UDC 517.934

ON STABILITY WITH RESPECT TO IMPULSIVE DISTURBANCES

E. A. BARBASHIN

§ 1. INTRODUCTION

Consider the differential equation

\[ \dot{x}=A(t)x+u(t), \tag{1.1} \]

where \(u(t)\) is a function defined on the numerical set \(0\leq t<\infty\), taking values in the Banach space \(E\), and \(A(t)\) is a linear operator mapping the space \(E\) into itself. Suppose that the function \(\|A(t)\|\) is integrable on every finite interval of the half-axis \(0\leq t<\infty\).

It is known [1] that there exists a unique solution of equation (1.1) satisfying the condition \(x(t_0)=x_0,\ t_0\geq 0\); this solution can be found by Cauchy’s formula

\[ x(t)=W(t,t_0)x_0+\int_{t_0}^{t} W(t,s)u(s)\,ds, \tag{1.2} \]

where \(W(t,t_0)\) is the Cauchy operator, i.e., the operator that is the solution of the problem

\[ \dot{W}(t,t_0)=A(t)W(t,t_0),\qquad W(t_0,t_0)=I \tag{1.3} \]

(\(I\) is the identity element of the space \(E\)).

There are two basic concepts in stability theory: stability in the sense of Lyapunov, i.e., stability with respect to changes in the initial conditions, and stability with respect to permanently acting disturbances.

It is easy to see that stability in the sense of Lyapunov can also be regarded as stability with respect to impulsive disturbances.

Indeed, consider the generalized function \(u(t)=x_0\delta(t-t_0)\), where \(x_0\) is some fixed element of the space \(E\), \(t_0\geq 0\), and \(\delta(t-t_0)\) is the well-known Dirac function. If the condition \(x(t)=0\) for \(t<t_0\) is fulfilled, then after the action of the indicated disturbance, as is not difficult to see, the solution of the problem

\[ \dot{x}=A(t)x+u(t),\qquad x(0)=0 \tag{1.4} \]

for \(t\geq t_0\) will have the form

\[ x(t)=W(t,t_0)x_0. \]

This means that the disturbance \(u(t)\) causes an instantaneous transition at the time \(t=t_0\) of the zero point to the point \(x_0\).

Consider the function \(e(t)\) (the unit-step function), defined as follows:

\[ e(t)=0 \quad \text{for } t<t_0,\qquad e(t)=1 \quad \text{for } t\geqslant t_0 . \]

It is easy to see that the solution of problem (1.4) can be represented by the formula

\[ x(t)=\int_0^t W(t,s)x_0\,de(s), \tag{1.5} \]

where the integral should be understood as a generalized Stieltjes integral (see § 2).

In the general case one may regard equation (1.1) as describing some element of an automatic control system, with the function \(u(t)\) corresponding to the input signal and the function \(x(t)\) to the output signal. Relation (1.5) gives us an example showing that one can adopt another, more general point of view. We may consider formula (1.5) as establishing a correspondence between the input distribution \(e(t)\) and the output distribution \(x(t)\). This new point of view not only makes it possible to substantially enlarge the class of input actions, but also makes it possible to construct a more coherent theory of stability, since under the indicated approach the typical discrepancies between the properties of input and output distributions disappear.

In this note only an elementary theory is constructed, unifying the cases of stability in the sense of Lyapunov and stability with respect to permanently acting disturbances. The generalized Stieltjes integral is used as the basic mathematical apparatus. The theory of generalized functions is not used in the present paper. Nevertheless, the indicated approach has made it possible to study an important class of impulsive actions of the form

\[ u(t)=u_0(t)+\sum_{k=1}^{\infty} a_k\delta(t-t_k), \]

where the function \(u_0(t)\) is locally integrable, and \(a_k\) are elements of the space \(E\). Below, necessary and sufficient conditions are given for boundedness of the solutions of problem (1.4). Estimates of solutions under the action of impulsive disturbances are obtained. In these estimates the total variation of the input distribution is used. The corresponding estimates are also obtained in the nonlinear case. Finally, in the last section of the paper, conditions are given for the existence and asymptotic stability of discontinuous periodic solutions.

The possibility of the approach indicated above was noted in our paper [2], where a special case was considered: the action of a finite number of impulses on a finite-dimensional system.

