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UDC 517.919 : 531.36
STABILITY OF GENERALIZED PROCESSES. II
S. T. ZAVALISHCHIN
INTRODUCTION
The present work is a continuation of papers [1, 2]. In article [2] questions of stability of generalized processes described by linear stationary equations were investigated. In the present work an attempt is made to develop the ideas of [1, 2] for the case of generalized processes that are solutions of nonstationary and nonlinear systems.
When nonstationary systems are considered (see § 1), it turns out that stability of one type or another is determined by the behavior of the operators
\[ \Lambda_p(t,s)=(-1)^p\frac{\partial^p}{\partial s^p} \int_s^t \frac{(t-\tau)^{p-1}}{(p-1)!}\,W(\tau,s)\,d\tau, \]
where \(p=1,2,\ldots\) and \(W(\tau,s)\) is the Cauchy operator. It is noteworthy that these operators coincide with the Cauchy operator if and only if the system describing the process is stationary.
The second section is devoted to the study of nonlinear systems. The concept of “nonlinearity” is given, and conditions ensuring the existence of solutions of such systems are derived. In conclusion, questions of stability of nonlinear systems are discussed; here the corresponding theorem is stated, which is an extension of Theorem 3 of [1] to generalized processes.
It should be noted that the paper often uses concepts and results of [1, 2] without explanation, but with the necessary references.
§ 1. STABILITY OF A NONSTATIONARY SYSTEM WITH RESPECT TO GENERALIZED ACTIONS
Let a system of differential equations be given
\[ \dot{\mu}=A(t)\mu+\eta, \tag{1.1} \]
in which \(A(t)\) is an \(n\times n\) matrix with infinitely differentiable elements, and \(\eta,\mu\) are generalized action and response, represented by elements of the space \(K'_+\). Along with system (1.1) we consider the homogeneous (unperturbed) system
\[ \dot{\mu}=A(t)\mu. \tag{1.2} \]
We shall denote the fundamental matrix of solutions of system (1.2) by \(U(t)\), and the Cauchy operator corresponding to system (1.2) by \(W(t,s)=U(t)U^{-1}(s)\).
In what follows we investigate the operator
\[ \mu=U(t)JU^{-1}(s)\eta, \tag{1.3} \]
where \(J\) is the operation of taking a definite integral [2]. Similarly to how this was done in [2], one can establish that the operator (1.3) maps the space \(K'_+\) into itself and that its values satisfy system (1.1). Relation (1.3), which for a stationary matrix \(A\) coincides with the Cauchy formula (1.3) of [2], will also be called the Cauchy formula.
By a solution of system (1.1) we shall mean the generalized process defined by the Cauchy formula (1.3).
Lemma 1. Let the action \(\eta\) be represented in integral form
\[ \left<\eta, x\right>_{x\in K(a)}=\int_0^a x^{(p_a)}(t)\,dn_a(t),\quad n_a\in V(a),\quad p_a\geq 0, \tag{1.4} \]
then the solution of system (1.1) has the following integral representation
\[ \left<\mu, x\right>_{x\in K(a)}=\int_0^a x^{(p_a-1)}(t)\,dm_a(t), \tag{1.5} \]
where
\[ m_a(t)=-\int_0^t \Lambda_{p_a}(t,s)\,dn_a(s) \tag{1.6} \]
and
\[ \Lambda_{p_a}(t,s)= \begin{cases} W(t,s), & p_a=0,\\[6pt] (-1)^{p_a}\dfrac{\partial^{p_a}}{\partial s^{p_a}} \displaystyle\int_s^t \dfrac{(t-\tau)^{p_a-1}}{(p_a-1)!}\,W(\tau,s)\,d\tau, & p_a\geq 1. \end{cases} \tag{1.7} \]
