UDC 62-503.4

CYBERNETICS AND CONTROL THEORY

Academician B. N. PETROV, G. A. STEPAN’YANTS

ON THE THEORY OF BRANCHING SYSTEMS

Consider a system of ordinary differential equations reduced to normal form:

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

where \(y, f\) are \(n\)-dimensional vector functions, and the right-hand sides \(f\) of system (1) belong to some locally compact set \(F\), which we shall call the space of right-hand sides. We divide the numerical line (the time axis) \(T\) by division points \(\tau_0,\tau_1,\ldots,\tau_k\) into a finite number of half-intervals \((\tau_0,\tau_1],(\tau_1,\tau_2],\ldots,(\tau_k,\infty)\), to each of which there is assigned some locally compact subset \(F_q\subset F\). We agree that, if \(f(y,t)\in F_q\), then system (1) has on the interval \((t_q,t_{q+1})\supset[\tau_q,\tau_{q+1}]\) a unique solution whose integral trajectory passes through a prescribed point \((y(\tau_q),\tau_q)\in Y\times T\), where \(Y\) is the phase space of system (1). The topological product \(E_q=F_q\times Y\) will be called the state space of the \(q\)-th section, and its element \(z_q=(f_q,y)\in E_q\) the state of system (1). Thus, the concept of a phase state \(y\in Y\) should be distinguished from the concept of a state \(z_q=(f_q,y)\in E_q\). The set \(E_0\) will be called the set of initial states.

Let probability measures \(P_{E0}(z_0), P_{F1}(f,z_0), P_{F2}(f,z_1),\ldots,\ldots,P_{fk}(f,z_{k-1})\) be given on the Borel sets of the locally compact spaces \(E_0,F_1,\ldots,F_k\), respectively, and let the measure \(P_{fq}(f,z_{q-1})\) depend, as on a parameter, on the state \(z_{q-1}\) of the \((q-1)\)-st section of the system.

To each state of the \(q\)-th section of the system \(z_q^0=(f_q^0,y^0)\in E_q\) we assign the solution \(y(t,z_q^0)\) of system (1) for \(f=f_q^0\) and initial conditions \(y^0=y(\tau_q)\), defined on the segment \([\tau_q,\tau_{q+1}]\).

If a probability measure \(P_{Eq}\) is given on the system of Borel sets \(B_q\) of the space \(E_q\), then the triple set \(\{E_q,B_q,P_{Eq}\}\) may be regarded as the basic probability space \((^1)\). If \(\{y(t,z_q^0):z_q^0\in E_q\}=y_q^*(t)\) is the set of solutions of system (1) corresponding to all possible points \(z_q^0\in E_q\), then the set \(y_q^*(t)\), together with the probability space \(\{E_q,B_q,P_{Eq}\}\), is a random process defined on the segment \([\tau_q,\tau_{q+1}]\). We shall call this random process the \(q\)-th section of the branching process of solutions of system (1).

To each sample function \(y(t;z_q^0)=y(t;(f_q^0,y^0))\) we assign the point \(z_q'=(f_q^0,y(\tau_{q+1},z_q^0))\), so that a prescribed continuous mapping of the set \(E_q\) into itself is obtained: \(z_q^0\to y(t,z_q^0)\to z_q'\). The probability distribution \(P_{Eq}\) on \(E_q\) thereby induces a new probability distribution \(P_{Eq}'\) on \(E_q\), namely, for any measurable set \(E_{q0}'\subset E_q\) we have \(P_{Eq}'(E_{q0}')=P_{Eq}(E_{q0})\), where \(E_{q0}\) is the set of all those points \(z_q^0\in E_q\) whose images \(z_q'\in E_{q0}'\) under the mapping \(z_q^0\to y(t,z_q^0)\to z_q'\).