§ 2. BASIC DEFINITIONS

Let \(E\) be a Banach space. Consider a function \(g(t)\), defined on the set of real numbers \(t\) and taking values in \(E\). Consider all possible partitions of the numerical interval \([\alpha,\beta]\) into a finite number of subintervals:

\[ \alpha=t_0\leqslant t_1\leqslant \cdots \leqslant t_n=\beta . \]

By definition, the quantity

\[ \sup \sum_{i=1}^{n}\|g(t_i)-g(t_{i-1})\|=\bigvee_{\alpha}^{\beta} g(t), \]

where the supremum is taken over all possible partitions of the interval \([a,\beta]\), is called the total variation of the function \(g(t)\) on the interval \([a,\beta]\). In this definition, of course, one may put \(\beta=\infty\). In this case we shall say that the function \(g(t)\) has bounded variation on the set \(a\leq t<\infty\), if \(g(t)\) has bounded variation on every finite part \([a,t]\) and the total variations

\[ \bigvee_{\alpha}^{t} g(t) \]

are bounded in their totality. By definition we have

\[ \bigvee_{\alpha}^{\infty} g(t)=\sup_{t>a}\bigvee_{\alpha}^{t} g(t). \]

It is known ([3], p. 73) that if the space \(E\) is Banach, then a function \(g(t)\) of bounded variation can have at most a countable number of discontinuity points, and at every point of the interval \([\alpha,\beta]\) there exist one-sided limits of this function.

We next give the definition of a generalized Stieltjes integral. Let \(U(t)\) be a linear operator, continuous in \(t\), mapping elements of the space \(E\) into \(E\), and let \(g(t)\) be a function of bounded variation on the interval \([\alpha,\beta]\). Form the integral sum of the form

\[ S_n=\sum_{k=1}^{n} U(\tau_k)\bigl(g(t_k)-g(t_{k-1})\bigr),\qquad t_{k-1}\leq \tau_k\leq t_k, \]

where the points \(t_k\) \((t_0=\alpha,\ t_n=\beta)\) form a partition of the interval \([\alpha,\beta]\). If there exists a limit of \(S_n\) as \(n\to\infty\) and \(\max_k |t_k-t_{k-1}|\to 0\), independent of the manner of partitioning the interval \([\alpha,\beta]\) and of the manner of choosing the points \(\tau_k\) on the subintervals, then we shall say that this limit is the generalized Stieltjes integral of the operator \(U(t)\) with respect to the function \(g(t)\). We introduce the notation

\[ \int_{\alpha}^{\beta} U(t)\,dg \]

for the indicated integral.

It is not difficult to verify that the inequality

\[ \left\|\int_{\alpha}^{\beta} U(t)\,dg\right\|\leq M\bigvee_{\alpha}^{\beta} g(t), \tag{2.1} \]

holds, where

\[ M=\sup_{\alpha\leq t\leq \beta}\|U(t)\|. \]

Consider the scalar function \(v(t)=\bigvee_{\alpha}^{t} g(t)\). It is easy to see that the inequality

\[ \left\|\int_{\alpha}^{\beta} U(t)\,dg\right\|\leq \int_{\alpha}^{\beta} \|U(t)\|\,dv, \tag{2.2} \]

holds, where the integral on the right-hand side is the ordinary Stieltjes integral with integrating function \(v(t)\).

§ 3. LINEAR PROBLEM

Let the law transforming the input distribution \(g(t)\) into the output distribution \(x(t)\) be given by the relation

\[ x(t)=\int_{0}^{t} W(t,s)\,dg, \tag{3.1} \]

where \(W(t,s)\) is a linear Cauchy operator mapping the space \(E\) into itself, while the functions \(g(t)\) and \(x(t)\) take values in \(E\). Obviously, the transformation (3.1) corresponds to problem (1.4).

Theorem 1. In order that to every function \(g(t)\) of bounded variation on the set \(0\leq t<\infty\) there correspond, by virtue of (3.1), a bounded function \(x(t)\), it is necessary and sufficient that the inequality
\[ \|W(t,t_0)\|\leq W_0<\infty \]
hold for all \(0\leq t_0\leq t<\infty\).

To show the necessity of the condition, consider the space \(L\) of functions \(u(t)\) such that
\[ \int_0^\infty \|u(t)\|\,dt \]
is finite. Obviously, in this case the functions
\[ g(t)=\int_0^t u(t)\,dt \]
will be functions of bounded variation on the set \(t\geq 0\). By the condition of the theorem, the solution of problem (1.4) must be bounded; hence, by virtue of Theorem 5.5 of [1], the uniform boundedness of \(W(t,t_0)\) follows.

The sufficiency of the condition of the theorem follows from inequality (2.1). From this inequality it follows that
\[ \|x(t)\|\leq W_0 \bigvee_0^\infty g(t), \]
whence we obtain the boundedness of \(x(t)\).

