Abstract
Full Text
UDC 519.27:531.395
MATHEMATICS
M. V. LEVIT
FREQUENCY CONDITIONS FOR ABSOLUTE STOCHASTIC STABILITY OF AUTOMATIC CONTROL SYSTEMS WITH RANDOM EXTERNAL ACTION
(Presented by Academician V. I. Smirnov on 5 V 1969)
1°. Consider the integral equation describing a control system with (n) nonlinear blocks (\varphi_j=\varphi_j(t,\sigma)):
[
\sigma_t=\alpha(t)+\int_0^t \Omega(t-\tau)\varphi_\tau\,d\tau,\qquad
\varphi_t=\varphi(t,\sigma_t),\qquad t\geqslant 0.
\tag{1}
]
Here (\alpha(t)), (\sigma_t) are vector random processes (of order (m)), (\varphi(t,\sigma)=|\varphi_j(t,\sigma)|) is a vector function of order (n), continuous in the aggregate of the variables (t) and (\sigma) ((t\geqslant 0,\ -\infty<\sigma_i<+\infty)), and (\Omega(t)) is a matrix function of order (m\times n). We shall assume that (\Omega(t)) is summable on the half-axis (\left(|\Omega(t)|\in L[0,+\infty)\right)^*), and that the measurable process ((^1)) (\alpha(t)) is, with probability 1 (w.p. 1), square-summable on every finite interval (\left(|\alpha(t)|\in L_2[0,T]\ \text{w.p. 1 for any } T>0\right)).
A solution of system (1) on the interval ([0,T]) will mean a pair of measurable processes (y_t=(\sigma_t,\varphi_t)), square-summable w.p. 1 on ([0,T']), (T'<T), and satisfying (1) almost everywhere on ([0,T]), w.p. 1.
By analogy with ((^2)), we shall say that the strengthened local existence theorem holds for system (1) if the following are true: (A) system (1) has a solution on some interval ([0,T_0]); (B) if this solution is square-summable on ([0,T_0]) w.p. 1, then it is extendable to some wider interval ([0,T_1]), (T_1>T_0).
The strengthened local existence theorem holds for a fairly broad class of systems (1). It holds, for example, in the case when (\varphi(t,\sigma)) satisfies a Lipschitz condition with respect to the argument (\sigma) with a constant independent of (t), and
[
\int_0^T E|\alpha(t)|^2\,dt<+\infty
]
for any (T>0^{**}). The proof is carried out by means of the contraction mapping principle for the operator
[
A(\sigma_t)=\int_0^T \Omega(t-\tau)\varphi(\tau,\sigma_\tau)\,d\tau+\alpha(t)
]
in the space (\mathfrak{E}[0,T]) of measurable random processes (\sigma_t) ((0\leqslant t\leqslant T)) such that
[
\int_0^T E|\sigma_t|^2\,dt<+\infty^{***}.
]
Denote by (\mathfrak{A}) the set of external actions (\alpha(t)) for which system (1) has a solution on the half-axis ([0,+\infty)). System (1) defines a mapping (U) of the set (\mathfrak{A}) onto the set (Y) of corresponding solutions (y_t=(\sigma_t,\varphi_t)). Suppose that nonnegative functions (r(\alpha)) and (\rho(y)) are given on (\mathfrak{A}) and (Y), respectively, possibly taking the value (+\infty), with (r(0)=0), (\rho(0)=0). We shall say that system (1) is stochastically stable
* Here and below,
[
|A|=\sqrt{\sum_{j,h}|a_{jh}|^2}
]
((A=|a_{jh}|)) is a rectangular matrix.
** Here and below, the symbol (E) denotes mathematical expectation.
*** Obviously, at the same time existence and uniqueness of the solution in (\mathfrak{E}[0,T]) will be proved. Below we also use the space (\mathfrak{E}[0,+\infty)) of measurable random processes (\sigma_t) ((0\leqslant t<+\infty)) such that
[
\int_0^{+\infty} E|\sigma_t|^2\,dt<+\infty.
]
with respect to the functions (r(\alpha)) and (\rho(y)), if the following hold: a) from the condition (r(\alpha)<+\infty) it follows that (\rho(y)<+\infty); b) from the condition (r(\alpha)\to 0) it follows that (\rho(y)\to 0).
