UDC 517.948.34

CAUCHY’S FORMULA AND STABILITY CONDITIONS FOR THE SOLUTION OF AN INTEGRO-DIFFERENTIAL EQUATION

L. P. BISYARINA

Consider the integro-differential equation

\[ \dot\varphi(t)=\lambda(t)\varphi(t)+\mu\int_{-\infty}^{t}K(t,s)\varphi(s)\,ds+g(t), \tag{1} \]

a particular case of which is the reactor kinetics equation [3]:

\[ \dot\varphi(t)T= \left[ \frac{k_{\mathrm{eff}}(t)-1}{k_{\mathrm{eff}}(t)} -\sum_{i=1}^{n}\beta_i \right]\varphi(t)+ \]

\[ +\sum_{i=1}^{n}\beta_i\lambda_i \int_{-\infty}^{t}\varphi(s)\exp[-\lambda_i(t-s)]\,ds+g(t). \tag{2} \]

For equation (1), in the present paper conditions are given for existence, uniqueness, and the form of the solution (Cauchy formula); conditions for stability and for the existence of an asymptotically stable periodic solution are considered.

I. Theorem 1. If the functions \(K(t,s)\) and \(\lambda(t)\) are such that the function \(R(t,\tau,\mu)\) satisfies, in the domain

\[ D=\{-\infty<s\leq \tau\leq t<+\infty,\ \mu\text{ is a parameter}\}, \]

the conditions:

1) \(R(t,\tau,\mu)\) is a continuous function together with its partial derivatives \(R'_t(t,\tau,\mu)\) and \(R'_\tau(t,\tau,\mu)\) in the domain \(D\);

2) there exists \(\lim_{\tau\to-\infty}R(t,\tau,\mu)=R(t,-\infty,\mu)\), which is a continuous function of the argument \(t\), having the continuous derivative \(R'_t(t,-\infty,\mu)\).

3) \(R(t,\tau,\mu)\) is a solution of the integro-differential equations:

\[ R'_t(t,\tau,\mu)= \mu\int_{\tau}^{t} \exp\left(-\int_{s}^{t}\lambda(\tau)\,d\tau\right) K(t,s)R(s,\tau,\mu)\,ds; \]

\[ 4)\quad R'_\tau(t,\tau,\mu)= -\mu\int_{\tau}^{t} \exp\left(\int_{s}^{\tau}\lambda(\tau)\,d\tau\right) K(s,\tau)R(t,s,\mu)\,ds; \]

\[ 5)\quad R'_t(t,-\infty,\mu)= \mu\int_{-\infty}^{t} \exp\left(-\int_{s}^{t}\lambda(\tau)\,d\tau\right) K(t,s)R(s,-\infty,\mu)\,ds; \]

\[ 6)\quad R(t,t,\mu)=1, \]

and the function \(g(t)\) is integrable in the domain \(D\), then equation (1), under the boundary condition \(\varphi(-\infty)=\varphi_0\), has a unique solution of the form

\[ \varphi(t)=\exp\left(\int_{-\infty}^{t}\lambda(\tau)\,d\tau\right)R(t,-\infty,\mu)\varphi_0+ \]

\[ +\int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu)g(\tau)\,d\tau . \tag{3} \]

Proof. Differentiating the left- and right-hand sides of equality (3) with respect to \(t\), and using properties 1)—6), we prove that the function of the form (3) is a solution of equation (1). To prove the uniqueness of the solution of equation (1), suppose that the function \(\varphi(t)\) is a solution of equation (1), and show that this solution has the form (3). For this, multiply both sides of the identity

\[ \dot{\varphi}(\tau)=\lambda(\tau)\varphi(\tau)+\mu\int_{-\infty}^{\tau}K(\tau,s)\varphi(s)\,ds+g(\tau) \]

by

\[ \exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu) \]

and integrate with respect to \(\tau\):

\[ \int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu)\dot{\varphi}(\tau)\,d\tau = \]

\[ = \int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)\lambda(\tau)R(t,\tau,\mu)\varphi(\tau)\,d\tau+ \]

\[ +\mu\int_{-\infty}^{t}\int_{-\infty}^{\tau}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)K(\tau,s)R(t,\tau,\mu)\varphi(s)\,ds\,d\tau+ \]

\[ +\int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu)g(\tau)\,d\tau . \]

