Full Text
UDC 517.934
STABILITY OF GENERALIZED PROCESSES. I
S. T. Zavalishchin
INTRODUCTION
Let a system of differential equations be given
\[ \dot{x}=Ax+u, \tag{0.1} \]
where \(x, u\) are \(n\)-dimensional vectors, \(A\) is an \(n\times n\) matrix.
In [3] possible methods are discussed for studying the behavior of solutions of the system (0.1) from the point of view of stability, namely the approach to the concept of Lyapunov stability and the approach to the concept of stability under constantly acting perturbations. It is noted that stability in the sense of Lyapunov can be included within the framework of the second approach if the domain of admissible perturbations is extended to functions of the type
\[ u(t)=u_0(t)+\sum_{i=1}^{\infty} a_i \delta(t-t_i). \tag{0.2} \]
In (0.2), \(u_0(t)\) is a locally integrable function, and \(\delta(t-t_i)\) is the Dirac function, i.e., the simplest generalized function concentrated at the point \(t_i\). In this case the system (0.1) is regarded as a link of an automatic-control circuit transmitting a distribution associated with the function (0.2). In this connection, it is of interest to study questions of stability in the case when the input function \(u(t)\) has, compared with (0.2), the more general structure of an arbitrary generalized function. It is clear that the solution of this question makes it possible to develop a theory which, from a unified point of view, encompasses the classical theory of stability. Moreover, the need to study a circle of such problems is also dictated by the fact that [4] the description of physical processes in the language of set functions, distributions, in particular generalized functions, more accurately reflects the actual nature of these processes.
It was in this form that the problem was posed to the author of this article by E. A. Barbashin. The work consists of two sections. The first of them is devoted to special questions of the theory of generalized functions directly connected with the problem posed; the second, to the problem itself.
§ 1. THE SPACE \(K'_+\)
- Let \(E_n\) be a normed Euclidean space of elements \(e\) with norm \(\|e\|_{E_n}\); let \(K\) be the union [2] of the countably normed spaces \(K(a)\) \((a=1,2,\ldots)\) [2] of infinitely differentiable functions different from zero only in the interval \((-a,a)\).
According to [2], by the space \(K'\) of generalized vector-functions we shall mean the collection of linear continuous vector-functionals \(\mu=(\mu_1,\ldots,\mu_n)'\) (\('\) denotes the transposition operation), defined on the space \(K\) with values in the space \(E_n\). For brevity below we shall also use, for \(\mu\), the terms generalized function, distribution. For the value of the generalized function \(\mu\) on the basic function \(x \in K\) we adopt the notation
\[
\langle \mu,x\rangle=(\langle \mu_1,x\rangle,\ldots,\langle \mu_n,x\rangle)'.
\]
If for a point \(t\) there exists a neighborhood \(O(t)\) such that \(\langle \mu,x\rangle=0\in E_n\) for every basic function \(x\) different from zero only within the neighborhood \(O(t)\), then the point \(t\) is called [1] inessential for the distribution \(\mu\). The essential points constitute the support of the distribution. We introduce
Definition 1.1. By the space \(K'_+\) we shall mean the collection of distributions whose supports are concentrated in the region \(t\geqslant 0\).
Let us note the obvious properties of the introduced space:
\(K'_+\) is a linear topological subspace of \(K'\);
\(K'_+\) is closed with respect to the operations of taking a derivative and (see item 2, § 1) an integral with variable upper limit;
the collection of locally integrable functions from \(K'_+\) is dense in the latter in the weak topology [2], as in the case of all of \(K'\).
