Full Text
ON A CERTAIN PROPERTY OF A LINEAR STABLE SYSTEM COMPLETELY CONTROLLED BY A RANDOM EXTERNAL ACTION
N. N. KRASOVSKII
The article considers an asymptotically stable linear system subjected to a scalar random action, by which it is completely controllable. It is shown that, with probability arbitrarily close to unity, every motion of the system, as time increases, will visit an arbitrary neighborhood of any point \(x^*\) of the phase space. In the case of incomplete controllability of the system, only points \(x^*\) from a certain linear subspace \(W\) of the phase space possess the indicated property.
§ 1. Consider the vector linear differential equation
\[ \frac{dx}{dt}=Ax+b\eta, \tag{1.1} \]
where \(x\) is an \(n\)-vector of phase coordinates \(\{x_i\}\); \(A\) is an \(n\times n\) matrix with constant coefficients \(a_{ij}\); \(b\) is a constant \(n\)-vector \(\{b_i\}\); \(\eta\) is a scalar random function of time. We shall assume that the function \(\eta(t)\) is a sequence of random impulses
\[ \eta(t)=\sum_{j=1}^{\infty}\gamma_j\delta(t-\tau_j). \tag{1.2} \]
Here \(\tau_j\) are the values of the random variable \(\tau\), distributed on the half-axis \(0\le t<\infty\) according to the Poisson law with frequency \(\lambda\). Consequently, the probability \(P_m(T)\) that an interval of time of length \(T\) contains \(m\) values \(\tau_j\) is determined by the equality ([1], p. 102)
\[ P_m(T)=\frac{(\lambda T)^m e^{-\lambda T}}{m!}. \tag{1.3} \]
The numbers \(\gamma_j\) in (1.2) are values of the random variable \(\gamma\) with the normal distribution law ([1], p. 119)
\[ p_\gamma(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{x^2}{2\sigma^2}}. \tag{1.4} \]
The distribution (1.4) has been chosen for definiteness. The arguments are carried out according to the same plan for any continuous distribution \(p_\gamma(x)\) with zero mathematical expectation
\[ M\{\gamma\}=0 \tag{1.5} \]
and finite variance
\[ M\{\gamma^2\}=\sigma^2. \tag{1.6} \]
The random variables \(\tau\) and \(\gamma\) are assumed independent. The symbol \(\delta(t)\) in (1.2) and below denotes the impulse \(\delta\)-function.
Let us clarify the meaning of the random motion \(x(t,x(t_0)=x^0)_{(1.1)}\) described by the stochastic equation (1.1). Suppose that in (1.2) some definite function
\[
\eta^\omega(t)=\sum \gamma_j^\omega \delta(t-\tau_j^\omega)
\]
has been realized. (Here and below the superscript \(\omega\) marks a certain realization of a random variable or random function.) Corresponding to the realizations \(\eta^\omega(t)\), in accordance with equation (1.1), there will be realizations \(x^\omega(t,x(t_0)=x^0)_{(1.1)}\) of the motion \(x(t,x(t_0)=x^0)_{(1.1)}\), and the equality \(x^\omega(t_0,x(t_0)=x^0)_{(1.1)}=x^0\) holds. On the intervals \(\tau_j^\omega<t\leq \tau_{j+1}^\omega\), the function \(x^\omega(t,x(t_0)=x^0)_{(1.1)}\) is a continuous solution of the system
\[ \frac{dx}{dt}=Ax, \tag{1.7} \]
and at \(t=\tau_j^\omega\) this function has a discontinuity
\[ \Delta_j x^\omega=b\gamma_j^\omega . \]
Consequently, we assume that the realizations \(x^\omega(t,x(t_0)=x^0)_{(1.1)}\) are piecewise-continuous functions of \(t\) which, at the points of discontinuity \(t=\tau_j^\omega\), are continuous from the left. Thus, to each set of realizations \(\{\tau_j^\omega,\gamma_j^\omega\}\) there corresponds a set of motions \(\{x^\omega(t,x(t_0)=x^0)_{(1.1)}\}\). To these sets of motions \(x^\omega(t,x(t_0)=x^0)_{(1.1)}\) one may assign probabilities equal to the probabilities of occurrence of the aggregate realizations \(\tau_j^\omega,\gamma_j^\omega\) that generated them. Such a visual representation of the random motion \(x(t,x(t_0)=x^0)_{(1.1)}\) is sufficient for what follows. Realizations \(\tau^\omega,\gamma^\omega\) of a more complicated nature than the sequences \(\{\tau_j^\omega,\gamma_j^\omega\}\) described above may be disregarded, assuming the probability of their occurrence to be zero ([2], p. 241).