After transforming the left-hand side of the equality obtained, on the basis of properties 3)—4), and canceling identical terms, we arrive at an expression of the form

\[ \varphi(t)-\exp\left(\int_{-\infty}^{t}\lambda(\tau)\,d\tau\right)R(t,-\infty,\mu)\varphi_0 = \]

\[ =\int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu)g(\tau)\,d\tau . \tag{4} \]

Remark. The function \(R(t,\tau,\mu)\) may be regarded as the sum of the series

\[ \sum_{n=0}^{\infty}N_n(t,\tau)\mu^n, \tag{5} \]

where

\[ N_n(t,\tau)=\int_{\tau}^{t}\int_{\tau}^{\tau_1}\exp\left(\int_{\tau_1}^{s}\lambda(\tau)\,d\tau\right)K(\tau_1,s)N_{n-1}(s,\tau)\,ds\,d\tau_1 . \tag{6} \]

\[ (n=1,\ 2,\ 3,\ldots,\ -\infty<\tau\leq t<+\infty) \]

and

\[ N_0(t,\tau)=1. \tag{7} \]

Therefore formulas (5)—(7) may be used for the approximate calculation of the resolvent. Starting from the definition of the resolvent \(R(t,\tau,\mu)\) as the sum of the series (5) and imposing on the functions \(K(t,s)\) and \(\lambda(t)\), and on their partial derivatives, certain conditions—continuity, boundedness, and uniform convergence of the improper integrals with respect to the parameter—one can prove that the series (5) converges for all finite values of the parameter \(\mu\) to the function \(R(t,\tau,\mu)\), which, in addition to properties 1)—6) of Theorem 1, will also satisfy two integral equations:

\[ R(t,\tau,\mu)=\mu\int_{\tau}^{t}\int_{\tau}^{\tau_1} \exp\left(\int_{\tau_1}^{s}\lambda(\tau)\,d\tau\right) K(\tau_1,s)R(s,\tau,\mu)\,ds\,d\tau_1+1; \]

\[ R(t,\tau,\mu)=\mu\int_{\tau}^{t}\int_{\tau}^{\tau_1} \exp\left(\int_{\tau_1}^{s}\lambda(\tau)\,d\tau\right) K(\tau_1,s)R(t,\tau_1,\mu)\,ds\,d\tau_1+1. \]

II. Suppose that some process is described by the equation

\[ \dot f(t)=\lambda_1(t)f(t)+\mu\int_{-\infty}^{t}K_1(t,s)f(s)\,ds+g_1(t) \tag{8} \]

and that its solution \(f(t)\) is known (calculated). We shall say that this process is characterized by the parameter-functions: \(\lambda_1(t)\), \(K_1(t,s)\), \(g_1(t)\), and \(\varphi_0^{(1)}\) (the boundary condition).

Very often one has to deal with similar processes, but characterized by somewhat different parameters:

\[ \lambda_2(t),\qquad K_2(t,s),\qquad g_2(t)\ \text{and}\ \varphi_0^{(2)}. \tag{9} \]

The question arises as to what changes in the parameters are admissible so that the process under consideration differs only slightly from the calculated one, i.e., so that the changes in the parameters may be neglected and all the computations need not be performed anew. In other words, the question is one of conditions for the stability of the solution of equation (8). Denoting by \(\varphi(t)=\psi(t)-f(t)\) and

\[ g(t)=[\lambda_2(t)-\lambda_1(t)]f(t)+[g_2(t)-g_1(t)]+ \]

\[ +\mu\int_{-\infty}^{t}[K_2(t,s)-K_1(t,s)]f(s)\,ds, \]