Let us note here the following: if \(\mu\in K'_+\) is the weak limit of a sequence of locally integrable functions \(\lambda_n\in K'\), then, generally speaking, \(\mu\) will not be the weak limit of the sequence \(\chi\lambda_n\) (\(\chi(t)\) is the Heaviside function, equal to zero for \(t<0\) and to one for \(t\geqslant 0\)); nevertheless, as is easily verified, the sequence
\[
\chi(t)\lambda_n\left(t-\frac{1}{n}\right)\in K'_+
\]
converges weakly to \(\mu\). The value \(\langle \mu,x\rangle\) (\(\mu\in K'_+\)) is determined by the basic function \(x\), considered only on the interval \((0,\infty)\), since \(\langle \mu,x\rangle=\langle \mu,y\rangle\) for any basic function \(y\) coinciding with \(x\) for \(t\geqslant 0\) by virtue of Definition 1.1. This fact makes it possible to extend the generalized function \(\mu\) to the set of functions \(x\), defined for \(t\geqslant 0\), infinitely differentiable, and finite on the right, by the formula
\[
\langle \mu,x\rangle=\langle \mu,y\rangle,
\]
where \(y\in K\) and \(y(t)=x(t)\) for \(t\geqslant 0\).
- We introduce on \(K'_+\) the operation of taking an integral with variable upper limit. In [1] a definition is given of the primitive of a distribution \(\mu\), which can be reduced to the following form:
\[ \int \mu\,dt=J\mu+C\qquad (C=\mathrm{const}), \]
where the operator \(J=\iota^*\), i.e. is adjoint to the operator \(\iota\), acting in the space of basic functions \(K\), according to the rule
\[ \iota x=\int_{-\infty}^{\infty}\left\{\int_{-\infty}^{t} \begin{vmatrix} c(\tau) & x(\tau)\\ c(s) & x(s) \end{vmatrix} \,d\tau\right\}\,ds. \tag{1.1} \]
In formula (1.1), \(c\in K\) and satisfies the condition
\[ \int_{-\infty}^{\infty} c(t)\,dt=1. \]
We formulate
Definition 1.2. By the integral with variable upper limit of the distribution \(\mu\) we shall mean the distribution \(J\mu\) under the condition that the function \(c(t)\) vanishes for \(t\geqslant 0\).
Let us note the following properties of the introduced operation:
by the definition of the function \(c(t)\)
\[ \iota x=\int_t^\infty x(\tau)\,d\tau \qquad (t\geqslant 0); \tag{1.2} \]
in the formula \(\langle J\mu,x\rangle=\langle \mu,\iota x\rangle\), in accordance with the fourth property of the space \(K'_+\), one may assume that \(\iota x\) is defined by relation (1.2);
\[ J\mu\in K'_+; \]
the introduced operation \(J\) coincides with the ordinary integral with variable upper limit for those locally integrable functions \(\lambda\in K'_+\) for which at least the following calculations are valid:
\[ \langle J\lambda,x\rangle=\langle \lambda,\iota x\rangle =\int_0^\infty\int_t^\infty x\,d\tau\,\lambda(t)\,dt = \]
\[ =\int_0^\infty x\,d\tau\int_0^t \lambda(\tau)\,d\tau +\int_0^\infty\int_0^\infty\int_0^t \lambda\,d\tau\,x(t)\,dt = \left\langle \int_0^t \lambda\,d\tau,\ x\right\rangle . \]
Here we have used the second of the properties now being formulated, the finiteness on the right of the function \(x\), and the continuity of the integral with variable upper limit of the locally integrable function \(\lambda\);
\[ \frac{d}{dt}J\mu=J\frac{d}{dt}\mu=\mu, \]
i.e., the operation \(J\) reconstructs a distribution from its derivative, which advantageously distinguishes it from the classical Lebesgue integral.
