Solvability and Construction of a Solution to the Fredholm Integral Equation of the First Kind ()
1. Introduction
The search for the necessary and sufficient conditions for the existence of a solution of the Fredholm integral equation of the first kind and the construction of its solution is one of the current unsolved problems in mathematics [1] [2] .
Many problems in natural sciences lead to the Fredholm integral equation of the first kind, where it is required to reconstruct the original phenomenon based on measurement results. Special cases of the Fredholm integral equation of the first kind include the Volterra integral equations of the first kind and Abel's integral equation.
Integrodifferential equations involving the Fredholm integral equation of the first kind, particularly including Volterra and Abel integral equations, serve as mathematical models for many phenomena in various scientific fields: biology [3] , medicine [4] , biophysics [5] , thermodynamics and biological processes [6] , mechanics and electrodynamics [7] , economics [8] and synergetics [9] .
The first work dedicated to distributed delay is the monograph [10] . A review of scientific research on differential equations with deviating arguments is contained in [11] . A qualitative theory of integrodifferential equations is presented in [12] . A review of numerical methods for solving integrodifferential equations can be found in [13] . The correct solvability of the initial problem of Volterra integral-differential equations is given in [14] . Linear homogeneous systems of integrodifferential and integral equations with Volterra and Fredholm matrix kernels with initial conditions equal to zero are considered in [15] . Nonlinear Volterra equations with loads and bifurcation parameters are described in [16] .
Based on the results of constructing ща the general solution of the Fredholm integral equation of the first kind with a fixed parameter, the following problems have been solved: boundary value problems of ordinary differential equations with phase and integral constraints without involving the Green’s function [17] [18] ; optimal control of dynamic systems with constraints without involving Pontryagin’s maximum principle [19] [20] ; controllability and performance of ordinary differential equations and parabolic equations with restrictions [21] [22] . The general theory of boundary value problems of dynamic systems is presented in [23] .
This work is a continuation of scientific research from [12] [17] - [23] . The scientific novelty of the results obtained in this article is the reduction of solvability and construction of a solution of the Fredholm integral equation of the first kind to an extremal problem in Hilbert space; construction of minimizing sequences and studies of their convergence; determination of weak limit points of minimizing sequences; creation of constructive theory of solvability and construction of solutions of integrodifferential equations with distributed delay in control.
2. Problem Definition
Let’s consider the Fredholm integral equation of the first kind
(1)
where
-known matrix of order n × m, the elements of the matrix
of the function
are measurable and belong to the class
at the set
function
– given,
– desired function, values
– fixed,
,
,
.
From (1), in particular, we get:
1) Volterra integral equation of the first kind
2) Abel integral equation
where
.
3) The Fredholm integral equation of the first kind are
where
is a fixed parameter.
The following tasks are solved:
Problem 1. Find the necessary and sufficient conditions for the existence of a solution to the integral Equation (1) for a given
.
Problem 2. Find a solution to the integral Equation (1) for a given
Problem 3. Find the necessary and sufficient conditions for the existence of a solution to the integral Equation (1) for a given
, when the desired function
. Either
or
Problem 4. Find a solution to the integral Equation (1) for a given
, when
, where
, is a bounded convex closed set in
.
Let us consider a controlled process described by an integral-differential equation of the following form
(2)
boundary value
(3)
where
are any fixed points?
Given data:
are given matrixes with piecewise continuous elements of orders n × n, n × m, n × m1 accordingly,
is a known matrix of order m1 × n1 with elements from
,
is a given control function
(4)
Definition 1. The solution of Equation (2) generated by the controls
,
with
is called the function
that satisfies the par
Therefore, the function
,
with the primary condition
satisfies the par of Newton-Leibnizand belongs to the class of continuous functions
,
.
Definition 2. Equation solution (2) the function
, generated by the controls
is called controlled equation, if the following controls
,
, that change the trajectory of the system (2)-(4) from any given point
into the moment of given time
, and to any desired end state
in timing
.
Problem 5. Find the necessary and sufficient conditions for controls of the equation (2) solution with the given (3), (4).
Problem 6. Find a pair
, that changes the trajectory of the system (2)-(4) from any given point
to the
, to any desired end state
in timing
.
Problem 7. Find a solution
corresponding to the pair
.
3. Solvability of the Fredholm Integral Equation of the First Kind
Consider the integral Equation (1). To solve problems 1, 2 it is necessary to investigate an extreme problem: to minimize the functional
(5)
with the condition of
(6)
where
is given function,
is the Euclidian norm.
Theorem 1. Let the core of operator
is measured and belongs to the class
in the rectangle
. then:
1) composed function (5) with the condition (6) is continuously differentiates on Fréchet, the gradient of functional
in any point of
is defined by the formula
(7)
2) gradient of function
satisfies the Lipschitz condition,
, for any u and
; (8)
3) composed functional (5) with the condition (6) is convex functional
for any
, (9)
if
,
;
4) second Fréchet derivative is equal to
(10)
5) if the following inequation is accomplished
(11)
the functional (5) with condition (6) is strongly convex.
Proof. As follows from (5), the functional
Then the excess functional
It follows that the Fréchet differential is equal to
Then, applying Fubini’s theorem to the variables of integration, we have
Then Fréchet’s derivative is
where
It follows that
is defined on formula (7). since
then
therefore
for any
. It follows this inequation (8).
Let us show that functional (5) with the condition (6) is convex.
For any
this inequation is true.
It means that functional (5) with condition (6) is convex.
Hence, the inequation (9) is solved. As it follows from (7)
Hence,
is defined by formula (10). From (10), (11) it follows that
with all
,
.