We shall call the class of systems (1) absolutely stochastically stable with respect to (r(\alpha)) and (\rho(y)) if every system in the class is stochastically stable with respect to (r(\alpha)) and (\rho(y)).*
In Theorems 1–4 below, sufficient conditions are given for the existence of solutions of system (1) on the half-axis ([0,+\infty)) and for the fulfillment, for any (T>0), of one of the inequalities
[
\int_0^T E|y_t|^2\,dt \leq M^2 \int_0^T E|\alpha(t)|^2\,dt + N\gamma + Q,
\tag{2}
]
[
\int_0^T E\bigl(|y_t|^2+|\dot{\sigma}_t|^2\bigr)\,dt
\leq
M^2 \int_0^T E\bigl(|\alpha(t)|^2+|\dot{\alpha}(t)|^2\bigr)\,dt + N\gamma + Q,
\tag{3}
]
in which the nonnegative constants (M, N, \gamma), and (Q) do not depend on (T). Using (2) and (3), it is easy to establish the stochastic stability of system (1) with various (r(\alpha)) and (\rho(y)). Namely:
Proposition 1. Suppose that for the solutions of system (1) on the half-axis ([0,+\infty)), (2) is satisfied. Then: (I) system (1) is stochastically stable with respect to the seminorms
[
r_1(\alpha)=\lim_{T\to\infty}\left(\frac{1}{T}\int_0^T E|\alpha(t)|^2\,dt\right)^{1/2},
\qquad
\rho_1(y)=\lim_{T\to\infty}\left(\frac{1}{T}\int_0^T E|y_t|^2\,dt\right)^{1/2};
]
(II) if (\gamma=Q=0), then the system is stochastically stable with respect to the norms
[
r_2(\alpha)=\sup_T\left(\frac{1}{T}\int_0^T E|\alpha(t)|^2\,dt\right)^{1/2},
\qquad
\rho_2(y)=\sup_T\left(\frac{1}{T}\int_0^T E|y_t|^2\,dt\right)^{1/2}
]
and the norms
[
r_3(\alpha)=\left(\int_0^{+\infty} E|\alpha(t)|^2\right)^{1/2},
\qquad
\rho_3(y)=\left(\int_0^{+\infty} E|y_t|^2\,dt\right)^{1/2}.
]
Proposition 2. Suppose that (\Omega(t)) and (\alpha(t)) (see Section 1) are absolutely continuous; then, from Section 1, the process (\sigma_t) is absolutely continuous. If (3) is satisfied, then assertions (I), (II) of Proposition 1 are valid, in which (r_j(\alpha)) and (\rho_j(y)) ((j=1,2,3)) are replaced respectively by (r_j(\tilde{\alpha})) and (\rho_j(\tilde{y})), where
[
\tilde{\alpha}(t)=\sqrt{|\alpha(t)|^2+|\dot{\alpha}(t)|^2},
\qquad
\tilde{y}(t)=\sqrt{|y_t|^2+|\dot{\sigma}_t|^2}.
]
Moreover, if (\alpha(t)\in \mathfrak{C}[0,+\infty)), (\dot{\alpha}(t)\in \mathfrak{C}[0,+\infty)), then (E|\sigma_t|^2\to 0) as (t\to\infty).
(2^\circ). Quite often, the specific properties of the nonlinear blocks of system (1) make it possible to assert that the solutions of the system in Section 1 satisfy, on the interval of existence, the relation
[
\int_0^t F(\varphi_\tau,\sigma_\tau,\dot{\sigma}_\tau)\,d\tau \geq -\gamma,
\tag{4}
]
where (F) is a real quadratic form of its arguments and (\gamma) is a nonnegative constant. Diverse examples of such relations are given in ((^4)), and the method of using them to study the stability of deterministic systems (1) is set forth in ((^2)). Following ((^2)), we extend each real quadratic form (F(\varphi,\sigma,\dot{\sigma})) (the vectors (\varphi,\sigma,\dot{\sigma}) have orders (n,m,m)) to complex values of the arguments, preserving Hermitian symmetry, and define the function (F(p,\bar{\varphi})=F(\bar{\varphi},-\chi(p)\bar{\varphi},-p\chi(p)\bar{\varphi})). Here (p=i\omega), (\omega\in(-\infty,+\infty)), (\bar{\varphi}) is a complex (n)-vector, and (-\chi(p)) is the Laplace transform of the function (\Omega(t)).
* For the first time, as far as the author knows, a similar definition of absolute stochastic stability was given in ((^3)). There, under certain additional assumptions concerning (\alpha(t)), theorems were proved that are special cases of Theorems 1 and 3 of the present paper.
Theorem 1. Suppose that for system (1) the strengthened local existence theorem is satisfied. Let the solutions of (1) with p. 1 satisfy relation (4), in which (F=F(\varphi,\sigma)) does not depend on (\dot\sigma). Let the following hold: ((A_1)) (F(0,\sigma)\geqslant 0) for all real (\sigma); ((B_1)) (\tilde F(p,\tilde\varphi)<0) for all complex (\tilde\varphi) and (p=i\omega,\ -\infty\leqslant\omega\leqslant+\infty). Then the solutions of system (1) are continuable to the half-axis ([0,+\infty)), and (2) is valid for (Q=0).