We now consider the space \(v\), whose elements are functions \(g(t)\) such that
\[ \|g(t)\|_v=\sup_{t\geq 0}\bigvee_t^{t+1} g(t)<\infty. \]

Theorem 2. Suppose that the condition
\[ \sup_{t\geq 0}\int_t^{t+1}\|A(t)\|\,dt<\infty \]
is satisfied.

In order that to every function \(g(t)\in v\) there correspond, by virtue of (3.1), a bounded function \(x(t)\), it is necessary and sufficient that the inequality
\[ \|W(t,t_0)\|\leq Be^{-\alpha(t-t_0)}\qquad (t_0\leq t<\infty), \tag{3.2} \]
hold, where \(\alpha>0\), \(B\geq 1\) do not depend on \(t_0\).

Let us prove the necessity of the condition. Consider the subspace of the space \(v\) consisting of all functions of the form
\[ g(t)=\int_0^t u(t)\,dt, \]
where \(u(t)\) are locally Bochner-integrable functions such that
\[ \sup_{t\geq 0}\int_t^{t+1}\|u(t)\|\,dt<\infty. \tag{3.3} \]
Since
\[ \bigvee_t^{t+1} g(t)\leq \int_t^{t+1}\|u(t)\|\,dt, \]
it is obvious that the functions \(g(t)\) belong to the space \(v\). On the other hand, the functions \(u(t)\) satisfying condition (3.3) belong to the space \(M\) considered in Massera’s work

and Schaeffer [1]. According to Theorem 5.3 of that work, from the boundedness condition for the function

\[ x(t)=\int_0^t W(t,s)\,dg=\int_0^t W(t,s)u(s)\,ds \]

condition (3.2) follows.

Let condition (3.2) now be satisfied. From inequality (2.2) it follows that

\[ \|x(t)\|\le Be^{-at}\int_0^t e^{as}\,dv, \]

where \(v(s)=\bigvee_0^s g(s)\).

Let us estimate the quantity \(\Phi(t)=e^{-at}\int_0^t e^{as}\,dv\). To this end, separate off from the number \(t>0\) its integer part, i.e., represent \(t\) in the form \(t=k+\tau\), where \(0\le \tau<1\), and \(k\) is an integer. Obviously, we have

\[ \Phi(t)\le e^{-ka}\left[\sum_{m=1}^{k+1}\int_{m-1}^{m} e^{am}\,dv\right]. \]

If

\[ \sup_{t\ge 0}\bigvee_t^{t+1}g(t)=\sup_{t\ge 0}\int_t^{t+1}dv=h<\infty, \]

then we obtain

\[ \Phi(t)\le he^{-ka}\sum_{m=1}^{k+1}e^{am}\le \frac{he^a}{1-e^{-a}}. \]

Thus, we have

\[ \|x(t)\|\le \frac{Bhe^a}{1-e^{-a}}, \tag{3.4} \]

which gives the required result.

§ 4. Auxiliary Lemmas

Lemma 1. Let \(u(t)\), \(f(t)\) be scalar nonnegative functions integrable on the interval \(t_0\le t\le t_0+T\), and let \(L\) be a positive constant. Then, if the inequality

\[ u(t)\le f(t)+L\int_{t_0}^{t}u(s)\,ds,\qquad t_0\le t\le t_0+T, \tag{4.1} \]

is satisfied, then the inequality

\[ u(t)\le f(t)+L\int_{t_0}^{t}e^{L(t-s)}f(s)\,ds \tag{4.2} \]

also holds.

In the case when \(u(t)\), \(f(t)\) are continuous, this lemma was obtained by Yu. M. Repin [4]. A more general result is contained in the book of M. G. Krein [5]. An analysis of the proofs given in the cited works readily makes it possible to establish the validity of the lemma also in the case under consideration, when \(u(t)\), \(f(t)\) are not necessarily continuous.

Lemma 2. If, under the assumptions of the preceding lemma, \(f(t)\) is a function of bounded variation on the interval \(t_0 \leq t \leq t_0+T\), then from inequality (4.1) it follows that

\[ u(t) \leq f(t_0)e^{L(t-t_0)}+\int_{t_0}^{t} e^{L(t-s)}\,df, \tag{4.3} \]

where the integral on the right-hand side is to be understood as a Stieltjes integral.

Indeed, inequality (4.3) follows directly from inequality (4.2), if for the integral appearing on the right-hand side one uses the formula of integration by parts, whose conditions of applicability are satisfied in the present case.