Thus, for example, the probability that on the interval \([t_0,t_0+T]\) the motion \(x(t,x(t_0)=x^0)_{(1.1)}\) is continuous is equal to \(e^{-\lambda T}\). Indeed, in accordance with (1.3) and (1.4), this is precisely the probability that on the interval \([t_0,t_0+T]\) not a single impulse \(\gamma\delta(t-\tau)\) with \(\gamma\ne 0\) is realized.
Along with system (1.1), we shall consider the controlled system
\[ \frac{dx}{dt}=Ax+bu, \tag{1.8} \]
where, unlike in (1.1), the function \(u(t)\) is assumed deterministic and may be chosen at will.
System (1.8) is called completely controllable [3] by the action \(u\) if, for any two points \(x^{(1)}\) and \(x^*\) and time interval \(t_1\leq t\leq t_*\) \((t_*>t_1)\), there exists a control \(u(t)\) that transfers system (1.8) from
\[
x(t_1)=x^{(1)}
\]
to the state
\[ x(t_*)=x^* . \tag{1.9} \]
We shall choose the control \(u(t)\) in the class of impulse functions of the form
\[ u(t)=\sum_{k=1}^{l}\beta_k\delta(t-\vartheta_k), \tag{1.10} \]
moreover, as in the case of system (1.1), we shall regard the motion \(x(t, x(t_1)=x^{(1)}, u)_{(1.8)}\) of system (1.8) at the points \(t=\vartheta_k\) as continuous from the left. In accordance with this, if in the control problem in (1.10) \(\vartheta_l=t_*\), then under \(x(t_*, x(t_1)=x^{(1)}, u)_{(1.8)}\) in condition (1.9) we shall understand \(x(t_*+0, x(t_1)=x^{(1)}, u)_{(1.8)}=x^*\).
Let us note that system (1.7) is asymptotically stable in the sense of Lyapunov [4]. System (1.8) is completely controllable (including in the class of impulse controls (1.10)) if the vectors
\[ b, Ab,\ldots,A^{n-1}b \tag{1.11} \]
are linearly independent [3, 5].
The purpose of the present article is, using only the basic concepts of probability theory, to show by methods of the qualitative theory of differential equations that, under the conditions of asymptotic stability of system (1.7) and complete controllability of system (1.8), the stochastic system (1.1) has the following property: whatever the initial state \(x(0)=x^0\), the random motion \(x(t,x(0)=x^0)_{(1.1)}\) of system (1.1) for \(t>0\) visits an arbitrary neighborhood of any point \(x^*\) with probability as close to unity as desired.
§ 2. Let us formulate the main theorem. We shall denote by \(\|x\|\) the Euclidean norm of the vector \(x\), i.e. \(\|x\|=(x_1^2+\ldots+x_n^2)^{1/2}\).
Theorem 2.1. Suppose the following conditions are fulfilled: 1. System (1.7) is asymptotically stable. 2. System (1.8) is completely controllable. Then, whatever the numbers \(\lambda>0\), \(\sigma>0\), \(\varepsilon>0\), \(p<1\) and the points \(x^0\), \(x^*\), the event \(\{\|x(t^*, x(0)=x^0)_{(1.1)}-x^*\|<\varepsilon\}\) (at least for one \(t=t^*>0\)) occurs with probability
\[ P\{\|x(t^*, x(0)=x^0)-x^*\|<\varepsilon\}>p. \tag{2.1} \]
We first prove two auxiliary assertions.