Proof. First note that the validity of the lemma in the case \(p_a=0\) follows directly from relation (1.5) of [2]. We pass to the case \(p_a\geq 1\). As in the case of a stationary matrix, a direct calculation leads to the relation
\[ \left<\mu,x\right>=\int_0^a y^{(p_a)}(s)\,dn_a(s), \tag{1.8} \]
where
\[ y(s)=\int_s^a W(t,s)x(t)\,dt. \]
Using Leibniz’s formula for the higher derivative of the product of two functions and collecting the terms at the derivatives of the basic function \(x(t)\), we obtain the expression
\[ y^{(p_a)}(s)=z(s)\{U^{-1}(s)\}^{(p_a)}-\sum_{\nu=0}^{p_a-1} Q_\nu x^{(\nu)}. \tag{1.9} \]
In relation (1.9) the following notation has been adopted:
\[ z(s)=\int_s^a U(t)x(t)\,dt,\quad Q_\nu=\sum_{i=\nu+1}^{p_a-1} C_{p_a}^{i}\,C_{i-1}^{\nu}\,U^{(i-1-\nu)}(U^{-1})^{p_a-i}. \tag{1.10} \]
Let us compute the integral (1.8) for each term of the sum (1.9). We transform the integral
\[ N=\int_0^a z(U^{-1})^{(p_a)}\,dn_a \]
to the form
\[ N=\int_0^a z\,d\int_0^s (U^{-1})^{(p_a)}\,dn(\tau) \]
and compute-
integral, the latter by parts. The expression obtained as a result is transformed in the same way as the integral \(N\). After all these operations we shall have the formula
\[ N=\int_{0}^{a} x(t)\,dm_{a,0}(t), \tag{1.11} \]
where
\[ m_{a,0}(t)=\int_{0}^{t} U(s)\int_{0}^{s}\bigl(U^{-1}(\tau)\bigr)^{(p_a)}\,dn_a(\tau). \tag{1.12} \]
Applying to the integral (1.11) the transformation (2.9) [2] for \(i=p_a-1\), we obtain the relation
\[ N=(-1)^{(p_a-1)}\int_{0}^{a} x^{(p_a-1)}(t)\,dm_{a,0}^{(1-p_a)}(t), \tag{1.13} \]
here the symbol \(m_{a,0}^{(1-p_a)}\) denotes the \(p_a-1\)-fold integral of the function (1.12) (the integration is carried out from zero). We transform the expression \(m_{a,0}^{(1-p_a)}\) in accordance with Cauchy’s formula for repeated integrals (2.15) [2]. The relation thereby obtained,
\[ m_{a,0}^{(1-p_a)}(t)=\int_{0}^{t}\frac{(t-\tau)^{(p_a-1)}}{(p_a-1)!}\,dm_{a,0}(\tau), \]
taking into account formula (1.12), is integrated by parts. The result of this operation is not difficult to reduce to the form
\[ m_{a,0}^{(1-p_a)}(t)= \int_{0}^{t}\int_{s}^{t} \frac{(t-\tau)^{p_a-1}}{(p_a-1)!}\, U(\tau)\bigl(U^{-1}(s)\bigr)^{(p_a)}\,dn_a(s). \tag{1.14} \]
Let us now turn to the remaining expressions obtained in integrating (1.8), taking into account the relation (1.9). We have
\[ \int_{0}^{a} Q_\nu x^{(\nu)}\,dn_a = \int_{0}^{a} x^{(\nu)}\,d\int_{0}^{t} Q_\nu(s)\,dn_a(s) = \]
\[ = (-1)^{p_a-1-\nu} \int_{0}^{a} x^{(p_a-1)}\,dm_{a,\nu}^{(\nu-p_a+1)}. \tag{1.15} \]
Here the transformation (2.9) [2] has been applied for \(i=p_a-1-\nu\). To compute \(m_{a,\nu}^{(\nu-p_a+1)}(t)\) we again use Cauchy’s formula for taking repeated integrals (2.15) [2]. We obtain the relation
\[ m_{a,\nu}^{(\nu-p_a+1)}(t) = \int_{0}^{t} \frac{(t-\tau)^{p_a-1-\nu}}{(p_a-1-\nu)!}\, Q_\nu(\tau)\,dn_a(\tau). \tag{1.16} \]
Thus, from relations (1.5), (1.8)—(1.10), (1.13)—(1.16) there follows the equality
\[ \Lambda_{p_a}(t,s) = (-1)^{p_a} \int_{s}^{t} \frac{(t-\tau)^{(p_a-1)}}{(p_a-1)!}\, U(\tau)\,d\tau\,\bigl(U^{-1}(s)\bigr)^{(p_a)} - \]
\[ -\sum_{\nu=0}^{p_a-1}(-1)^{p_a-\nu}\frac{(t-s)^{p_a-1-\nu}}{(p_a-1-\nu)!}\,Q_\nu(s). \tag{1.17} \]
It is not difficult to see that the expression standing on the right-hand side of formula (1.17) is the \(p_a\)-th derivative with respect to the argument \(s\) of the function
\[ (-1)^{p_a}\int_s^t \frac{(t-\tau)^{p_a-1}}{(p_a-1)!}\,U(\tau)\,d\tau\,U^{-1}(s). \]
The latter completes the proof of the lemma.