To each measurable set \(E_{(q+1)0}\subset E_{q+1}\) we assign the measurable set \(F_{(q+1)0}\subset F_{q+1}\), equal to the set of all functions \(f_{q+1}\) corresponding to points \(z_{q+1}\in E_{(q+1)0}\), and the set \(E_{q0}'\subset E_q\) of points \(z_q'=(f_q,y(\tau_{q+1}))\in E_q\) whose phase states \(y(\tau_{q+1})\) coincide with the phase states of the points \(z_{q+1}\in E_{(q+1)0}\). We put, by definition,

\[ P_{E(q+1)}(E_{(q+1)0})=\int_{E'_{q0}}\int_{F_{(q+1)0}} dP_{F(q+1)}(f,z_q)dP_{Eq}(z_q). \tag{2} \]

Definition 1. We shall call the system (1) branching if the half-intervals \((\tau_0,\tau_1],\ldots,(\tau_k,\infty)\), the sets \(E_0,F_1,\ldots,F_k\), and the probability measures \(P_{E0},P_{F1},\ldots,P_{Fk}\) are given. The probability measures \(P_{Eq}(z_q)\), defined on the Borel subsets of \(E_q=F_q\times Y\) by relation (2), will be called the probability distribution on the space of initial states of the \(q\)-th segment.

By a realization of the branching process of solutions \(y_\xi(t)\) we shall mean any continuous real-valued function defined on \([\tau_0,\infty)\) for which there exists a sequence of sample functions \(y_{\xi0}(t),y_{\xi1}(t),\ldots,\ldots,y_{\xi k}(t)\) of the solution segments of the branching system (1) such that \(t\in(\tau_q,\tau_{q+1}]\Rightarrow y_\xi(t)=y_{\xi q}(t)\).

The asymptotic behavior of branching systems depends on their properties on each branching segment.

Definition 2. A branching system is called stable at the point \(z_0\in E_0\) with respect to the attracting set \(W\subset Y\) if, for every neighborhood \(U\) of the set \(W\), there exist a neighborhood \(V\) of the point \(z_0\) and a \(\tau\) such that, for any realizations \(y_{0\xi}(t)\) and \(y_\xi(t)\) of the branching process of solutions corresponding respectively to the point \(z_0\) and to an arbitrary point \(z\in V\), the difference \((y_\xi(t)-y_{0\xi}(t))\in U\) for any \(t>\tau\).

Definition 3. A branching system is called stable in the strong sense at the point \(z_{q0}\in E_q\) on the \(q\)-th half-interval \((\tau_q,\tau_{q+1}]\) with respect to the attracting set \(W_q\subset Y\), for an admissible initial scatter \(W_{q0}\subset Y\), if for every neighborhood \(U\) of the set \(W_q\) there exist a neighborhood \(U'\) of the set \(W_{q0}\) and a \(\tau\in(\tau_q,\tau_{q+1}]\) such that, for any sample function \(y(t,z_q)\) for which the difference \([\Pr_Y(z_q)-\Pr_Y(z_{q0})]\in U'\) for all \(t\ge \tau\), the difference \([y(t,z_q)-y(t,z_{q0})]\in U\).

Definition 4. The set \(E_{00}\) of all points \(z_0\in E_0\) at which the branching system is stable is called the stable set.

In a natural way one introduces the definition of a stable set for each branching segment.

The following theorem holds, establishing the connection between the stability of a branching system and its stability on each branching segment.

Theorem 1. For stability of a branching system at the point \(z_0\) with respect to the attracting set \(W\), it is sufficient that the following conditions be fulfilled:

1. For an arbitrary \(q\)-th branching segment the stable set \(E_{q0}\) is nonempty, so that at each of its points \(z_{q0}\) the system is stable in the strong sense on the half-interval \((\tau_q,\tau_{q+1}]\), and the attracting set \(W_q\) and the admissible initial scatters \(W_{q0}\) of each branching segment are related by the relation \((W_{q-1}-W_{q-1})\subset W_{q0}\), where \((W_{q-1}-W_{q-1})\) denotes, as usual, the set of differences of all possible pairs of points from \(W_{q-1}\).

2. There exists a realization \(y_{0\xi}^{0}(t)\) of the branching process of solutions corresponding to the initial state \(z_0\) such that the point \((f_q,y)\in E_{q0}\) for any \(f_q\in F_q\) and any \(y\) for which \([y-y_{0\xi}^{0}(\tau_q)]\in W_{q-1}\).