Lemma 2.1. Suppose system (1.7) is asymptotically stable. Then, whatever the numbers \(\lambda>0\), \(\sigma>0\), \(q<1\) and the point \(x^0\), one can specify a number \(N\) such that the event \(\{\|x(t,x(0)=x^0)_{(1.1)}\|<N\}\) for every \(t\ge 0\) occurs with probability
\[ P\{\|x(t,x(0)=x^0)_{(1.1)}\|<N\}>q. \tag{2.2} \]
We prove the lemma. If system (1.7) is asymptotically stable, then one can construct a Lyapunov function, a positive definite quadratic form,
\[ v(x)=\sum_{i,j=1}^{n}\alpha_{ij}x_i x_j, \tag{2.3} \]
whose derivative \(dv/dt\) by virtue of equation (1.7) satisfies the equality [4]
\[ \frac{dv}{dt} = \sum_{i=1}^{n}\frac{\partial v}{\partial x_i} \sum_{j=1}^{n}a_{ij}x_j = -\|x\|^2. \tag{2.4} \]
Let us compute the mean value ([6], p. 224) \(dM\{v\}/dt\) of the derivative \(dv(x^\omega(t,x(\vartheta)=x)_{(1.1)})/dt\) of the function \(v\) (2.3) along the random motion \(x(t)\) of system (1.1), issuing at some instant \(\vartheta\) from a fixed point \(x\). We obtain ([6], p. 225)
\[ \frac{dM\{v\}}{dt} = \sum_{i=1}^{n}\frac{\partial v}{\partial x_i} \sum_{j=1}^{n}a_{ij}x_j+ \]
\[ +\frac{1}{2}\sum_{i,j=1}^{n}\frac{\partial^{2}v}{\partial x_i \partial x_j}\, b_i b_j \lambda \sigma^{2} = -\|x\|^{2}+\nu \quad (\nu=\mathrm{const}). \tag{2.5} \]
Consider the function
\[ V(t)=M\{v(x(t,x(0)=x^{0})_{(1.1)})\}, \tag{2.6} \]
equal to the mathematical expectation of the random values \(v(x^\omega(t,x(0)=x^0)_{(1.1)})\), computed over all realizations \(x^\omega(t,x(0)=x^0)_{(1.1)}\). The function \(V(t)\) is differentiable, and the equality holds ([6], p. 225)
\[ \frac{dV}{dt} = M\left\{\frac{dM\{v\}}{dt}\right\} = -M\{\|x(t,x(0)=x^0)_{(1.1)}\|^{2}\}+\nu . \tag{2.7} \]
The derivation of equality (2.7) is based on the formula of iterated mathematical expectations ([11], p. 159) and on the Markov property ([11], pp. 283–284) of the process \(x(t)_{(1.1)}\).
The positive definite form \(v(x)\) (2.3) satisfies the estimates
\[ e_1\|x\|^{2}\le v(x)\le e_2\|x\|^{2} \]
\[ (e_1>0,\ e_2>0\ —\ \mathrm{const}). \tag{2.8} \]
From (2.7) and (2.8) there follows the inequality
\[ \frac{dV}{dt}\le -\frac{V(t)}{e_2}+\nu . \tag{2.9} \]
Integrating inequality (2.9), we see that, for any choice of the initial condition \(V(0)=M\{v(x^0)\}=v(x^0)\) and for any \(\nu>0\), one can specify a number \(Q[x^0,\nu]\) such that
\[ V(t)<Q[x^0,\nu]\quad \text{for all } t\ge 0. \tag{2.10} \]
From (2.6), (2.8), and (2.10) there follows the inequality
\[ M\{\|x(t,x(0)=x^0)_{(1.1)}\|^{2}\}<\frac{Q}{e_1} \tag{2.11} \]
or, according to Chebyshev’s inequality ([11], p. 187),
\[ P\{\|x(t,x(0)=x^0)_{(1.1)}\|<N\}>q \]
provided that \(N^{2}>Q/e_1(1-q)\). The lemma is proved.
