha \in A\) and for all \(\tau \in T_d\),

\[ \mathfrak{P}\{F_\tau[z(t,\alpha),\, t \leqslant \tau;\, t \in T] \in B\} \geqslant 1-\varepsilon. \]

The choice of different \(\lambda\), \(\mu\), \(d\), and \(\varepsilon_0\) makes it possible to obtain different kinds of stability. Thus, if \(\varepsilon_0 = 0\), then it is required that for any \(\varepsilon > 0\) there exist a domain \(A\) in the parameter space from the class \(\mathcal{A}_\lambda\) such that, for any \(\alpha \in A\) and \(\tau \in T_d\), the event \(F_\tau \in B\) occur with probability \(\geqslant 1-\varepsilon\); if \(\varepsilon_0 = -0\), then the existence of such a domain is required ...

for which, when $\tau \in T_d$, this event occurs with probability 1; if $\varepsilon_0 > 0$, then the definition requires that in the class $\mathcal A_\lambda$ there exist a set for which the event $F_\tau \in B$ is fulfilled with sufficiently high probability $(\geq 1-\varepsilon_0)$. The choice of particular $d$ and $\varepsilon_0$ is determined by the nature of the problem under study. Let us illustrate this with several examples.

1. Let the system be described by the differential equations $dz/dt=\Phi(z,t)$. As the parameters of the system we shall take the initial deviations $z(0)\equiv \alpha$; $T=[0,\infty)$; the set $\Lambda$ is the $n$-dimensional space, where $n$ is the dimension of the vector $z$; the functional $F_\tau\{z(t,\alpha),\,t\leq\tau\}=\|z(\tau,\alpha)\|$, where $\|\cdot\|$ is some norm. The class $\mathcal B_\mu$ consists of all possible sets $B_b$, $b>0$, of nonnegative functions defined on $[0,\infty)$ and bounded by the number $b$; $\mathcal A_\lambda$ is the class of sets containing some neighborhood of zero; $\varepsilon_0=-0$, $d=0$. In this case the definition of $(\lambda,\mu)$-stability reduces to the definition of Lyapunov stability of the equilibrium at the origin of coordinates $^{(1)}$.

2. If, in the preceding example, the class $\mathcal B_\mu$ consists of a single set $B$ of nonnegative functions bounded by a fixed number $b$, and $\mathcal A_\lambda$ is the class of sets containing a certain fixed set, then the definition of $(\lambda,\mu)$-stability reduces to the definition of practical stability $^{(1)}$. In general, by choosing the properties $\lambda$ and $\mu$ and the parameters $\alpha$ in an appropriate way, one can obtain any definition of stability given in $^{(1)}$.

3. Now let the system be described by the differential equations

\[ dz/dt=\Phi[z,t,y(t,\beta)], \qquad \Phi(0,t,y)=0, \]

where $y(t,\beta)$ is a parametric family of random processes, $\beta$ is a parameter taking values from some set $\Lambda_1$ $^{(2)}$. As the parameter of the system we take the pair $(z(0),\beta)$, specified on the direct product of the sets $\Lambda_1\times\Lambda_2\equiv\Lambda$, where $z(0)\in\Lambda_2$. $\mathcal A_\lambda$ is the class of sets containing some neighborhood of the set $\{z(0)=0\}$ in the space $\Lambda$; $\varepsilon_0=0$, $d=1$. The class $\mathcal B_\mu$ and the functional $F_\tau$ are the same as in example 1. In this case the definition of $(\lambda,\mu)$-stability reduces to the definition, given in $^{(2)}$, of stability in probability of the solution $z=0$ of the given system of differential equations.

1. Consider queueing systems described in $^{(3)}$. Let the functional of the process of system operation be $F_\tau=W(\tau)$—the virtual waiting time until the start of service. The class $\mathcal B_\mu$ consists of one set $B$ of uniformly bounded functions on $T$; $F_\tau \in B$, if there exists $B_b\subset B$ such that $F_\tau\in B_b$; $\varepsilon_0=0$. The class $\mathcal A_\lambda$ consists of the one-point fixed set $\{\alpha\}$. The set $\Lambda$ may be of arbitrary nature. In this case the definition of $(\lambda,\mu)$-stability reduces to the definition of substability given by Loynes.

As is known, sufficiently general models of complex systems are aggregative systems $^{(4)}$. Let us consider a class of aggregative systems—autonomous piecewise linear aggregates $^{(5)}$. The internal state $z(t)$ for a piecewise linear aggregate has the following structure: $z(t)\equiv(i,\eta_i)$, $i=1,2,\ldots$, are the principal states of the aggregate, and $\eta_i$ is a vector of dimension $m_i$, whose components are called supplementary coordinates and vary in the domain $\Gamma_i$ of an $m_i$-dimensional space according to the law $d\eta_i/dt=v_i$. When the vector $\eta_i$ reaches the boundary $\gamma_i$ of the domain $\Gamma_i$ at the moment $t$, the internal state of the aggregate changes in accordance with the transition function

\[ P_\alpha(G_j/\eta_i)=\mathscr P\{z(t+0)=(j,\eta_j),\,\eta_j\in G_j\subset\Gamma_j/z(t)= \]
\[ =(i,\eta_i),\,\eta_i\in\gamma_i,\,\alpha\in\Lambda\}. \]

In [5] it was indicated that, for aggregates, one can single out a class of functionals of practical interest such that these functionals are additional coordinates and change according to laws analogous to the laws governing the change of the remaining coordinates. Below we study the stability of aggregates, which may be called practical stability by analogy with the concept introduced for differential equations [1], and give a method for determining the region of stability.