we obtain an equation of the form

\[ \dot\varphi(t)=\lambda_2(t)\varphi(t)+\mu\int_{-\infty}^{t}K_2(t,s)\varphi(s)\,ds+g(t), \]

where the deviation of the function \(\varphi(t)\) from the zero solution shows how much the solution \(\psi(t)\) of equation (8) with parameters (9) deviates from the calculated solution \(f(t)\), while the function \(g(t)\) characterizes the magnitude of this deviation. Thus the question of stability of the solution of equation (1) is reduced to the investigation of the stability of the trivial solution of the equation

\[ \dot\varphi(t)=\lambda(t)\varphi(t)+\mu\int_{-\infty}^{t}K(t,s)\varphi(s)\,ds. \tag{10} \]

Definition 1. The zero solution of equation (10) is called stable under permanently acting perturbations if, for any \(\varepsilon>0\), one can specify \(\delta>0\) and \(h>0\) such that from \(|\varphi_0|<\delta\), \(\sup_{-\infty<t<+\infty}|g(t)|<h\), it follows that \(\|\varphi(t)\|<\varepsilon\) for \(-\infty<t<+\infty\), where \(\varphi(t)\) is a solution of equation (1).

Definition 2. The zero solution of equation (10) is called stable under permanently acting perturbations bounded on the average if, for any \(\varepsilon>0\), one can specify such \(\delta>0\) and \(h>0\) that from \(|\varphi_0|<\delta\) and
\[ \int_{-\infty}^{+\infty}|g(\tau)|\,d\tau<h \]
it follows that \(\|\varphi(t)\|<\varepsilon\) for \(-\infty<t<+\infty\), where \(\varphi(t)\) is a solution of equation (1).

Definition 3. The zero solution of equation (10) is called stable under permanently acting perturbations bounded in the mean square if, for any \(\varepsilon>0\), one can specify such \(\delta>0\) and \(h>0\) that from \(|\varphi_0|<\delta\) and
\[ \left(\int_{-\infty}^{+\infty}g^2(\tau)\,d\tau\right)^{1/2}<h \]
it follows that \(\|\varphi(t)\|<\varepsilon\) for \(-\infty<t<+\infty\), where \(\varphi(t)\) is a solution of equation (1).

Theorem 2. Suppose that the conditions of Theorem 1 hold. If, in addition, the following conditions are satisfied:
\[ a)\quad \|R(t,\tau,\mu)\|\leq M, \]
\[ b)\quad \exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)\leq B_0\exp[-\beta(t-\tau)]\leq q>0, \]
where
\[ B_0>0,\qquad \beta>0,\qquad -\infty<\tau\leq t<+\infty, \]
\[ c)\quad |\varphi_0|<\frac{\varepsilon(1-p)}{qM},\quad \text{where } p<1, \]
and one of the conditions:
\[ A)\quad \sup_{-\infty<t<+\infty}|g(t)|<\frac{\varepsilon\beta p}{B_0M}, \]
\[ B)\quad \int_{-\infty}^{+\infty}|g(\tau)|\,d\tau<\frac{\varepsilon p}{qM}, \]
\[ C)\quad \left(\int_{-\infty}^{+\infty}g^2(\tau)\,d\tau\right)^{1/2}<\frac{\varepsilon p\sqrt{2\beta}}{B_0M}, \]
then the solution (3) of equation (1) is stable.

Proof. We write the solution of equation (1) by formula (3):
\[ \varphi(t)=\exp\left(\int_{-\infty}^{t}\lambda(\tau)\,d\tau\right)R(t,-\infty,\mu)\varphi_0+ \]
\[ +\int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu)g(\tau)\,d\tau. \]
Taking into account that
\[ \|\varphi(t)\|=\sup_{-\infty<t<+\infty}|\varphi(t)|, \]

\[ \|R(t,\tau,\mu)\|=\sup_{-\infty<\tau<t<+\infty}|R(t,\tau,\mu)|, \]

we estimate the solution on the basis of conditions \(a)—c)\):

\[ |\varphi(t)|<\varepsilon(1-p)+B_0M\int_{-\infty}^{t}\exp[-\beta(t-\tau)]|g(\tau)|\,d\tau . \]

Then, if any of the conditions \(A), B)\), or \(C)\) is satisfied, we shall have \(\|\varphi(t)\|<\varepsilon\), i.e., the solution of equation (1) is stable.