Remark. Let the matrix \(A\) of system (1.1) be constant. Then it is not difficult to establish that expression (1.17) coincides with formula (2.17) [2]. Hence it follows that in the case of the stationary system (1.2) the operator \(\Lambda_{p_a}(t,s)\) does not depend on \(p_a\) and coincides with the Cauchy operator. However, it is easy to indicate systems (1.1) with variable matrix for which this fact is not true. It turns out that a deeper result holds, which is formulated by
Lemma 2. The operator \(\Lambda_p(t,s)\) \((p=1,2,\ldots)\) coincides with the Cauchy operator if and only if the matrix \(A\) is constant.
Proof. We prove necessity. Suppose that the operator \(\Lambda_{p_a}(t,s)\) coincides with the Cauchy operator. Putting \(p_a=1\), we shall have
\[ W(t,s)=\Lambda_1(t,s)=E-\int_s^t U(\tau)\bigl(U^{-1}(s)\bigr)^{(1)}\,d\tau. \tag{1.18} \]
Differentiate both sides of the identity (1.18) with respect to the argument \(t\). As a result we again obtain the identity
\[ \dot U(t)U^{-1}(s)=-U(t)\bigl(U^{-1}(s)\bigr)^{(1)}. \tag{1.19} \]
Relation (1.19) is brought to the form
\[ U^{-1}(t)\dot U(t)=-\bigl(U^{-1}(s)\bigr)^{(1)}U(s), \]
whence it follows that \(U^{-1}(t)\dot U(t)=C\), where \(C\) is a constant matrix. Thus, the matrix \(U(t)\) satisfies the differential-matrix equation \(\dot U'=C'U'\) (prime denotes transposition), solving which with the initial condition \(U'(0)=E\), we obtain \(U(t)=e^{Ct}\). Substituting the value found for the matrix \(U(t)\) into system (1.2), we obtain \(A(t)=C\). Necessity is established.
Sufficiency was proved in the remark.
We now proceed to discuss questions of stability of the generalized process described by system (1.2). In [2] the concepts of \(V\)-stability and \(v\)-stability were introduced; their content does not depend on the concrete structure of the differential system, since it reflects general properties of operators acting in the space \(K_+'\). Therefore we have the possibility of speaking about \(V\)-stability (\(v\)-stability) of both the nonstationary system (1.2) and (see § 2) the nonlinear system (2.3).
Let us first note that in the classical case \(p_a=0\) \((a=1,2,\ldots)\) \(V\)-stability coincides with stability in the sense of Lyapunov and thus is an extension of this concept to the case of generalized processes. At the same time, \(v\)-stability is an extension of ordinary exponential stability, but already under the condition
\[ \sup_{t\ge 0}\int_t^{t+1}\|A(\tau)\|\,d\tau<\infty. \]
The facts listed are established analogously to the way it was found in [2] that, in the classical case for a constant matrix, \(V\)-stability and \(\mathfrak v\)-stability coincide, respectively, with boundedness of the Cauchy operator and exponential stability.