Let us now consider the controlled system (1.8) and estimate the variation of its motion \(x(t,x(t_1)=x^{(1)},u)_{(1.8)}\) under variation of the control \(u(t)\) (1.10) and of the initial condition \(x^{(1)}\). Suppose that an interval \([t_1,t_1+T]\) is chosen, on which two controls are given
\[ u^{(i)}(t)=\sum_{k=1}^{l}\beta_k^{(i)}\,\hat{\delta}(t-\vartheta_k^{(i)}-t_1) \]
\[ (i=1,2;\ \vartheta_k^{(1)}\in[0,T],\ \vartheta_k^{(2)}\in[0,T)). \tag{2.12} \]
By Cauchy’s formula for solutions of nonhomogeneous systems ([7], p. 172), we have
\[ \begin{aligned} &x\bigl(t_1+T,\ x(t_1)=x^{(2)},\ u^{(2)}\bigr)_{(1.8)} \\ &\quad - x\bigl(t_1+T+0,\ x(t_1)=x^{(1)},\ u^{(1)}\bigr)_{(1.8)} \\ &= F[T]\bigl(x^{(2)}-x^{(1)}\bigr) \\ &\quad + \int_0^T F[T-\vartheta]\,b \left\{ \sum_{k=1}^{l} \left[\beta_k^{(2)}\delta\bigl(\vartheta-\vartheta_k^{(2)}\bigr) -\beta_k^{(1)}\delta\bigl(\vartheta-\vartheta_k^{(1)}\bigr)\right] \right\}\,d\vartheta . \end{aligned} \tag{2.13} \]
From (2.13), relying on the continuity of the fundamental matrix \(F[t]\) of system (1.7), we are convinced of the validity of the following lemma.
Lemma 2.2. Let the numbers \(T\), \(\beta_k^{(1)}\), and \(\vartheta_k^{(1)}\) \((k=1,\ldots,l)\) be chosen. Then, for any \(\varepsilon>0\), one can indicate a number \(\Delta>0\) such that the inequality
\[ \begin{aligned} \bigl\|\, &x\bigl(t_1+T,\ x(t_1)=x^{(2)},\ u^{(2)}\bigr)_{(1.8)} \\ &-x\bigl(t_1+T+0,\ x(t_1)=x^{(1)},\ u^{(1)}\bigr)_{(1.8)}\,\bigr\| < \varepsilon, \end{aligned} \tag{2.14} \]
will hold provided only that
\[ \|x^{(2)}-x^{(1)}\|<\Delta,\quad |\beta_k^{(1)}-\beta_k^{(2)}|<\Delta,\quad |\vartheta_k^{(2)}-\vartheta_k^{(1)}|<\Delta \tag{2.15} \]
\[ (k=1,\ldots,l), \]
whatever the number \(t_1\) may be.