Theorem 3. Suppose that the conditions of Theorem 1 and conditions \(a), b)\) of Theorem 2 hold. If, in addition, the functions \(K(t,s)\), \(\lambda(t)\), and \(g(t)\) occurring in equation (1) are periodic with period \(\omega\), and if one of the conditions

\[ A_1)\qquad \sup_{-\infty<t<+\infty}|g(t)|<h_0<\frac{\delta\beta}{B_0M}, \]

\[ B_1)\qquad \int_{-\infty}^{+\infty}|g(\tau)|\,d\tau<h_1<\frac{\delta}{qM}, \]

\[ C_1)\qquad \left(\int_{-\infty}^{+\infty}g^2(\tau)\,d\tau\right)^{1/2}<h_2<\frac{\sqrt{2\beta}}{B_0M}, \]

is satisfied, then in the domain \(D\) there exists an asymptotically stable periodic solution of equation (1) under the boundary condition \(|\varphi_0|<\delta>0\).

Proof. We first prove that there exists a solution of equation (1) such that, for some number \(\delta>0\), one can specify a number \(T\) so large that from the inequality \(|\varphi_0|<\delta\) there follows the existence of the inequality \(|\varphi(t)|<\delta\) for \(t>T\), i.e., we prove that equation (1) has the property of dissipative stability [1].

We use the estimate for the solution of equation (1):

\[ |\varphi(t)|\leq B_0M\exp(-\beta t)|\varphi_0| +B_0M\int_{-\infty}^{t}\exp[-\beta(t-\tau)]|g(\tau)|\,d\tau \]

or

\[ |\varphi(t)|\leq \Phi_1(t)+\Phi_2(t), \]

where

\[ \Phi_1(t)=B_0M\exp(-\beta t)|\varphi_0|, \]

\[ \Phi_2(t)=B_0M\int_{-\infty}^{t}\exp[-\beta(t-\tau)]|g(\tau)|\,d\tau . \]

Take

\[ t=T=\frac{1}{\beta}\ln(B_0MN) \]

(the number \(N\) will be specified later) and estimate \(\Phi_1(t)\) and \(\Phi_2(t)\) for \(t>T\):

\[ \Phi_1(t)<B_0M\exp(-\beta\ln T)|\varphi_0|<\frac{\delta}{N}. \]

Now suppose that condition \(A_1)\) is satisfied; then

\[ \Phi_2(t)<B_0Mh_0\int_{-\infty}^{t}\exp[-\beta(t-\tau)]\,d\tau<\delta . \]

If condition \(B_1)\) is satisfied, then

\[ \Phi_2(t) < qMh_1 < \delta . \]

Finally, suppose that inequality \(C_1)\) holds:

\[ \Phi_2(t) < B_0Mh_2\left(\int_{-\infty}^{t}\exp[-2\beta(t-\tau)]\,d\tau\right)^{1/2}<\delta . \]

Thus,

\[ \sup_{-\infty<t<+\infty}\Phi_2(t)<\delta \]

(there were two strengthened inequalities). Consequently, one can choose an integer \(N\) so large that

\[ \sup_{-\infty<t<+\infty}\Phi_2(t)<\frac{N-1}{N}\,\delta . \]

Then

\[ \|\varphi(t)\|\leq \Phi_1(t)+\Phi_2(t)<\frac{\delta}{N}+\frac{N-1}{N}\,\delta=\delta \quad \text{for } t>T . \]