§ 5. THE NONLINEAR PROBLEM

Let the law of transformation of the input action be given by the equation

\[ x(t)=\int_{0}^{t} W(t,s)R(x,s)\,ds+\int_{0}^{t} W(t,s)\,dg. \tag{5.1} \]

It is easy to see that, in the case of a differentiable function \(g(t)\), problem (4.1) is equivalent to the problem

\[ \dot{x}=A(t)x+R(x,t)+u(t), \qquad x(0)=0, \tag{5.2} \]

where \(u(t)=\dot{g}(t)\).

Assume that conditions (3.2) are satisfied and

\[ \|R(x,t)\| \leq L\|x\| \tag{5.3} \]

in the domain \(D:\ \|x\|\leq H,\ 0\leq t<\infty\). Assume also that the inequality

\[ \lambda=\alpha-BL>0 \tag{5.4} \]

holds.

Theorem 3. Let

\[ \sup_{t\geq 0}\ \bigvee_{t}^{t+1} g(t)=h. \]

If conditions (3.2), (5.3), and (5.4) are satisfied, then the estimate

\[ \|x(t)\|\leq B\,\frac{he^{\lambda}}{1-e^{-\lambda}} \]

is valid.

Indeed, obviously, we have

\[ \|x(t)\|\leq BLe^{-\alpha t}\int_{0}^{t} e^{\alpha s}\|x(s)\|\,ds +Be^{-\alpha t}\int_{0}^{t} e^{\alpha s}\,dv, \]

where

\[ v(s)=\bigvee_{0}^{s} g(s). \]

For the function \(\varphi(t)=e^{\alpha t}\|x(t)\|\), the inequality

\[ \varphi(t)\leq BL\int_{0}^{t}\varphi(s)\,ds +B\int_{0}^{t} e^{\alpha s}\,dv \]

holds.

By virtue of Lemma 2 we have

\[ \varphi(t)\leq Be^{BLt}\int_0^t e^{\lambda s}\,dv, \]

whence it follows that

\[ \|x(t)\|\leq Be^{-\lambda t}\int_0^t e^{\lambda s}\,dv . \]

Using now estimate (3.4), we obtain the required result.

Theorem 3 is a theorem on stability with respect to perturbations which, in a particular case, may also be instantaneous.

§ 6. EXISTENCE AND STABILITY OF DISCONTINUOUS PERIODIC SOLUTIONS

Consider the equation

\[ x(t)=W(t,0)x_0+\int_0^t W(t,s)R(x,s)\,ds+\int_0^t W(t,s)\,dg, \tag{6.1} \]

where \(R(x,t)\) has the property \(R(0,t)=0\) for \(t\geq 0\).

Equation (6.1) corresponds, obviously, to problem (5.2), with the only difference that now \(x(0)=x_0\). Suppose that the Cauchy operator \(W(t,s)\) satisfies the condition

\[ W(t+1,s+1)=W(t,s). \tag{6.2} \]

In addition, suppose that \(R(x,s)\) and \(g(s)\) are also periodic functions of \(s\), with period equal to 1. Obviously, condition (6.2) is equivalent to the requirement of periodicity of the operator function \(A(t)\). Let us also note that if the period of the functions indicated above is equal to a number \(\omega\), different from 1, then the change of time \(t=\omega\tau\) brings us to the case under consideration.

Suppose that conditions (3.2), (5.4) are still fulfilled. We replace condition (5.3) by the requirement that the Lipschitz condition be fulfilled in the domain \(D\),

\[ \|R(x,t)-R(y,t)\|\leq L\|x-y\|. \tag{6.3} \]

Theorem 4. Let \(\|x_0\|\leq \delta\), where \(\delta=H/2B\), and let

\[ \sup_{t\geq 0}\bigvee_t^{t+1} g(t)=h<\rho\,\frac{\delta}{B}e^{-\lambda}(1-e^{-\lambda}),\qquad 0<\rho<1. \tag{6.4} \]

Then for the solution of equation (6.1) the estimate \(\|x(t)\|\leq H\) is valid. Moreover, one can indicate an \(x_0\) from the domain \(\|x\|\leq\delta\) such that the corresponding solution \(x(t)\) will be periodic and asymptotically stable. Every other solution determined by an initial point from the domain \(\|x\|\leq\delta\) will be attracted to the indicated periodic motion.