Relying on Lemmas 2.1 and 2.2, let us verify the validity of Theorem 2.1. Let the numbers \(\lambda\), \(\sigma\), \(\varepsilon\), \(p\) and the points \(x^0\), \(x^*\) be given. Choose the number \(q\) from the condition
\[ q=p^{1/2}. \tag{2.16} \]
In accordance with Lemma 2.1, for \(x^0\), \(\lambda\), \(\sigma\), and \(q\) (2.16), choose the number \(N\) so that inequality (2.2) is satisfied. Fix some number \(T\) and, in accordance with the results of [5] mentioned in § 1, construct a control \(u^{(1)}(t)\) (2.12) solving problem (1.9) for \(x^{(1)}=0\), \(t_*=t_1+T\), and \(x^*\), coinciding with \(x^*\) under the conditions of Theorem 2.1 (\(t_1\) is as yet an undetermined number). By the number \(\varepsilon\) and by the chosen \(T\) and \(u^{(1)}\), determine, in accordance with Lemma 2.2, the number \(\Delta\) ensuring inequality (2.14). System (1.7) is asymptotically stable. Consequently, one can indicate a number \(T^0\) such that for the motion \(x\bigl(t,\ x(t_0)\bigr)_{(1.7)}\) of this system, for any \(t_0\), the inequality
\[ \bigl\|x\bigl(t_0+T^0,\ x(t_0)\bigr)_{(1.7)}\bigr\|<\Delta, \tag{2.17} \]
will hold, provided only that
\[ \|x(t_0)\|\le N . \tag{2.18} \]
Consequently, if on the interval \([t_0,\ t_0+T^0+T)\) the control \(u(t)=\gamma^\omega(t)\) is chosen so that
\[ \gamma^\omega(t)=0\quad \text{for } t_0\le t<t_0+T^0, \]
\[ \gamma^\omega(t)=\sum_{k=1}^{l}\gamma_k^\omega\,\delta\bigl(t-\tau_k^\omega\bigr) \quad \text{for } t_0+T^0\le t<t_0+T^0+T \tag{2.19} \]
\[ \bigl(\tau_k^\omega \in [\,t_0+T^0\le t<t_0+T^0+T\,]\bigr), \]
\[ |\gamma_k^\omega-\beta_k^{(1)}|<\Delta,\quad |\tau_k^\omega-t_0-T^0-\vartheta_k^{(1)}|<\Delta \tag{2.20} \]
\[ (k=1,\ldots,l), \]
then, under condition (2.18), the inequality
\[ \|x(t_0+T^0+T,\ x(t_0),\ \gamma^\omega)_{(1.8)}-x^*\|<\varepsilon . \tag{2.21} \]
holds. Consider the sequence of numbers
\[ T_0=0,\quad T_i=(T^0+T)i\quad (i=1,\ 2,\ \ldots). \]
Let us call \(A_j\) the event consisting in the fact that, on the interval \([T_j,T_{j+1})\), in system (1.1) there is realized a function \(\gamma^\omega(t)\) satisfying conditions (2.19), (2.20). The events \(A_j\) are mutually independent and have equal positive probabilities \(P(A_j)\), which can be computed from (2.20) and (1.3), (1.4).
By Bernoulli’s theorem ([1], p. 96), one can indicate a number \(m\) such that during the time interval \(T_m\) the event \(A_j\) \((j=0,\ \ldots,\ m-1)\) occurs at least once with probability
\[ P\left\{\sum_{j=0}^{m-1} A_j\right\}>p^{\frac12}, \tag{2.22} \]
where \(p\) is the number from the conditions of Theorem 2.1.
Let us call \(B_j\) the event consisting in the fact that the inequality
\[ \|x(T_j,\ x(0)=x^0)_{(1.1)}\|<N, \tag{2.23} \]
holds, where \(x^0\) is the point from the conditions of the theorem, and \(N\) is the number from Lemma 2.1. By Lemma 2.1 the inequality
\[ P\{B_j\}>q=p^{\frac12} \tag{2.24} \]
holds.
Let us call \(E_j\) the event which consists in the occurrence of the event \(A_j\) and, at the same time, in the nonoccurrence of the events \(A_i\) for \(i=j+1,\ldots,m-1\). The events \(B_j\) and \(E_j\) are independent.