Next choose the number

\[ T=m\omega>\frac{1}{\beta}\ln(B_0MN), \]

where \(m\) is an integer, and two boundary values \(\varphi_0\) and \(\psi_0\) from the domain of boundary values, less than \(\delta>0\) in absolute value. The difference of the solutions corresponding to these boundary conditions will be equal to

\[ \varphi(t)-\psi(t)= \exp\left(\int_{-\infty}^{t}\lambda(\tau)\,d\tau\right) R(t,-\infty,\mu)(\varphi_0-\psi_0) \]

and

\[ \|\varphi(t)-\psi(t)\|\leq B_0M\exp(-\beta t)|\varphi_0-\psi_0| \]

for

\[ t>T>\frac{1}{\beta}\ln(B_0MN), \]

or

\[ \|\varphi(t)-\psi(t)\|<\frac{1}{N}|\varphi_0-\psi_0|, \quad \text{where } N>1 . \]

On the basis of the contraction mapping principle, in the domain \(D\) there exists a function \(\varphi(t)\) with boundary condition \(|\varphi_0|<\delta\), such that the following equalities hold:

\[ \varphi(T)=\varphi(-\infty)=\varphi_0, \qquad \varphi(\omega+T)=\varphi(\omega). \]

Since the functions entering the equation are periodic with period \(\omega\), it follows that \(\varphi(\omega)=\varphi_0\). Asymptotic stability follows from the inequality

\[ \|\varphi(t)-\psi(t)\|< B_0M\exp(-\beta t)|\varphi_0-\psi_0|. \]

III. Let us now consider an equation of the form

\[ \varphi(t)=\int_{-\infty}^{t}H(t,s,\varphi(s))\,ds+V(t,\varphi(t)). \tag{11} \]

We separate, according to some rule, from the functions \(H(t,s,\varphi(s))\) and \(V(t,\varphi(t))\) the term proportional to \(\varphi(t)\), and from the function \(V(t,\varphi(t))\) the free term; then equation (11) takes the form

\[ \dot{\varphi}(t)=\lambda(t)\varphi(t)+\mu\int_{-\infty}^{t}K(t,s)\varphi(s)\,ds+ \]

\[ +\int_{-\infty}^{t}Z(t,s,\varphi(s))\,ds+V(t,\varphi(t))+g(t), \tag{12} \]

where

\[ Z(t,s,0)=V(t,0)=0. \]

Theorem 4. Let the functions \(\lambda(t)\), \(K(t,s)\), \(R(t,\tau,\mu)\) satisfy the conditions of Theorem 1, and suppose the following conditions hold:

\[ a_1)\quad \|R(t,\tau,\mu)\|\leq M, \]

\[ b_1)\quad \exp\left(\int_{\tau}^{t}\{\lambda(\tau)+M[L(t)+N(\tau)]\}\,d\tau\right) \leq B_0\exp[-\beta(t-\tau)]\leq q<\frac{1}{M}, \]

where

\[ B_0>0,\qquad \beta>0,\qquad L(t)>0,\qquad N(t)>0, \]

\[ c_1)\quad \int_{-\infty}^{+\infty}\left|Z(t,s,\varphi(s))-Z(t,s,\psi(s))\right|\,ds \leq L(t)|\varphi(t)-\psi(t)|, \]

\[ d_1)\quad \left|V(t,\varphi(t))-V(t,\psi(t))\right| \leq N(t)|\varphi(t)-\psi(t)|, \]

and the function

\[ f(t)=F(t,\varphi(t))=\int_{-\infty}^{t}Z(t,s,\varphi(s))\,ds+V(t,\varphi(t))+g(t) \]

is integrable with respect to \(t\). Then in the domain \(D\) there exists a unique solution of equation (12) with the boundary condition \(\varphi(-\infty)=\varphi_0\).