3. For the last \((k\)-th) segment, any neighborhood of the set \(W\) may be taken as the attracting set.

Proof. a) For an arbitrary \(q\), for every neighborhood \(U(W_q)\) of the attracting set \(W_q\), there exists such a neighborhood \(U[y_{0\xi}^{0}(\tau_q)]\) of the point \(y_{0\xi}^{0}(\tau_q)\in Y\), containing all points \(y\) for which \([y-y_{0\xi}^{0}(\tau_q)]\in W_{q-1}\), that for any initial states of the \(q\)-th segment \(z_{qa}=(f_{qa},y_a)\) and \(z_{qb}=(f_{qb},y_b)\), for which the phase states \(y_a\in\)

\(\in U[y_{0\xi}^{0}(\tau_q)]\), \(y_b \in U[y_{0\xi}^{0}(\tau_q)]\), the difference \([y_a(\tau_{q+1}) - y_b(\tau_{q+1})] \in U(W_q)\) for any \(f_q \in F_q\).

Indeed, by the conditions of the theorem, from the relation \([y_a - y_{0\xi}^{0}(\tau_q)] \in W_{q-1}\) it follows that, for any \(f_q \in F_q\), the point \(z_{qa} = (f_q, y_a) \in E_{q0}\); and since \(W_{q0} \supset (W_{q-1} - W_{q-1})\), for every neighborhood \(U(W_q)\) of the set \(W_q\) there is a neighborhood \(U(W_{q0})\) of the set \(W_{q0}\) such that, for any \(z_{qa} = (f_q, y_a) \in E_{q0}\), the relation \([y_a(\tau_q) - y_b(\tau_q)] \in U(W_{q-1})\) entails \([y_a(\tau_{q+1}) - y_b(\tau_{q+1})] \in U(W_q)\).

On the other hand, if \((y_a - y_{0\xi}^{0}) \in W_{q-1}\), \((y_b - y_{0\xi}^{0}) \in W_{q-1}\), then \((y_a - y_b) \in (W_{q-1} - W_{q-1})\); therefore, for any neighborhood \(U(W_{q-1} - W_{q-1})\) of the set \((W_{q-1} - W_{q-1})\), there is a neighborhood \(U(W_{q-1})\) of the set \(W_{q-1}\) such that, if \(U(W_{q-1})\) contains both \((y_a - y_{0\xi}^{0})\) and \((y_b - y_{0\xi}^{0})\), then \((y_a - y_b) \in U(W_{q-1} - W_{q-1})\).

The constructions given prove assertion a), if one takes into account that the set \(W_{q0}\) contains the origin of the coordinates of the phase space, and every neighborhood of the set \(W_{q0}\) is a neighborhood of \((W_{q-1} - W_{q-1})\), since, by assumption, \(W_{q0} \supset (W_{q-1} - W_{q-1})\).

For the initial interval, the conditions of item a) are fulfilled by virtue of the continuous dependence of solutions of differential equations on initial conditions and on the form of the right-hand side.

b) Since, by assumption, \((W_{q-1} - W_{q-1}) \subset W_{q0}\), on the basis of item a), for the neighborhood \(U(W_{q-1})\) there is a neighborhood \(U(W_{q-2})\) such that assertion a) is valid for the neighborhoods \(U(W_{q-2})\) and \(U(W_{q-1})\), and hence also for the neighborhoods \(U(W_{q-2})\) and \(U(W_{q0})\). By induction we obtain that, for every neighborhood \(U(W_{k-1})\), there is a neighborhood \(V(z_0)\) of the point \(z_0\) such that, for \(z \in V(z_0)\), the difference \([y_\xi(\tau_k, z) - y_\xi(\tau_k, z_0)] \in U(W_{k0})\).

By virtue of item 3 of the theorem, as the set \(W_k\) one may take an open neighborhood of the set \(W\); and since an open set is a neighborhood of each of its points, there are \(t_k > \tau_k\) and a neighborhood \(U(W_k)\) such that, for any two points \(y_a, y_b \in W_{k0}\), the difference \([y_a(t) - y_b(t)] \in W_k\) for any \(f_a \in F_k\), \(f_b \in F_k\), \(t > t_k\). Together with the preceding result, the relation obtained proves the theorem.