Consider the event
\[ G=\sum_{j=0}^{m-1} B_jE_j . \tag{2.25} \]
The event \(G\) occurs if and only if, for at least one \(j=0,\ldots,m-1\), the events \(E_j\) and \(B_j\) occur simultaneously. Since the events \(E_i\) and \(E_j\) for \(i\ne j\) are incompatible, and the events \(E_j\) and \(B_j\) are independent, we have
\[ P\{G\}=\sum_{j=0}^{m-1} P\{B_j\}P\{E_j\}, \tag{2.26} \]
or, by virtue of (2.24),
\[ P\{G\}>p^{\frac12}\sum_{j=0}^{m-1} P\{E_j\} =p^{\frac12}P\left\{\sum_{j=0}^{m-1} E_j\right\}. \tag{2.27} \]
But the events \(\sum_{j=0}^{m-1} A_j\) and \(\sum_{j=0}^{m-1} E_j\) are equivalent; therefore, from (2.22) and (2.27) it follows that
\[ P\{G\}>p. \tag{2.28} \]
Thus, one may assert that with probability greater than \(p\), on the interval \([0,T_m]\) the following situation will occur at least once: the point \(x(T_j, x(0)=x^0)_{(1.1)}\) will satisfy inequality (2.23) (\(j=0\) or \(j=1,\ldots,j=m-1\)), and on the following interval \([T_j,T_{j+1})\) an action \(\eta^\omega(t)\) satisfying conditions (2.19), (2.20) will then be realized. But, by the choice of all the quantities mentioned above, this also means that for at least one \(T_j\) (\(j\leq m\)) the point \(x(T_j, x(0)=x^0)_{(1.1)}\) lies in the \(\varepsilon\)-neighborhood of the point \(x^*\) with probability greater than \(p\). The theorem is proved.
§ 3. We shall show that in the case of instability of system (1.7) the assertion constituting the content of Theorem 2.1 may fail to hold. The following assertion is true.
Theorem 3.1. Let system (1.7) be unstable, and suppose that among the eigenvalues of the matrix \(A\) there is at least one number \(\rho_j\) with positive real part. Then one can specify numbers \(p>0\), \(\varepsilon>0\), \(\lambda>0\), \(\sigma>0\), and a point \(x^0\) such that the probability of the event \(\{\|x(t,x(0)=x^0)_{(1.1)}\|>\varepsilon\) for all \(t\geq 0\}\) satisfies the inequality
\[ P\{\|x(t,x(0)=x^0)_{(1.1)}\|>\varepsilon \text{ for all } t\geq 0\}>p . \tag{3.1} \]
Consequently, here, in contrast to Theorem 2.1, the random motion \(x(t,x(0)=x^0)_{(1.1)}\) will not, with probability arbitrarily close to one, approach arbitrarily close to the point \(x^*=0\).
Let us verify the validity of Theorem 3.1. Under the conditions of the theorem one can specify a quantity \(y\), composed of the coordinates \(x_i\) of system (1.8) and satisfying the condition
\[ \frac{dy}{dt}=ay\,(a>0-\mathrm{const}). \tag{3.2} \]
If the number \(\rho_j\) with \(\operatorname{Re}\rho_j>0\) is real, then as \(y\) in (3.2) we take the linear combination \(y=l_1x_1+\cdots+l_nx_n\), satisfying the condition
\[ \frac{dy}{dt}=\rho_j y . \tag{3.3} \]
In the case of complex \(\rho_j\), as \(y\) we choose the quantity
\[ y=\sqrt{y_1^2+y_2^2},\qquad y_i=\sum_{j=1}^{n} l_j^{(i)}x_j, \tag{3.4} \]
where the quantities \(l_j^{(i)}\) determine the linear combinations \(y_i\) (\(i=1,2\)) of the coordinates \(x_j\), satisfying the equations:
\[ \frac{dy_1}{dt}=(\operatorname{Re}\rho_j)y_1-(\operatorname{Im}\rho_j)y_2, \]
\[ \frac{dy_2}{dt}=(\operatorname{Im}\rho_j)y_1+(\operatorname{Re}\rho_j)y_2. \tag{3.5} \]
The quantity \(y\) on motions of system (1.1) satisfies the inequality
\[ \frac{dy}{dt}\geq ay-\varkappa\|b\|\,|\eta(t)|, \tag{3.6} \]
where \(\chi\) is some positive constant. Transform inequality (3.6) by the change of variable
\[ z=e^{-at}y, \tag{3.7} \]
then
\[ \frac{dz}{dt}\gg -\chi\|b\|e^{-at}|\eta(t)|. \tag{3.8} \]
Consider the line given in the plane \(\{t,z\}\) by the equation
\[ z=z^0(t)=e^{-\frac{ah(k+1)}{2}} \quad \text{for } hk<t\le h(k+1)\quad (k=0,1,\ldots), \tag{3.9} \]
where \(h\) is some positive number.