The following theorems give sufficient conditions for the generalized process (1.3) to possess the types of stability under consideration. The statement of necessary conditions is complicated by the specificity of the structure of the operators \(\Lambda_{p_a}(t,s)\) in comparison with the Cauchy operator.
Theorem 1.1. Let the operators \(\Lambda_{p_a}(t,s)\) be bounded, i.e., satisfy the condition
\[
\|\Lambda_{p_a}(t,s)\|\leq \Lambda_0<\infty
\qquad (0\leq s\leq t<\infty),
\tag{1.20}
\]
where \(\Lambda_0\) does not depend on \(p_a\). Then the generalized process (1.3) is \(V\)-stable.
Theorem 2.1. Let the operators \(\Lambda_{p_a}(t,s)\) satisfy the inequality
\[
\|\Lambda_{p_a}(t,s)\|\leq \Lambda_0 e^{-\alpha(t-s)}
\qquad (0\leq s\leq t<\infty),
\tag{1.21}
\]
where \(0<\alpha,\ \Lambda_0<\infty\) and they do not depend on \(p_a\). Then the generalized process (1.3) is \(\mathfrak v\)-stable.
As for the proof of these assertions, we note the following. In establishing Theorems 2.1 and 2.2 of [2], in the sufficiency part only properties of the Cauchy operator of the type of properties (1.20) and (1.21), respectively, were used. Further, Lemma 1 leads to relation (1.6), which is obtained formally from relation (2.6) of [2] by replacing the Cauchy operator by the operator \(\Lambda_{p_a}(t,s)\). It is now clear that the proof of the formulated theorems can be obtained formally, respectively, from the proofs of the sufficiency of Theorems 2.1 and 2.2 of [2] by replacing the Cauchy operator by the operator \(\Lambda_{p_a}(t,s)\).
Let us proceed to the study of generalized nonlinear differential systems. Below, in § 2, the question of the existence of solutions of such systems is investigated, and the \(\mathfrak v\)-stability of the generalized processes described by them is discussed.
§ 2. NONLINEAR GENERALIZED SYSTEMS. EXISTENCE OF SOLUTIONS. STABILITY WITH RESPECT TO GENERALIZED DISTURBANCES
Let a nonlinear operator \(P(\mu,t)\) act in the space \(K'_+\) of generalized functions, and suppose that its values again belong to the space \(K'_+\). Let us write for \(\mu\) and \(P(\mu,t)\) the corresponding integral representations [2]
\[
\left\langle \mu,x\right\rangle_{x\in K(a)}
=
\int_0^a x^{(p_a-1)}(t)\,dm_a(t)
\qquad (p_a\geq 0),
\]
\[
\left\langle P(\mu,t),x\right\rangle_{x\in K(a)}
=
\int_0^a x^{(r_a)}(t)\,dR_a(t)
\qquad (r_a\geq -1).
\tag{2.0}
\]
In what follows we shall assume that the operator \(P(\mu,t)\) is continuous in the weak topology [3] and that the following conditions are fulfilled:
\[
r_a\leq p_a-1,
\tag{2.1}
\]
\[
R_a(t)=R_{\bar a}(m_a,t),
\tag{2.2}
\]
the latter of which means that the function \(R_a(t)\), for fixed \(a\), depends only on the function \(m_a(t)\) considered on the interval \((0,a)\), and on
\[
\bar a=(a,p_1,\ldots,p_a).
\]
Now let us consider the nonlinear generalized system
\[ \dot{\mu}=A(t)\mu+P(\mu,t)+\eta . \tag{2.3} \]
Here \(A(t)\) is an \(n\times n\) nonstationary matrix with infinitely differentiable elements; \(\eta,\mu\) are, respectively, a generalized action and response, represented by elements of the space \(K_{+}'\). From the point of view of the existence of a solution, it is not difficult to establish the equivalence of system (2.3) to the system of integral equations
\[ \mu=U(t)JU^{-1}(s)\eta+U(t)JU^{-1}(s)P(\mu,s), \tag{2.4} \]
where \(J\) is the operation of taking the integral with variable upper limit.