Theorem 2. Suppose that for system (1) all the conditions of Theorem 1 are satisfied, except for ((A_1)). Let it be known that (|\varphi_t|\leqslant \mathrm{const}) with p. 1, and that (\Omega(t)) satisfies the relation
[
\lim_{T\to\infty}\int_0^T \int_{T-\tau}^{\infty}|\Omega(t)|\,dt\,d\tau<+\infty *.
]
Then the solutions of system (1) are continuable to the half-axis ([0,+\infty)), and (2) is valid.
Theorem 3. Suppose that in system (1) (\Omega(t)) and (\alpha(t)) (with p. 1) are absolutely continuous, moreover (|\Omega(t)|\in L[0,+\infty)), and (|\dot\alpha(t)|\in L_2[0,T]), (T>0) with p. 1. Let the strengthened local existence theorem be satisfied. Let the solutions of system (1) with p. 1 satisfy relation (4). Let the following hold: ((A_2)) (F(0,\sigma,\dot\sigma)\geqslant 0) for all real (\sigma) and (\dot\sigma); ((B_2)) (\tilde F(p,\tilde\varphi)<0) for all complex (\tilde\varphi) and (p=i\omega,\ -\infty\leqslant\omega\leqslant+\infty). Then the solutions of system (1) are continuable to the half-axis ([0,+\infty)), and (3) is valid for (Q=0).
Theorem 4. Suppose that for system (1) all the conditions of Theorem 3 are satisfied, except for ((A_2)). Let it be known that (|\varphi_t|\leqslant \mathrm{const}) with p. 1 and
[
\lim_{T\to\infty}\int_0^T \int_{T-\tau}^{\infty}\bigl(|\Omega(t)|+|\dot\Omega(t)|\bigr)\,dt\,d\tau<+\infty .
]
Then the solutions of (1) are continuable to the half-axis ([0,+\infty)), and (3) is valid.
3°. We shall briefly indicate the main points in the proofs of the formulated assertions.
Proposition 1 is obvious. In Proposition 2 only the last assertion requires proof. Note that from the conditions (\alpha(t)\in\mathfrak C[0,+\infty)), (\dot\alpha(t)\in\mathfrak C[0,+\infty)) and inequality (3) it follows that (\sigma_t\in\mathfrak C[0,+\infty)), (\dot\sigma_t\in\mathfrak C[0,+\infty)). Then from
[
E\bigl(|\sigma_t|^2-|\sigma_0|^2\bigr)=2\int_0^t E(\sigma_\tau,\dot\sigma_\tau)\,d\tau
]
we obtain that there exists a finite limit (\lim_{t\to\infty}E|\sigma_t|^2), and this limit is equal to zero.
Theorems 1–4 are a probabilistic analogue of the results obtained in ((^2)) for deterministic systems (1). Therefore the proof of many points is based on the same ideas and is carried out analogously.
Proof of Theorems 1 and 2. As in ((^2)), if the transfer function (\chi(i\omega)) and the quadratic form (F(\varphi,\sigma)) satisfy condition ((B_1)), then there exists a constant (\delta>0) such that
[
\tilde F(i\omega,\tilde\varphi)+2\delta(1+|\chi(i\omega)|^2)|\tilde\varphi|^2\leqslant 0.
]
Consider the random variable
[
I(T)=\int_0^T \bigl[F(\varphi_\tau,\xi_\tau)+2\delta(|\varphi_\tau|^2+|\xi_\tau|^2)\bigr]\,d\tau .
]
Here (T) belongs to the interval of existence of the solution (y_t=(\sigma_t,\varphi_t)) of system (1) , (\xi_1=\sigma_t-\alpha(t)=\Omega(t)\varphi_t) .
Put (\varphi_t^T=\varphi_t) for (0T) or (t<0), (\xi_t^T=\Omega(t)*\varphi_t^T). Then
[
I(T)=J(T)-\int_T^{+\infty}\bigl(F(0,\xi_\tau^T)+2\delta|\xi_\tau^T|^2\bigr)\,d\tau,\quad \text{where } J(T)=
]
* Under the limit sign there is a function nondecreasing in (T) and, consequently, the limit exists.
** The existence of such an interval is guaranteed by the strengthened local existence theorem.
** The symbol () denotes the convolution operation.
[
= \int_{-\infty}^{+\infty} \left[ F(\varphi_\tau^{T}, \xi_\tau^{T}) + 2\delta (|\varphi_\tau^{T}|^{2} + |\xi_\tau^{T}|^{2}) \right] d\tau .