- We introduce in the space \(K'_+\) a number of special topologies necessary for posing and studying the question of stability of generalized differential equations. A distribution \(\mu\), being a linear functional on the space \(K\) (the union of the spaces \(K(a)\)), will be a linear functional on each space \(K(a)\), and moreover will possess the property of continuity by the definition of the continuity of \(\mu\) as a functional on \(K\). In [2] its representation in integral form is indicated:
\[ \underset{x\in K(a)}{\langle \mu,x\rangle} = \int_{-a}^{a} x^{(p)}(t)\,dm(t), \tag{1.3} \]
where \(m\) belongs to the space of functions of bounded variation on the interval \([-a,a]\) and \(p\geqslant 0\). Obviously, in our case \((\mu\in K'_+)\) formula (1.3) can be rewritten in the form
\[ \underset{x\in K(a)}{\langle \mu,x\rangle} = \int_0^a x^{(p)}\,dm. \tag{1.4} \]
Here already \(m\in V(a)\)—the space of functions of bounded variation on the interval \([0,a]\). The number \(p\) in formulas (1.3) and (1.4) will be taken to be the smallest. Then for \(p=0\) the representations (1.3) and (1.4) do not distinguish between the case when \(\mu\) is an “essential” generalized function (for example, a delta-function) and when \(\mu\) is a function of bounded variation on each interval \((-a,a)\) or \((0,a)\); the space of the latter we denote by \(V(\infty)\). However, in the case of the space \(K'_+\) it is not difficult to establi-
to derive, by integration by parts, the relation for distributions belonging to the space \(V(\infty)\),
\[ \underset{x\in K(a)}{<\mu,x>}=\int_0^a x^{(-1)}\,d(-\mu),\qquad \mu\in V(\infty). \tag{1.5} \]
In formula (1.5) the notation \(x^{(-1)}=tx\) has been adopted. We note that the representation (1.5) is analogous in form to the representation (1.4) and does not hold for “proper” generalized functions. Hence, by taking \(p\) in formula (1.4) to vary from \(-1\), we include, with the above remarks concerning \(x^{(-1)}\), the representation (1.5) in the representation (1.4), and the latter will then distinguish the case when the distribution \(\mu\) belongs to the space \(V(\infty)\) from the case when \(\mu\) does not belong to \(V(\infty)\). After this preliminary remark we proceed to the definition of special topologies in the space \(K'_+\).
Consider an arbitrary sequence \(\mu_1,\mu_2,\ldots,\;(\mu_k\in K'_+)\). Using the representation (1.4), we shall have
\[ \underset{x\in K(a)}{<\mu_k,x>}=\int_0^a x^{(p_{ka})}\,dm_{ka}. \tag{1.6} \]
Suppose that the following condition is satisfied:
\[ p_{ka}\leqslant p_a<\infty \qquad (k=1,2,\ldots). \tag{1.7} \]
Definition 1.3. The sequence \(\{\mu_k\}\) will be called \(B\)-convergent (\(v\)-convergent, \(V\)-convergent) if, as \(k\to\infty\), uniformly in \(a\),
\[ \sup_{0\leq t\leq a}\|m_{ka}\|_{E_n} \quad \left( \max_{0\leq l\leq a}\bigvee_l^{\,l+1} m_{ka},\; \bigvee_0^{\,a} m_{ka} \right) \]
tends to zero. The space \(K'_+\), endowed with these convergences, will be denoted respectively by \(BK'_+\), \(vK'_+\), \(VK'_+\).
We proceed to discuss the topologies introduced. It is obvious that the topology of the space \(VK'_+\) is stronger than the topologies of the spaces \(vK'_+\) and \(BK'_+\). We shall next establish that the topologies of the spaces \(vK'_+\) and \(VK'_+\) are stronger than the topology of weak convergence. For this, in view of the remark made above, it is enough to verify this fact for the topology of the space \(vK'_+\). The latter follows from the estimates
\[ \left\| \underset{x\in K(a)}{<\mu_k,x>} \right\|_{E_n} = \left\| \int_0^a x^{(p_{ka})}\,dm_{ka} \right\|_{E_n} \leqslant M_{ka}\bigvee_0^a m_{ka} \leqslant \]
\[ \leqslant M_{ka}\,a\max_{0\leq l\leq a}\bigvee_l^{\,l+1} m_{ka}, \]
where
\[ M_{ka}=\max_{0\leq t\leq a}|x^{(p_{ka})}|\leqslant M_a, \]
which, in accordance with condition (1.7), is finite and does not depend on \(k\). Let us clarify the meaning of the convergences introduced for the space \(V(\infty)\). As noted above, in this case \(p_{ka}=-1\) \((k,a=1,2,\ldots)\), \(m_{ka}=-\mu_k\) for \(t\in(0,a)\). Hence it follows that a \(B\)-convergent sequence, starting from some index, belongs to the space \(B\) of bounded functions and tends to zero in its norm
\[ \|\mu\|=\sup_{t>0}\|\mu(t)\|_{E_n},\qquad \mu\in B; \tag{1.8} \]
a \(v\)-convergent sequence, starting from some index, belongs to the space \(v\) [3] and tends to zero in its norm
\[ \|\mu\|=\sup_{t>0} V_t^{t+1}\mu,\qquad \mu\in v; \tag{1.9} \]
a \(V\)-convergent sequence, starting from some index, belongs to the space \(V(0,\infty)\) and tends to zero in its norm
\[ \|\mu\|=V_0^\infty\mu,\qquad \mu\in V(0,\infty). \tag{1.10} \]
Let the operator \(\Omega\) map the space \(K_+^\prime\) into itself. We introduce a number of notions of continuity of the operator \(\Omega\).