To find a solution of system (2.4), we apply the method of successive approximations. We obtain the recurrence relation
\[ \mu_{l+1}=U(t)JU^{-1}(s)\eta+U(t)JU^{-1}(s)P(\mu_l,s). \tag{2.5} \]
The sequence \(\mu_0,\mu_1,\ldots\) so defined will be weakly convergent if ([3], p. 68), for any fixed \(a\) and any fixed basic function \(x\in K(a)\), there exists the limit
\[ \lim_{l\to\infty}\langle \mu_l,x\rangle = \lim_{l\to\infty}\int_0^a x^{(p_a-1)}(t)\,dm_{l,a}(t). \tag{2.6} \]
The latter will hold if ([4], p. 254) the total variations of the functions \(m_{l,a}(t)\) are bounded in the aggregate,
\[ \bigvee_0^a m_{l,a}\le V_0<\infty\quad (l=1,2,\ldots)^{*} \tag{2.7} \]
and, as \(l\to\infty\), tend to the limiting function
\[ m_a(t)=\lim_{l\to\infty}m_{l,a}(t). \tag{2.8} \]
We shall assume that in the classical case \(p_a=0\) the solution of system (2.4) exists and is unique. The conditions ensuring this circumstance can be found in the corresponding courses on differential equations. We now pass to the formulation and proof of conditions for the existence and uniqueness of the solution of system (2.4) in the case where \(p_a\ge 1\).
Theorem 2.1. Let the operators \(\Lambda_{p_a}(t,s)\), for all \(p_a=0,1,\ldots\), satisfy inequality (1.21). If the nonlinear operator \(P(\mu,t)\) is continuous in the weak topology and possesses properties (2.1), (2.2), and the functions \(R_a(m_a,t)\) satisfy the Lipschitz condition
\[ \bigvee_0^a\left[R_a(m_{1,a},t)-R_a(m_{2,a},t)\right] \le q\bigvee_0^a\left[m_{1,a}-m_{2,a}\right] \tag{2.9} \]
\[ (p_a\ge 1) \]
with a constant \(q<\dfrac{\alpha}{\Lambda_0}\) (see Theorem 1.2, § 1), then the solution of system (2.4) exists and is unique.
\[
{}^{*}
\]
Here and below, the symbol \(\displaystyle \bigvee_0^a m\) denotes the total variation of the function \(m(t)\) on the interval \([0,a]\).
Proof. Taking into account the properties (2.1), (2.2) of the operator \(\mathrm P(\mu,t)\), we apply Lemma 1 to relation (2.4). As a result we obtain the recurrent formula
\[ m_{l+1,a}(t)=-\int_0^t \Lambda p_a(t,s)\,dn_a(s)+\mathrm M(m_{l,a}), \tag{2.10} \]
where the operator \(\mathrm M(m_{l,a})\) acts in the space \(V(a)\) of functions of bounded variation on the interval \([0,a]\) and is defined by the relation
\[ \mathrm M(m_{l,a})=\int_0^t\int_0^\tau \Lambda p_{a-1}(\tau,s)\,dR_a^{-}(m_{l,a},s)\,d\tau . \tag{2.11} \]
It is evident that the sequence \(m_{0,a}, m_{1,a}, \ldots\), defined by formula (2.10), arises when solving, by the method of successive approximations, the system of integral equations
\[ m_a(t)=-\int_0^t \Lambda p_a(t,s)\,dn_a(s)+\mathrm M(m_a(s)). \tag{2.12} \]
The operator appearing on the right-hand side of relation (2.12) acts in the space \(V(a)\) and, moreover, satisfies the contraction-mapping principle ([5], p. 44).