Definition 2. An aggregate functions stably with respect to \(\rho \geqslant 0\) if, for given regions \(B_i \subset \Gamma_i,\ i=1,2,\ldots,\) there exists a subset \(A\) in the set of parameters \(\lambda\), possessing property \(\lambda\), such that for any \(a \in A\)

\[ \mathfrak{P}\{\eta(t)\in \bigvee_i B_i t<\infty\}\geqslant 1-\rho . \]

Here \(\eta(t)\) is understood as the vector \(\eta_i(t)\), if at time \(t\) the aggregate is in the \(i\)-th principal state. It is obvious that Definition 2 is a particular case of Definition 1, with \(\varepsilon_0=\rho=0\). Suppose that the aggregate parameters include the initial conditions \(\eta(0)\), and by \(\alpha\) we shall understand all the remaining parameters of the aggregate. Without loss of generality let us consider the case where, for some \(i\in I\), \(B_i=\Gamma_i\), while for the others \(B_i\) is the empty set. This can always be done if we assume that, when \(\eta_i\) reaches the boundary \(B_i\) lying inside \(\Gamma_i\), the aggregate passes into a certain fictitious absorbing state \(\phi\). Consider the aggregate at the moments of its transition from one principal state to another. At these moments the states of the aggregate \(z=(i,\eta_i)\) form a Markov process with a discrete parameter, with \(\eta_i\in\gamma_i\). Let \(\chi_\alpha^{(n)}(z)\) be the probability that within \(n\) transitions the aggregate will leave the set of admissible states \(I\) at least once, under the condition that \(z(0)=z=(i,\eta_i),\ i\in I,\ \eta_i\in\gamma_i\). Then the functions \(\chi_\alpha^{(n)}\) satisfy the recurrence relation

\[ \chi_\alpha^{(1)}[(i,\eta_i)] = \sum_{j\notin I} P_\alpha(\Gamma_j/\eta_i), \]

\[ \chi_\alpha^{(n+1)}[(i,\eta_i)] = \sum_{j\in I}\int_{\gamma_j} \chi_\alpha^{(n)}[(j,\eta_j)]P_\alpha(C(d\sigma_j)/\eta_i) +\chi_\alpha^{(1)}[(i,\eta_i)], \tag{1} \]

where \(d\sigma_i\) is an elementary “area element” on the hypersurface \(\gamma_i\), and

\[ C(d\sigma_i)=\{\eta_i:\eta_i\in\Gamma_i,\eta_i=\eta_i^*+kv_i,\eta_i^*\in d\sigma_i,\ k\in(-\infty,0]\} \]

is the part of the cylinder constructed on \(d\sigma_i\) that lies in the region \(\Gamma_i\).

The function

\[ \chi_\alpha[(i,\eta_i)] = \lim_{n\to\infty}\chi_\alpha^{(n)}[(i,\eta_i)] \]

gives the probability that the aggregate will not leave the admissible region if \(z(0)=(i,\eta_i)\). Then the region

\[ A_1=\{a,\ (i,\eta_i):\chi_\alpha[(i,\eta_i)]<\rho\} \]

is the region of stability if it satisfies condition \(\lambda\). Thus, the stability problem has been reduced to the estimation of a certain probability. If we assume that \(\chi_\alpha[(i,\eta_i)]=1\) for \(i\notin I\), then the equation for \(\chi_\alpha\) can be written in the form

\[ \chi_\alpha(z)=E_z\chi_\alpha, \tag{2} \]

where \(E_z\chi_\alpha\) is the conditional mathematical expectation of the function \(\chi_\alpha\). It is easy to show that \(\chi_\alpha\) is the minimal nonnegative solution of (2). Relations (1) can be written in a more convenient form if the aggregate is first reduced to the canonical form [5].

Solving equation (2) may prove difficult, although it allows one to find the entire region of stability. In this connection, the following method is proposed for estimating the function \(\chi_\alpha(z)\), which is a certain analogue of Lyapunov’s method.

Theorem 1. Let there exist a function \(V_\alpha(z)\geqslant 0\) such that \(V_\alpha[(i,\eta_i)]\geqslant 1,\ i\notin I;\ E_zV_\alpha - V_\alpha(z)\leqslant 0,\ i\in I\) (it is assumed that \(E_zV_\alpha\) exists). Then \(V_\alpha(z)\geqslant \chi_\alpha(z)\) for all \(\alpha,z\).