Definition 1.4. We shall call the operator \(\Omega\) \(v\)-continuous (\(V\)-continuous) if it is continuous as a mapping into the space \(BK_+^\prime\) of the space \(vK_+^\prime\) (\(VK_+^\prime\)).
The meaning of the introduced continuities in the case of the space \(V(\infty)\) and of an arbitrary operator \(\Omega\) is established by the remarks on the meaning of convergences for the space \(V(\infty)\). Obviously, a \(V\)-continuous operator maps a sequence belonging to the space \(V(0,\infty)\) and converging to zero in its norm (1.10) into a sequence which, starting from some index, belongs to the space \(B\) and converges to zero in its norm (1.8). A completely analogous fact also holds for a \(v\)-continuous operator, with the replacement in the preceding statement of the space \(V(0,\infty)\) by the space \(v\). For linear operators \(\Omega\) a deeper fact is valid, namely the following.
Theorem 1.1. Let the operator \(\Omega\) be linear. Then in the case of the space \(V(\infty)\), a \(V\)-continuous operator \(\Omega\) maps the space \(V(0,\infty)\) into the space \(B\), and a \(v\)-continuous one maps the space \(v\) into the space \(B\).
Proof. We prove the first part of the assertion. Suppose the contrary, i.e., that there exists \(\eta_0\in V(0,\infty)\) such that \(\mu_0=\Omega\eta_0\notin B\). Consider the sequence \(\eta_k=\dfrac{1}{k}\eta_0\). Obviously, \(\eta_k\in V(0,\infty)\) and tends to zero in its norm (1.10). To this sequence there corresponds the sequence
\[
\mu_k=\Omega\eta_k=\frac{1}{k}\Omega\eta_0=\frac{1}{k}\mu_0.
\]
From the considerations on the meaning of continuity for an arbitrary operator \(\Omega\) it follows that, starting from some index \(k_0\), \(\mu_k\in B\) \((k\ge k_0)\), and, consequently, \(\mu_0=k_0\mu_{k_0}\in B\), which contradicts the supposition. Thus the validity of the first part of the theorem is established. The validity of the second part is established analogously. The theorem is proved.
§ 2. STABILITY OF GENERALIZED DIFFERENTIAL EQUATIONS
Let a system of differential equations be given
\[ \dot\mu=A\mu+\eta, \tag{2.1} \]
where \(A\) is an \(n\times n\) matrix with constant elements, and \(\eta(t)\) is a vector-function belonging to the space \(K_+^\prime\). Along with the system (2.1) we shall consider the homogeneous system
\[ \dot\mu=A\mu, \tag{2.2} \]
the fundamental matrix of which we denote by \(U(t)=e^{At}\), and the corresponding Cauchy operator by \(W(t,s)=U(t)U^{-1}(s)\).
Consider the relation
\[ \mu=U(t)JU^{-1}(s)\eta(s)=C\eta . \tag{2.3} \]
If in expression (2.2) \(\eta\in K'_+\), then, using the closedness of the space \(K'_+\) with respect to the operation \(J\) (taking an integral with a variable upper limit), one may assert that also \(\mu\in K'_+\). Thus, the operator \(C\) maps the space \(K'_+\) into itself. Further, by direct substitution of (2.3) into system (2.1), taking into account the last property of the operation \(J\) (see item 2, § 1), one can verify that the generalized vector-function (2.2) satisfies system (2.1). Moreover, using the fourth property of the operation \(J\), we arrive at the fact that formula (2.3) in the ordinary case coincides with the well-known Cauchy formula. In what follows, we shall attach the name Cauchy to formula (2.3), and by a solution of system (2.1) we shall understand the generalized process corresponding, by virtue of system (2.1), to a vector-function \(\eta\in K'_+\) and defined by the Cauchy formula.