We shall prove this fact. The difference of the values of the operator under consideration on the elements \(m_{1,a}, m_{2,a}\in V(a)\) coincides with the difference of the values of the operator \(\mathrm M\) on the same elements. Let us compute the variation of the latter. Using relation (2.11) and the formula for taking the variation of an absolutely continuous function ([4], p. 279), we obtain
\[ \Delta = V_0^a[\mathrm M(m_{1,a})-\mathrm M(m_{2,a})] = \int_0^a \varphi(t)\,dt . \tag{2.13} \]
In relation (2.13) the following notation has been adopted:
\[ \varphi(t)= \left\| \int_0^t \Lambda p_{a-1}(t,s)\, d\bigl[R_a^{-}(m_{1,a},s)-R_a^{-}(m_{2,a},s)\bigr] \right\|_{E_n}. \tag{2.14} \]
Next, applying to the function (2.14) the estimate for the Stieltjes integral
\[ \left\| \int_a^b y(s)\,dm(s) \right\|_{E_n} \le \int_a^b \|y\|\,d\,V_0^s m \tag{2.15} \]
and taking into account inequality (1.21), we obtain the estimate
\[ \varphi(t)\le \Lambda_0 e^{-\alpha t}\psi(t). \tag{2.16} \]
In inequality (2.16) it is denoted that
\[ \psi(t)=\int_0^t e^{\alpha s}\,d\,V_0^s \bigl[R_a^{-}(m_{1,a},s)-R_a^{-}(m_{2,a},s)\bigr]. \tag{2.17} \]
Now from relations (2.13) and (2.16) there follows the estimate
\[ \Delta \le \Lambda_0\int_0^a e^{-\alpha t}\psi(t)\,dt . \tag{2.18} \]
The integral standing on the right-hand side of inequality (2.18) is integrated by parts; as a result we obtain the estimate
\[ \Delta \leqslant \frac{\Lambda_0}{\alpha} \left[-\psi(a)e^{-\alpha a}+\int_0^a e^{-\alpha t}\,d\psi(t)\right]. \tag{2.19} \]
Taking into account relation (2.17) in inequality (2.19), we obtain the estimate
\[ \Delta \leqslant \frac{\Lambda_0}{\alpha} \left\{-\psi(a)e^{-\alpha a} +\bigvee_0^a\left[R_a^-(m_{1,a},s)-R_a^-(m_{2,a},s)\right]\right\}. \tag{2.20} \]
Let us estimate \(\psi(a)\) from below, replacing \(e^{\alpha s}\) in formula (2.17) by its least value, equal to one. Using the resulting estimate
\[ \psi(a)\geqslant \bigvee_0^a\left[R_a^-(m_{1,a},s)-R_a^-(m_{2,a},s)\right] \]
in inequality (2.20), we obtain the estimate
\[ \Delta \leqslant \frac{\Lambda_0}{\alpha}(1-e^{-\alpha a}) \bigvee_0^a\left[R_a^-(m_{1,a},s)-R_a^-(m_{2,a},s)\right]. \tag{2.21} \]
Thus, from inequalities (2.21) and (2.9) there follows the final estimate
\[ \bigvee_0^a\left[\mathrm{M}(m_{1,a})-\mathrm{M}(m_{2,a})\right] \leqslant q\,\frac{\Lambda_0}{\alpha}(1-e^{-\alpha a}) \bigvee_0^a[m_{1,a}-m_{2,a}], \]
which means that the operator standing on the right-hand side of formula (2.12) satisfies the contraction-mapping principle ([5], p. 44). Hence it follows that the sequence of functions \(m_{0,a}, m_{1,a},\ldots\), determined by the recurrence relation (2.10), converges in the norm of the space \(V(a)\) to a function \(m_a\), which again belongs to this space and satisfies system (2.12). Therefore the sequence under consideration has properties (2.7) and (2.8). Thus, the sequence \(\mu_0,\mu_1,\ldots\) converges weakly to some generalized function \(\mu\), which, by virtue of the weak continuity of the operator \(\mathrm{P}(\mu,t)\), is a solution of the integral system (2.4). This completes the proof of the theorem.