Let, at \(t=0\), the point \(x(0)=x^0\) be chosen so that \(z(0)=1\). We shall assume that the realizations \(x^\omega(t,x(0)=x^0)_{(1.1)}\), which generate the realizations \(z^\omega(t)\) of the random variable \(z(t)\) (3.7), are terminated as soon as the inequality
\[ z^\omega(t)>z^0(t) \tag{3.10} \]
is violated.
Estimate the probability of termination of the realizations \(z^\omega(t)\) of the random variable \(z(t)\) on the interval
\[ kh<t\le (k+1)h. \tag{3.11} \]
The mathematical expectation \(M_h\{|\Delta z^-|\}\) of the total downward jump of \(z(t)\) on the interval under consideration (3.11), by virtue of (1.3), (1.4), and (3.8), is estimated by the quantity
\[ M_h\{|\Delta z^-|\}\le \frac{\chi\|b\|e^{-akh}}{\sigma\sqrt{2\pi}} \int_{-\infty}^{\infty}|\zeta|e^{-\frac{\zeta^2}{2\sigma^2}}\,d\zeta \times \]
\[ \times \int_0^h \lambda\,d\vartheta =\chi^0 h\lambda\sigma e^{-akh} \quad (\chi^0>0\text{ — const}). \tag{3.12} \]
(Positive \(\Delta z^+\) we ignore, thereby increasing the estimate of \(M_h\{|\Delta z^-|\}\).)
In order that, on the interval (3.11), a termination of \(x^\omega(t,x(0)=x^0)_{(1.1)}\) occur, it is necessary that the magnitude of the total jump \(|\Delta z^\omega-|\) on (3.11) be no less than
\[ |\Delta z^0|=e^{-\frac{akh}{2}}\left(1-e^{-\frac{ah}{2}}\right). \tag{3.13} \]
From (3.12) and (3.13), by Chebyshev’s inequality, we conclude that the probability \(P(k)\) of termination of the realization \(x^\omega(t,x(0)=x^0)_{(1.1)}\) on the interval (3.11) satisfies the inequality
\[ P(k)\le \frac{\chi^0 h\lambda\sigma e^{-akh}} {e^{-\frac{akh}{2}}\left(1-e^{-\frac{ah}{2}}\right)} = \chi^{(1)}\lambda\sigma e^{-\frac{a}{2}kh} \tag{3.14} \]
\[ (\chi^{(1)}>0\text{ — const}). \]
It follows from (3.14) that the probability \(P\{z^\omega(t)>z^0(t)\ (\text{for all }t\ge 0)\}\) that the realizations \(x^\omega(t,x(0)=x^0)_{(1.1)}\) with \(z(0)=1\) never leave the domain (3.10) satisfies the inequality
\[ P\{z^\omega(t)>z^0(t)\ (\text{for all } t \geqslant 0)\}\geqslant 1-\sum_{k=0}^{\infty}\chi^{(1)}\lambda\sigma e^{-\frac{akh}{2}} . \tag{3.15} \]
It follows from (3.15) that one can choose \(\lambda>0\) and \(\sigma>0\) so that the inequality
\[ P\{z^\omega(t)>z^0(t)\ (\text{for all } t \geqslant 0)\}>1-p \tag{3.16} \]
is satisfied.
But inequality (3.16), by the choice of \(z^0(t)\) (3.9) and by (3.7), means that one can specify \(\varepsilon^0>0\) for which
\[ P\{y^\omega(t)>\varepsilon^0\ (\text{for all } t \geqslant 0)\}>1-p, \]
and this means precisely the validity of Theorem 3.1.
§ 4. In this section we consider the case in which system (1.8) is not completely controllable. We shall again assume system (1.7) to be asymptotically stable. If the system is not completely controllable, then the vectors \(A^k b\) in (1.11) are linearly dependent [3].