Thus, in the language of automatic control theory, the operator \(C\) establishes the correspondence between the “input” and the “output” of the element described by system (2.1). The sequel is devoted to studying the character of this transformation from the point of view of stability, the notion of which will be formulated below. First we establish the validity of the following proposition.
Lemma. Let \(\eta\) be represented in the form
\[ \left<\eta,x\right>_{x\in K(a)} = \int_{0}^{a} x^{(p_a)}\,dn_a, \qquad n_a\in V(a),\qquad p_a\geq 0, \tag{2.4} \]
then the solution of system (2.1) is represented in the form
\[ \left<\mu,x\right>_{x\in K(a)} = \int_{0}^{a} x^{(p_a-1)}\,dm_a, \tag{2.5} \]
where
\[ m_a(t)=-\int_{0}^{t} W(t,s)\,dn_a(s)=C_*n_a . \tag{2.6} \]
Proof. First note that the validity of the lemma in the case \(p_a=0\) \((a=1,2,\ldots)\) follows directly from relation (1.5).
We proceed to the proof of the lemma for \(p_a\geq 1\). Taking into account the second property of the operation \(J\), direct calculation leads to the relation
\[ \left<\mu,x\right> = \int_{0}^{a} y^{(p_a)}\,dn_a, \tag{2.7} \]
where
\[ y(s)=\int_{s}^{a} W(t,s)x(t)\,dt . \]
Compute \(y^{(p_a)}\):
\[ y^{(p_a)} = (-1)^{p_a}A^{p_a}y + \sum_{i=0}^{p_a-1}(-1)^{i+1}A^i x^{(p_a-1-i)} . \tag{2.8} \]
By integration by parts it is not difficult to establish the equalities
\[ \int_0^a x^{(p_a-1-i)}\,dn_a = (-1)^i\int_0^a x^{(p_a-1)}\,dn_a^{(-i)}, \tag{2.9} \]
where the symbol \(n_a^{(-i)}\) denotes the \(i\)-fold integral of \(n_a\) (integration is carried out from 0), and
\[ \int_0^a y\,dn_a=\int_0^a x\,dn_{a0}, \tag{2.10} \]
where
\[ n_{a0}=\int_0^s\int_0^t W(t,\tau)\,dn(\tau)\,dt. \tag{2.11} \]
Proceeding with the right-hand side of equality (2.10) analogously to the case of (2.9), we shall have
\[ \int_0^a y\,dn_a = (-1)^{p_a-1}\int_0^a x^{(p_a-1)}\,dn_{a0}^{(1-p_a)} . \tag{2.12} \]
Thus, from relations (2.7)—(2.9) and (2.12) there follows the equality
\[ \langle \mu,x\rangle = -\int_0^a x^{(p_a-1)}\,dm_{a0}, \tag{2.13} \]
where the notation is adopted
\[ m_{a0} = \sum_{i=0}^{p_a-1} A^i n_a^{(-i)} + A^{p_a} n_{a0}^{(1-p_a)} . \tag{2.14} \]
The further transformations of expression (2.14) will be connected with the application of Cauchy’s formula
\[ g^{(-i)} = \int_0^s \frac{(s-\tau)^i}{i!}\,dg, \qquad g(-0)=0. \tag{2.15} \]
Applying formula (2.15) to each term of the sum (2.14), the latter can be brought to the form
\[ m_{a0}=\int_0^s X(s,t)\,dn_a(t), \tag{2.16} \]
where
\[ X = \sum_{i=0}^{p_a-1} A^i \frac{(s-t)^i}{i!} + A^{p_a}e^{-At} \int_t^s \frac{(s-\tau)^{p_a-1}}{(p_a-1)!}\,e^{A\tau}\,d\tau . \tag{2.17} \]
The equality \(X(s,t)=W(s,t)\) is established after carrying out successive integration by parts of the integral occurring in relation (2.17). Hence, taking into account expressions (2.16) and (2.13), relations (2.5) and (2.6) follow. The lemma is proved.