]
Since (|\varphi_t^{T}| \in L(-\infty,+\infty) \cap L_2(-\infty,+\infty)) a.s., it follows that (|\xi_t^{T}| \in L(-\infty,+\infty) \cap L_2(-\infty,+\infty)) a.s.; moreover, for their Fourier transforms the equality (\tilde{\xi}^{T} = -\chi(i\omega)\tilde{\varphi}^{T}) holds a.s. Applying Parseval’s equality to (J(T)), we obtain that, by the choice of (\delta), (J(T) \leq 0) a.s. Consequently,
[
I(T) \leq - \int_{T}^{+\infty} F(0,\xi_\tau^{T})\,d\tau
]
a.s. Similarly to (2), it is easy to show the existence of constants (k_1>0), (k_2>0) such that, for any (\varepsilon>0), the inequality
[
-F(\varphi,\sigma-\alpha)-\varepsilon (|\varphi|^{2}+|\sigma-\alpha|^{2})
\leq -F(\varphi,\sigma) + (k_1+k_2/\varepsilon)|\alpha|^{2}
]
is valid. Take (\varepsilon=\delta), (k=k_1=k_2/\delta). We obtain
[
\delta \int_{0}^{T} (|\varphi_\tau|^{2}-|\xi_\tau|^{2})\,d\tau
= I(T) - \int_{0}^{T} \left[ F(\varphi_\tau,\xi_\tau) + \delta (|\varphi_\tau|^{2}+|\xi_\tau|^{2}) \right] d\tau
]
[
\leq - \int_{T}^{+\infty} F(0,\xi_\tau^{T})\,d\tau
- \int_{0}^{T} F(\varphi_\tau,\sigma_\tau)\,d\tau
+ k \int_{0}^{T} |\alpha(\tau)|^{2} d\tau
]
a.s. Applying to the last inequality relation (4) for the form (F(\varphi,\sigma)), we obtain that, under the conditions of Theorem 1 or 2, a.s. for all (T) from the interval of existence of the solution the relation
[
\delta \int_{0}^{T} (|\varphi_\tau|^{2}+|\xi_\tau|^{2})\,d\tau
\leq \gamma - \int_{T}^{+\infty} F(0,\xi_\tau^{T})\,d\tau
+ k \int_{0}^{T} |\alpha(\tau)|^{2} d\tau .
]
Let the conditions of Theorem 2 be satisfied. Then
[
\left| \int_{T}^{+\infty} F(0,\xi_\tau^{T})\,d\tau \right|
\leq \mathrm{const} \int_{0}^{T} \int_{T-\tau}^{+\infty} |\Omega(t)|\,dt\,d\tau
\leq \mathrm{const}.
]
Therefore there exist nonnegative constants (M, N), and (Q), independent of (T), such that a.s.
[
\int_{0}^{T} (|\varphi_\tau|^{2}+|\sigma_\tau^{2}|)\,d\tau
\leq M^{2} \int_{0}^{T} |\alpha(t)|^{2}\,dt + N\gamma + Q .
\tag{5}
]
Under the conditions of Theorem 1, however, from ((A_1)) there follows inequality (5) with (Q=0). From (5) and the strengthened local existence theorem it follows that the solutions can be continued to the half-axis ([0,+\infty)). Since (\varphi_t), (\sigma_t), and (\alpha(t)) are measurable processes, estimate (2) follows from (5).
Proof of Theorems 3 and 4. Let the conditions of Theorem 3 be satisfied. By the strengthened local existence theorem, system (1) has a solution (y_t) on some interval. Differentiating the first equation of system (1), we obtain that the process (\hat{y}_t=(\hat{\sigma}_t,\varphi_t)), where
[
\hat{\sigma}_t =
\begin{pmatrix}
\sigma_t\
\dot{\sigma}_t
\end{pmatrix},
]
will be a solution of a system of the form (1), in which the roles of (\alpha(t)) and (\Omega(t)) are played by
[
\hat{\alpha}(t)=
\begin{pmatrix}
\alpha(t)\
\dot{\alpha}(t)
\end{pmatrix}
\quad \text{and} \quad
\hat{\Omega}(t)=
\begin{pmatrix}
\Omega(t)\
\dot{\Omega}(t)
\end{pmatrix}.
]
For the system thus obtained all the conditions of Theorem 1 are valid. Then from (5), for some (M) and (N), it follows a.s. that
[
\int_{0}^{T} (|\varphi_t|^{2}+|\sigma_t|^{2}+|\dot{\sigma}t|^{2})\,dt
\leq M^{2} \int)\,dt + N\gamma .}^{T} (|\alpha(t)|^{2}+|\dot{\alpha}(t)|^{2
]
Hence, as before, the solutions (y_t) can be continued to the half-axis ([0,+\infty)), and (3) is valid. Theorem 4 follows in the same way from Theorem 2.
Leningrad State University
named after A. A. Zhdanov
Received
21 IV 1969
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