Let \(W\) denote the subspace of the \(n\)-dimensional vector space \(\{x\}\) generated by the vectors (1.11). The subspace \(W\) is an invariant space of the systems (1.1) and (1.8), i.e., any motion \(x^\omega(t,x(t_0)=x^0)_{(1.1)}\) or \(x(t,x(t_0)=x^0,u)_{(1.8)}\) that begins at a point \(x^0\) of \(W\) does not leave the subspace \(W\) for any \(t\). Moreover, whatever the point \(x^*\) in \(W\) and the interval \([t_1,t_*]\), there exists a control \(u(t)\) (1.10) solving problem (1.9) for \(x^{(1)}=0\). The latter assertion follows from the results of the paper [5]. Indeed, according to [5], for the existence of a control (1.10) for given \(x^{(1)}=0\) and \(x^*\), it is sufficient that the inequality
\[ \min_{\rho}\max_t \left( \left| \sum_{i=1}^{n}\rho_i \sum_{j=1}^{n} f_{ij}(t_*-t)b_j \right| \right)>0 \tag{4.1} \]
be satisfied, under
\[ \sum_{i=1}^{n}\rho_i x_i^* = 1,\qquad t_1\leqslant t\leqslant t_*, \]
where \(f_{ij}(t)\) are the elements of the fundamental matrix \(F(t)\) of system (1.7). But under the condition \(x^*\in W\), i.e., for \(x^*=\sum_{j=0}^{n-1} l_j A^j b\), the quantity (4.1) cannot be equal to zero, since otherwise the equalities
\[ (s\cdot x^*)=(s\cdot b)l_1+\ldots+(s\cdot A^{n-1}b)l_n=1, \]
\[ \left[ \frac{d^k(s\cdot F(t_*-t)b)}{dt^k} \right]_{t=t_*} = (-1)^k(s\cdot A^k b)=0 \quad (k=0,\ldots,n-1), \]
would hold simultaneously, which is impossible.
Thus, for \(x^*\in W\) inequality (4.1) is satisfied and, consequently, the problem (1.9) under consideration is solvable in the form (1.10).
Using the mentioned properties of the subspace \(W\) and arguing similarly to how this was done in the proof of Theorem 2.1, one can verify the validity of the following assertion.
Theorem 4.1. Suppose the following conditions are satisfied: 1. System (1.7) is asymptotically stable. 2. System (1.8) is not completely controllable. Then
a) whatever the numbers \(\lambda>0\), \(\sigma>0\), \(\varepsilon>0\), \(p<1\), the point \(x^0\) and the point \(x^* \in W\), the event \(\{\|x(t^*;\,x^0=x(0))_{(1.1)}-x^*\|<\varepsilon\}\) (at least for one \(t=t^*>0\)) occurs with probability
\[ P\{\|x(t^*;\,x(0)=x^0)_{(1.1)}-x^*\|<\varepsilon\}\ge p; \]
b) if the point \(x^*\) does not lie in the space \(W\), then for \(x^0 \in W\) the motion \(x(t, x(0)=x^0)_{(1.1)}\), while remaining in \(W\), does not approach the point \(x^*\) as \(t \to 0\).
References
- B. V. Gnedenko, A Course in Probability Theory. Gostekhizdat, 1954.
- J. L. Doob, Stochastic Processes. IL, 1956.
- R. E. Kalman, On the general theory of control systems. Proceedings of the First IFAC Congress, vol. 1. Publishing House of the Academy of Sciences of the USSR, 1961.
- A. M. Lyapunov, The General Problem of the Stability of Motion. Gostekhizdat, 1950.
- N. N. Krasovskii, PMM, 23, issue 4, 1959.
- N. N. Krasovskii, PMM, 24, issue 2, 1962.
- V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations. 2nd ed. Gostekhizdat, 1949.
Received by the editors
November 21, 1964
Sverdlovsk Branch of the V. A. Steklov Institute of Mathematics