Let us consider the case \(p_a=0\) \((a=1,2,\ldots)\). Here \(\eta\) is the derivative of a function \(n\in V(\infty)\), and moreover \(n_a(t)=n(t)\) \((0\le t\le a,\ a=1,2,\ldots)\).
Moreover, as has already been noted repeatedly, \(m_a(t)=-\mu(t)\) \((0\le t\le a;\ a=1,2,\ldots)\).
Thus, from the general scheme we have singled out the case studied in [3], where the stability problem is solved from the point of view of the nature of the transformation \(n\to\mu\). The lemma proved makes it possible to generalize the approach indicated in [3], namely: we shall solve the stability problem for the generalized process (2.3) by relying on the transformation \(n_a\to m_a\) \((a=1,2,\ldots)\), effected by formula (2.6). Thereby the classical case will be embraced within a single framework, when the generalized vector-function \(\eta\) is the derivative of some function of bounded variation on each interval \((0,a)\) \((a=1,2,\ldots)\).
We now introduce the notions of stability for a generalized process determined by Cauchy’s formula.
Definition 2.1. The generalized process (2.3) will be called \(V\)-stable if the operator \(C\), defined by Cauchy’s formula, is \(V\)-continuous.
Definition 2.2. The generalized process (2.3) will be considered \(v\)-stable if the operator \(C\) is \(v\)-continuous.
Let us establish the meaning of the stability notions formulated for the case \(p_a=0\) \((a=1,2,\ldots)\).
Suppose that the solution (2.3) is stable in accordance with Definition 2.1. Then the operator \(C\) maps a \(V\)-convergent sequence \(\eta_k\) into a \(B\)-convergent sequence \(u_k=C\eta_k\). By virtue of definition (1.5), the sequence \(n_k\), corresponding to the sequence \(\eta_k\) by formula (1.4), turns out to be \(V\)-convergent simultaneously with the sequence \(\eta_k\), if each \(n_k\) is regarded as a generalized function and the representation obtained from formula (1.4) with \(p=-1\) is used for it. For \(\mu_k\) such a reduction is not needed, since, by assumption, \(\mu_k\in V(\infty)\).
Thus, the operator \(C_*\), acting according to formula (2.6) and mapping a \(V\)-convergent sequence into a \(B\)-convergent one, is \(V\)-continuous, and therefore, by Theorem 1.1, maps the space \(V(0,\infty)\) into the space \(B\). The application of Theorem 1 from [3] ensures the boundedness of the Cauchy operator \(W(t,s)\) for \(0\le s\le t<\infty\), which means the stability of the homogeneous system (2.2) in the ordinary sense. Conversely, if system (2.2) is stable in the ordinary sense, then from the boundedness of the Cauchy operator and Theorem 1 from [3] it follows that the operator \(C_*\) maps the space \(V(0,\infty)\) into the space \(B\).
In accordance with theorem (4.2) [5], the operator \(C_*\) will be bounded, and therefore, as is not difficult to show, \(V\)-continuous. Hence, with the aid of arguments connected with the proof of the direct assertion, there follows the \(V\)-continuity of the operator \(C\), i.e. the \(V\)-stability of the generalized process (2.3).
Thus, the \(V\)-stability of the generalized process (2.3) in the classical case \(p_a=0\) \((a=1,2,\ldots)\) is equivalent to the ordinary stability of the homogeneous system (2.2). Similarly, using only Theorem 2 from [3] instead of Theorem 1, one can establish the equivalence of \(v\)-stability to the ordinary exponential stability of system (2.2) in the classical case.
Theorems 2.1 and 2.2, which we formulate below, establish the validity of these results also in the general case.
Theorem 2.1. The generalized process determined by Cauchy’s formula (2.3) is \(V\)-stable if and only if the inequality holds
\[
\|W(t,t_0)\|\le W_0<\infty \qquad (0\le t_0\le t).