Proof. \(V_\alpha(z) \geq E_z V_\alpha \geq \chi_\alpha^{(1)}(z)\). Using (1), by induction we show that \(V_\alpha(z) \geq \chi_\alpha^{(n)}(z)\). Consequently, \(V_\alpha(z) \geq \chi_\alpha(z)\).

It is often of interest to estimate the probability \(\chi_\alpha^{(n)}(z)\). For this purpose one may use the inequality

\[ \chi_\alpha^{(n)}(z) \leq \chi_\alpha^{(k)}(z) + (n-k)\sup_z\left[\chi_\alpha^{(k+1)}(z)-\chi_\alpha^{(k)}(z)\right], \quad k<n, \tag{3} \]

which follows from relations (1). A more general theorem is also valid.

Theorem 2. Let a Markov process with discrete parameter be given, taking at time \(n\) the value \(\zeta_n\) from the space \(X\) and having transition probability at time \(n\) \(p_\zeta^{(n)}(A)\), \(A \subset X\). Define the random time \(\tau(\zeta)=\inf\{n:\zeta(n)\notin Q,\zeta(0)=\zeta\}\) of the first exit from the set \(Q \subset X\), if at the initial time the process was at the point \(\zeta\). Let \(V(\zeta)\geq 0\) be a function on \(X\); \(V(\zeta)\geq 1\) for \(\zeta\in X\setminus Q\); \(E_\zeta^{(n+1)}V - V(\zeta)\leq \Delta_{n+1}\), \(\Delta_{n+1}\geq 0\), \(0\leq n\leq N-1\), \(\zeta\in Q\), where

\[ E_\zeta^{(n)}V=\int_X p_\zeta^{(n)}(d\xi)V(\xi). \]

Then

\[ \chi^{(N)}(\zeta)\leq V(\zeta)+\mathcal{E}\left\{\sum_{i=1}^{\min(\tau,N)}\Delta_i \,/\, \zeta(0)=\zeta\right\}\leq V(\zeta)+\sum_{i=1}^N \Delta_i. \]

Proof. Denote by \(N_1(\zeta)=\min[\tau(\zeta),N]\). Obviously,

\[ \chi^{(N)}(\zeta)=\sum_{i=1}^N \int_Q p_\zeta^{(1)}(d\xi_1)\cdots \int_Q p_{\xi_{i-2}}^{(i-1)}(d\xi_{i-1})\int_{X\setminus Q}p_{\xi_{i-1}}^{(i)}(d\xi_i)V(\xi_i). \tag{4} \]

Since

\[ \int_{X\setminus Q} p_{\xi_{i-1}}^{(i)}(d\xi_i)V(\xi_i) = E_{\xi_{i-1}}^{(i)}V-\int_Q p_{\xi_{i-1}}^{(i)}(d\xi_i)V(\xi_i), \tag{5} \]

we have

\[ \chi^{(N)}(\zeta)\leq V(\zeta)+\sum_{k=1}^N \Delta_k\left(\sum_{i\geq k}\mathcal{P}\{N_1=i\}\right); \tag{6} \]

\[ \sum_{k=1}^N \Delta_k\left(\sum_{i\geq k}\mathcal{P}\{N_1=i\}\right) = \sum_{i=1}^N \mathcal{P}\{N_1=i\}\left(\sum_{k=1}^i \Delta_k\right) = \mathcal{E}\left\{\sum_{i=1}^{N_1}\Delta_i \,/\, \zeta(0)=\zeta\right\}. \tag{7} \]

It follows from (4)—(7) that

\[ \chi^{(N)}(\zeta)\leq V(\zeta)+\mathcal{E}\left\{\sum_{i=1}^{N_1}\Delta_i \,/\, \zeta(0)=\zeta\right\}. \]

Since \(\sum_{i=1}^{N_1}\Delta_i\leq \sum_{i=1}^N\Delta_i\), we have

\[ \mathcal{E}\left\{\sum_{i=1}^{N_1}\Delta_i \,/\, \zeta(0)=\zeta\right\}\leq \sum_{i=1}^N\Delta_i. \]

The theorem is proved.

An analogous theorem, but under more stringent restrictions on the function \(V\), was proved in \({}^{6}\). The computed examples show that inequality (3) gives good estimates already for small \(k\), and therefore it is advisable to take \(\chi^{(k)}(z)\) for some \(k\) as the function \(V(z)\).

The author expresses gratitude to Corresponding Member of the Academy of Sciences of the USSR N. P. Buslenko for advice and constant attention to the present work.

Received
31 II 1967

REFERENCES

1. J. La-Salle, S. Lefschetz, Investigation of Stability by Lyapunov’s Direct Method, Moscow, 1964.
2. I. Ya. Katz, N. N. Krasovskii, PMM, 24, 6, 809 (1960).
3. R. M. Loynes, Proc. Cambridge Phil. Soc., 58, p. 3, 497 (1962).
4. N. P. Buslenko, Izv. AN SSSR, Technical Cybernetics, No. 5, 7 (1963).
5. I. N. Kovalenko, Izv. AN SSSR, Technical Cybernetics, No. 6, 3 (1964).
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