We now turn to the study of the stability of the generalized process described by the nonlinear system (2.3). Let \(\eta\in D=\{\eta:p_a=p,\ a\geqslant1;\ p\geqslant0\}\).
Theorem 2.2. Let the operators \(\Lambda_{p_a}(t,s)\) \((p_a=0,1,\ldots)\) satisfy inequality (1.21). If the nonlinear operator \(\mathrm{P}(\mu,t)\) has properties (2.1), (2.2), and the function \(R_a^-(m_a,t)\)—the superposition of \(m_a(t)\), \(t\) for \(\mu\in D\) and
\[ \left\|R_a^-(m_a,t)\right\|_{E_n} \leqslant L\left\|m_a(t)\right\|_{E_n} \quad (t\geqslant0), \tag{2.22} \]
where \(\lambda=\alpha-L\Lambda_0>0\), then the generalized process described by the nonlinear system (2.1) is \(v\)-stable with respect to \(D\).
Proof. For the value of the generalized function \(\mathrm{P}(\mu,t)\) on the basic function \(x\in K(a)\), we use the integral expression
\[ \langle \mathrm{P}(\mu,t),x\rangle = -\int_0^a x^{(p_a)}(s)\,d\int_0^s R_a^-(m_a,\tau)\,d\tau . \tag{2.23} \]
Representation (2.23) has been obtained by applying transformation (2.9) [2] to formula (2.0), in which \(r_a=p_a-1\). With the aid of Lemma 1 from integral system (2.4), taking into account relation (2.23), one can pass to the following relation between the functions \(m_a,\eta_a,R_a^-(m_a,t)\):
\[ m_a(t)=-\int_0^t \Lambda_{p_a}(t,s)\,dn_a(s)+\int_0^t \Lambda_{p_a}(t,s)R_{\bar a}^{-}(m_a,s)\,ds. \tag{2.24} \]
Let us estimate the norm of the function (2.24). To estimate the first integral on the right-hand side of formula (2.24), we shall use inequalities (2.15) and (2.21). In estimating the second integral, we take into account inequalities (1.21) and (2.22). As a result we obtain the following inequality:
\[ \|m_a(t)\|_{E_n}\leq \Lambda_0 Le^{-\alpha t}\int_0^t e^{\alpha s}\|m_a(s)\|_{E_n}\,ds+ \]
\[ +\Lambda_0e^{-\alpha t}\int_0^t e^{\alpha s}\,d\bigvee_0^s n_a. \tag{2.25} \]
Thus, we have obtained the inequality with which we dealt in proving the sufficiency of Theorem 2 [1]. Therefore, proceeding with estimate (2.25) as was done in the indicated theorem, we arrive at the inequality
\[ \|m_a(t)\|_{E_n}\leq \Lambda_0\,\frac{e^\lambda}{1-e^{-\lambda}}\max_{0\leq l\leq a}\bigvee_l^{\,l+1} n_a. \tag{2.26} \]
From estimate (2.26) there follows the inequality
\[ \sup_{0\leq t\leq a}\|m_a(t)\|_{E_n}\leq \Lambda_0\,\frac{e^\lambda}{1-e^{-\lambda}}\max_{0\leq l\leq a}\bigvee_l^{\,l+1} n_a. \tag{2.27} \]
Let the sequence of disturbances \(\eta_1,\eta_2,\ldots\) \(v\)-converge [2]. Then estimate (2.27), in which the parameters \(\Lambda_0,\lambda\) do not depend on \(p_a\), ensures \(B\)-convergence of the corresponding sequence of reactions \(\mu_1,\mu_2,\ldots\).
The theorem is proved.
In conclusion, the author thanks E. A. Barbashin for valuable advice and constant attention to this work.
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Received by the editors
March 28, 1966
Sverdlovsk Branch of the Mathematical
Institute named after V. A. Steklov