\tag{2.18}
\]
Proof. The necessity of condition (2.18) follows from the coincidence of the notion of stability (2.1) in the classical case with ordinary stability.
Sufficiency. Let us estimate expression (2.6):
\[ \|m_{ka}(t)\|_{E_n} = \left\|\int_0^t W(t,s)\,dn_{ka}(s)\right\|_{E_n} \leq \sup_{0\leq s\leq t\leq a}\|W(t,s)\|\bigvee_0^a n_{ka}. \tag{2.19} \]
Thus, taking into account inequalities (2.18) and (2.19), we have
\[ \sup_{0\leq t\leq a}\|m_{ka}(t)\|_{E_n} \leq W_0\bigvee_0^a n_{ka}. \tag{2.20} \]
Consequently, the convergence to zero, uniformly with respect to \(a\), of \(\bigvee_0^a n_{ka}\) entails the same for the left-hand side of (2.20), which means the \(V\)-continuity of the operator \(C\). This completes the proof of the theorem.
Theorem 2.2. In order that the generalized process defined by the Cauchy formula (2.3) be \(v\)-stable, it is necessary and sufficient that the inequality
\[ \|W(t,t_0)\|\leq W_0 e^{-\alpha(t-t_0)} \qquad (0\leq t_0\leq t), \tag{2.21} \]
hold, where \(W_0,\alpha>0\) and do not depend on \(t_0\).
Proof. The necessity of condition (2.21), analogously to the case of Theorem 2.1, follows from the coincidence in the classical case of the notion of \(v\)-stability (2.2) with ordinary exponential stability (2.21). We prove
Sufficiency. Let us estimate expression (2.6), using the known estimate
\[ \left\|\int_a^b x\,dn\right\|_{E_n} \leq \int_a^b \|x\|_{E_n}\,dg, \]
where \(g(t)=\bigvee_0^t n\). We have
\[ \|m_{ka}(t)\|_{E_n} = \left\|\int_0^t W(t,s)\,dn_{ka}(s)\right\|_{E_n} \leq \int_0^t \|W(t,s)\|\,dg_{ka}(s), \tag{2.22} \]
here \(g_{ka}(s)=\bigvee_0^s n_{ka}\). From inequalities (2.18) and (2.22) there follows the estimate
\[ \sup_{0\leq t\leq a}\|m_{ka}(t)\|_{E_n} \leq W_0\int_0^t e^{-\alpha(t-s)}\,dg_{ka}(s). \tag{2.23} \]
Following the ideas of [3], the integral appearing on the left-hand side of inequality (2.23) can be estimated by the quantity
\[ \frac{e^\alpha}{1-e^{-\alpha}}\max_{0<l<a}\bigvee_l^{\,l+1} n_{ka}. \]
Taking this and inequality (2.23) into account, we have the inequality
\[ \sup_{0\leq t<a}\|m_{ka}(t)\|_{E_n} \leq W_0\frac{e^\alpha}{1-e^{-\alpha}}\max_{0<l<a}\bigvee_l^{\,l+1} n_{ka}. \tag{2.24} \]
Thus, the convergence to zero, uniformly with respect to \(a\), of the quantity
\[ \max_{0<l<a}\bigvee_l^{\,l+1} n_{ka} \]
ensures the same for the quantity appearing on the right-hand side
inequality (2.24). Therefore the operator \(C\) is \(v\)-continuous, which completes the proof of the theorem.
The author thanks E. A. Barbashin and A. M. Il’in for valuable advice and their constant attention to this work.
References
-
Gelfand I. M., Shilov G. E. Generalized Functions and Operations on Them, issue 1. Fizmatgiz, 1959.
-
Gelfand I. M., Shilov G. E. Spaces of Basic and Generalized Functions, issue 2. Fizmatgiz, 1958.
-
Barbashin E. A. Differential Equations, 2, No. 7, 1966.
-
Koshlyakov N. S. Differential Equations of Mathematical Physics. Fizmatgiz, 1962.
-
Massera J. L., Schäffer J. J. Annals of Mathematics, 67, No. 3, 517—573, 1958.
Received by the editors
March 9, 1966
Sverdlovsk Branch
of the V. A. Steklov
Mathematical Institute