For the proof, let us note that from (6.1) it follows that

\[ \|x(t)\|\leq Be^{-\alpha t}\|x_0\|+BL\int_0^t e^{-\alpha(t-s)}\|x(s)\|\,ds +B\int_0^t e^{-\alpha(t-s)}\,dv, \tag{6.5} \]

where

\[ v(s)=\bigvee_0^s g(s). \]

Consider the function \(\varphi(t)=e^{\alpha t}\|x(t)\|\). According to (6.5), we have

\[ \varphi(t)\leqslant B\delta+B\int_0^t e^{\alpha s}\,d\nu+BL\int_0^t\varphi(s)\,ds, \]

whence, by Lemma 2, it follows that

\[ \varphi(t)\leqslant B\delta e^{BLt}+B\int_0^t e^{BLt}e^{\lambda s}\,d\nu. \]

Taking into account the notation (5.4), we obtain

\[ \|x(t)\|\leqslant B\delta e^{-\lambda t}+Be^{-\lambda t}\int_0^t e^{\lambda s}\,d\nu. \]

Using the estimate of Theorem 3, we write

\[ \|x(t)\|\leqslant B\delta e^{-\lambda t}+B\cdot \frac{he^\lambda}{1-e^{-\lambda}}, \]

whence, by (6.4), it follows that

\[ \|x(t)\|\leqslant B\delta e^{-\lambda t}+\rho\delta\leqslant (B+\rho)\delta<H, \]

since one may always assume that \(B>1>\rho\).

On the other hand, for \(T>\frac{1}{\lambda}\ln\frac{B}{1-\rho}\) we obtain \(\|x(T)\|\leqslant\delta\), and, consequently, every point \(x_0\) of the ball \(\|x\|\leqslant\delta\) again passes in time \(T\) into this ball. It is not difficult to see that in the present case the transformation \(x_0\to x(T)\) is continuous and single-valued. We shall show that this transformation satisfies the conditions of the contraction mapping principle.

If \(x_0,y_0\) are any two elements of the ball \(\|x\|\leqslant\delta\), then, according to (6.1), we have

\[ \|y(T)-x(T)\|\leqslant \|W(T,0)\|\,\|y_0-x_0\|+ \]

\[ +\int_0^T \|W(T,s)\|\,\|R(y,s)-R(x,s)\|\,ds. \]

Taking into account conditions (3.2) and (6.3), we obtain

\[ \|y(T)-x(T)\|\leqslant Be^{-\alpha T}\|y_0-x_0\| +BL\int_0^T e^{-\alpha(T-s)}\|y(s)-x(s)\|\,ds. \]

Introducing the notation \(\varphi(t)=e^{\alpha t}\|y(t)-x(t)\|\), we obtain

\[ \varphi(T)\leqslant B\|y_0-x_0\|+BL\int_0^T \varphi(s)\,ds, \]

whence, by Lemma 2, it follows that \(\varphi(T)\leqslant Be^{BLT}\|y_0-x_0\|\), or, equivalently,

\[ \|y(T)-x(T)\|\leqslant Be^{-\lambda T}\|y_0-x_0\|. \tag{6.6} \]

Since, by the choice of \(T\), we have \(Be^{-\lambda T}<1\), we arrive at the conclusion that the contraction mapping principle [6] can be applied.

By the indicated principle, in the ball \(\|x\|\leqslant\delta\) there exists a unique point \(z(0)\) such that \(z(T)=z(0)\). Since the number \(T\) may be regarded as an inte-

then the indicated point \(z(0)\) corresponds to a periodic motion. Let us show that the period of this motion is exactly equal to 1. Indeed, consider the point \(z(1)\); obviously, by periodicity we have \(z(1)=z(T+1)\), and hence \(z(1)\) is also a fixed point of the mapping considered above. But since the fixed point must be unique, it must be that \(z(0)=z(1)\).

It follows further from inequality (6.6) that the concluding part of the theorem is valid.

References

1. Massera J. L., Schäffer J. J. Annals of Mathematics, 67, No. 3, 517—573, 1958.
2. Barbashin E. A. PMM, 25, issue 2, 276—283, 1961.
3. Hille E., Phillips R. Functional Analysis and Semi-Groups. IL, 1962.
4. Repin Yu. M. PMM, 21, issue 2, 253—261, 1957.
5. Krein M. G. Lectures on the theory of stability of solutions of differential equations in a Banach space. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1964.
6. Krasnosel’skii M. A. Topological Methods in the Theory of Nonlinear Equations. GITTL, 1956.

Received by the editors
March 2, 1966

Sverdlovsk Branch
of the V. A. Steklov
Mathematical Institute