Proof. Equation (12) is equivalent to the integral equation

\[ \varphi(t)=\exp\left(\int_{-\infty}^{t}\lambda(\tau)\,d\tau\right)R(t,-\infty,\mu)\varphi_0+ \]

\[ +\int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu)F(\tau,\varphi(\tau))\,d\tau. \]

We shall show that the operator \(A(t)\), acting on the function \(\varphi(t)\), with boundary condition \(\varphi_0\), is a contraction operator. For this purpose consider this operator applied to the difference of the functions \(\varphi(t)\) and \(\psi(t)\), respectively with boundary conditions \(\varphi_0\) and \(\psi_0\):

\[ \varphi(t)-\psi(t)= \exp\left(\int_{-\infty}^{t}\lambda(\tau)\,d\tau\right)R(t,-\infty,\mu)(\varphi_0-\psi_0)+ \]

\[ +\int_{-\infty}^{t} \exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)R(t,\tau,\mu) \left\{[V(\tau,\varphi(\tau))-V(\tau,\psi(\tau))]+\right. \]

\[ \left. +\int_{-\infty}^{\tau}[Z(\tau,s,\varphi(s))-Z(\tau,s,\psi(s))]\,ds \right\}\,d\tau. \]

Then, using properties \(a_1)—d_1)\) of the theorem, we obtain

\[ |\varphi(t)-\psi(t)| \leq \exp \left(\int_{-\infty}^{t}\lambda(\tau)\,d\tau\right) M|\varphi_0-\psi_0|+ \]

\[ +\int_{-\infty}^{t}\exp\left(\int_{\tau}^{t}\lambda(\tau)\,d\tau\right)M[N(\tau)+L(\tau)]|\varphi(\tau)-\psi(\tau)|\,d\tau. \]

On the basis of Lemma 1 of [2] we shall have

\[ |\varphi(t)-\psi(t)| \leq M\exp\left(\int_{-\infty}^{t}\{\lambda(\tau)+M[L(\tau)+N(\tau)]\}\,d\tau\right)\times \]

\[ \times |\varphi_0-\psi_0| \leq Mq|\varphi_0-\psi_0|. \]

Thus,

\[ \|A\|\leq Mq<1. \]

Then, by the contraction mapping principle, there exists such a function \(\varphi(t)\), and moreover a unique one, with boundary condition \(\varphi_0\), which satisfies equation (12).

Lemma 1 (on the estimate of the solution). If the conditions of Theorem 4 hold, then the solution of equation (12) satisfies the estimate

\[ |\varphi(t)|\leq \Phi_1(t)+\Phi_2(t), \]

where

\[ \Phi_1(t)=B_0M\exp(-\beta t)\varphi_0, \]

\[ \Phi_2(t)=B_0M\int_{-\infty}^{t}\exp[-\beta(t-\tau)]g(\tau)\,d\tau. \]

The proof is carried out using Cauchy’s formula (3) and Lemma 1 of [2].

Theorem 5. Let the conditions of Theorem 4 hold. If, in addition, one of the conditions A), B), or C) of Theorem 2 is satisfied, then the solution of equation (12) is stable under the boundary condition

\[ |\varphi_0|<\frac{\varepsilon(1-p)}{qM}. \]

Theorem 6. Let the conditions of Theorem 4 hold. If, in addition, the functions \(K(t,s)\), \(\lambda(t)\), \(g(t)\), \(Z(t,s,\varphi(s))\), and \(V(t,\varphi(t))\) are periodic with period \(\omega\), and if one of the conditions \(A_1)\), \(B_1)\), or \(C_1)\) of Theorem 3 is fulfilled, then in the domain \(D\) there exists an asymptotically stable periodic solution of equation (13) under the boundary condition \(|\varphi_0|<\delta>0\).

The proofs of Theorems 5 and 6 are carried out in the same way as those of Theorems 2 and 3.

References

1. E. A. Barbashin, L. P. Bisyаrina, Izv. vuzov, Matematika, No. 3 (34), 3—14, 1963.
2. L. Kh. Liberman, Izv. vuzov, Matematika, No. 3 (4), 142—151, 1958.
3. L. N. Usachev, An equation for the value of neutrons in a reactor. Materials of the Soviet Delegation at the International Conference on the Peaceful Uses of Atomic Energy. Geneva, 1955. Reactor Construction and Theory of Reactors. Publishing House of the Academy of Sciences of the USSR, 1955.

Received by the editors
March 25, 1966

Sverdlovsk State
Pedagogical Institute