This theorem makes it possible to estimate the stability of branching systems, reducing the stability analysis to the successive determination of the set of admissible initial deviations of each system describing the motion on the individual intervals.

Theorem 1 also makes it possible to estimate the probability of stable operation of a system. To do this, we proceed as follows. To each set \(F_q\) we assign its measurable subset \(F_{q0}(z_0)\), depending, possibly, on the initial state \(z_0 \in E_0\) and having the property that, for system (1), sufficient stability conditions are satisfied at the point \(z_0\) if each \(F_q \subset F\) is replaced by the set \(F_{q0}(z_0) \subset F_q\). In the case where such sets do not exist, put \(F_{q0}(z_0) = \varnothing\). Defining probability measures of the sets \(F_{q0}(z_0)\) and knowing the probability distribution on the space of initial states, one can estimate the probability of stable operation of the system.

Example. Suppose there is a first-order branching system whose equation is \(\dot y = f(y,t)\), with two branching intervals corresponding to the half-intervals \((0,\tau_1]\), \((\tau_1,\tau_2]\), and \((\tau_2,\infty)\). Let the set \(F\) consist of the single element \(f_0(y,t) = -ay\), \(a > 0\); let the set \(F_1\) consist of constant functions \(f(y) = x = \mathrm{const}\), \(x \in (-b_1,b_2)\); and let the set \(F_2\) be the set of single-valued continuous functions \(f(y) = f^{0}(y) + \delta\), \(\delta \in (-\varepsilon,\varepsilon)\), with \(f^{0}(y) = y\) for \(|y| < C\), and \(f^{0}(y) = C\,\operatorname{sgn} y - y\) for \(|y| \ge C\). An arbitrary realization \(y_\xi(t)\) of the branching process of solutions, corresponding to \(y(0) = y^{0}\), can be written in the form \(y_\xi(t) = y^{0}e^{-at}\) for \(t \in (0,\tau_1)\), and \(y_\xi(t) = y^{0}e^{-a\tau_1} + (t-\tau)x\), \(x \in (-b_1,b_2)\), for \(t \in (\tau_1,\tau_2]\). It is verified directly that on

on the penultimate (first) branching segment the system is strongly stable at every point with respect to the set $W_1=[-b_1(\tau_2-\tau_1),\, b_2(\tau_2-\tau_1)]$, and on the last (second) segment the system is strongly stable at every point with respect to the set $W_2=[-\varepsilon,\varepsilon]$, with initial phase state $y(\tau_2)\in(-C,+C)$. In this case the set of admissible initial spreads $W_{20}$ can be represented in the form of the interval $W_{20}=(y(\tau_2)-C,\ y(\tau_2)+C)$, and the set $(W_1-W_1)=[-(b_1+b_2)(\tau_2-\tau_1)+(b_1-b_2)(\tau_2-\tau_1)]$.

If, as the realization $y_{\xi0}(t)$ entering into the formulation of the theorem, one chooses the realization corresponding to $y(0)=y^0$, $x=0$, and $f_2(y)=f^0(y)$, so that
$y_\xi(t)=(y^0 e^{-a\tau_1}+C\,\operatorname{sgn} y^0)e^{t-\tau_2}-C\,\operatorname{sgn} y^0$ for $t\in(\tau_2,\infty)$, then the sufficient stability condition corresponding to the relation $W_{20}\supset(W_1-W_1)$ will be the inequality $(b_1+b_2)(\tau_2-\tau_1)\le C-|y^0 e^{-a\tau_1}|$. In the case where the probability distribution on $F_2$ is given as the uniform distribution on the interval $(-\varepsilon,\varepsilon)$ of the random variable $\delta$, this same inequality is a sufficient condition for the probability of stable operation to be not less than $0.5$ for the attracting set $W=[-\varepsilon/2,\varepsilon/2]$.

Moscow Aviation Institute
named after S. Ordzhonikidze

Received
19 II 1969

REFERENCES

1. K. Ito, Probability Processes, vol. 1, IL, 1960.