It means, that functional (5) with condition (6) is strongly convex в
. The Theorem is proven.
Lemma 1. Let this
be a solution to the optimal control problem (5), (6). For the Fredholm integral equation of the first kind (1) to have a solution, it is necessary and sufficient that the value
.
Proof. As follows from the optimal control problem (5), (6), the value
, if and only if
, for all
.
It means the function
,
is a solution to the integral Equation (1). The Lemma is proven.
Thus, for the existence of a solution of the integral Equation (1), it is necessary and sufficient that the value of
(the solution of problem 1).
Lemma 2. Let this
be a solution for optimal control problem: minimize the functional
(12)
For the Fredholm integral equation of the first kind (1) with
has a solution, it is necessary and sufficient that the value of
.
Proof. The value
,
, then and only then, when
, with all
. Hence
is the integral equation’ solution (1) with this condition
(problem 3 solution). The Lemma is proven.
4. Construction of the Solution of the Fredholm Integral Equation of the First Kind
Consider the integral Equation (1). The solution to problem 2 follows from the following theorem.
Theorem 2. Let for the extreme problem (5), (6) the sequence
with algorithm
where
is defined by formula (7) with
,
is the starting point.
Then:
1) the numerical sequence is strictly decreasing
the limit
;
If the set
is limited, then:
2) the sequence
is a minimizing
,
;
3) the sequence
weakly converges to the set
, where
4) the following estimate of the convergence rate is valid
(13)
5) if inequality (11) is fulfilled, then the sequence is
strongly converges to the
. The following convergence estimates are valid
(14)
where
,
are constant from (8), (11)accordingly;
6) For the Fredholm integral equation of the first kind (1) to have a solution, it is necessary and sufficient that the value
,
. In this case
is a solution for integral Equation (1);
7) if the value is
, then integral Equation (1) doesn’t have a solution with the given
.
Proof. Since
, then
,
,
from the inclusion of
it follows, that
Then
. Hence, the numerical sequence is
strictly decreasing and
. The first statement of the theorem is proved.
Since the funcitonal
is convex with the
, then set of
. Hence the
is bounded convex closed set in
weakly bicompact. Convex continuously differentiable functional
is weakly semicontinuous from below on the set
. Hence the set
,
is a null set,
, the following inequation is valid.
where D is a the
diameter. Notice, that
Hence, on the set
the lower bound of the functional
in point
, the sequence
is minimized. Hence, the second statement of the theorem is proved.
third statement of the theorem follows from the inclusion of
,
that is weakly semi-compact set,
,
. Hence,
при
.
From inequations
where
npu
estimate follows (13), where
. The fourth statement of the theorem is proved.
If inequality (11) is satisfied, then the functional (5) under condition (6) is strongly convex. Then
It follows that
, where
. Hence,
then
, where
. Hence the estimate (14) follows. The fifth statement is proved.
As follows from (6), the value of
, with all
. Sequence
is minimizing for any starting point
, where
.
If
, thus
. Thus, the integral Equation (1) has a solution if and only if the value
, where
is the solution to the integral Equation (1). If value
, mo
, thus
is not a solution of the integral Equation (1). The theorem is proved.
Example 1. An integral equation is given
(15)
The optimization problems (5), (6) will be written in the following form
The gradient of the functional
Lipschitz constant:
.
Sequence:
,
,
.
The sequence
converges to the element
,
. The value
, the solution of the integral equation (15) is
,
. It is easy to find that
.
5. Solvability of the Fredholm Integral Equation of the First Kind with Constraint
Consider the integral equation (1), when
, where
is a bounded convex closed set in
.
The solution to problem 3,4 can be obtained from the solution of the optimization problem as
minimize the functional:
(16)
with condition
(17)
Theorem 3. Let the operator kernel be measurable
and belongs to
in the rectangle
Then:
1) the functional (16) under condition (17) is continuously differentiable by Fréchet, the gradient of the functional
in any point
defined by the formula
2) the gradient of the functional
satisfies the Lipschitz condition
for all
;
3) functional (16) with conditions (17) is convex.
The proof of the theorem is similar to the proof of Theorem 1.
Theorem 4. Let for the optimization problem (16), (17) the last sequence
on algorithm
if
,
,
or
,
.
Thus:
1) The numerical sequence
is strictly decreasing.
2)
,
;
If, in addition, the set
is limited, then:
3) sequence
is minimizing
4)
,
when
,
5) so that the integral Equation (1) under the condition
has a solution, it is necessary and sufficient that the value of
.
The proof of the theorem is like the proof of Theorem 2.
6. Controllability of a Linear Integra-Differential System
Let us consider the solutions of problems 5 - 7 for the process described by a linear system of integrodifferential Equations (2) under conditions (3), (4). To solve Problems 5 - 7, it is necessary to investigate the controllability of an auxiliary system of the following kind
, (18)
(19)
(20)
Let’s introduce the notation
,
,
. Then the system (18)-(20) will be written as
, (21)
(22)
(23)
where
matrix of order
.
Solution of differential Equation (21) with the initial condition
has the following form
(24)
where
,
is the fundamental matrix of solutions of a linear homogeneous system
,
,
,
is the unit matrix of order
.
From given (24) considering, that
,
, we have
(25)
where
. Thus, the solution of the boundary problem (21)-(23) is a solution of the integral equation (25).
Theorem 5. For the existence of control
, which translates the trajectory of equation (21) from any starting point
at any given time to any desired end state
, it is necessary and sufficient that the matrix
(26)
order of
(matrix) to be positively determined, where (
) is a transposition symbol. At the same time, the control
(27)
is the solution of the differential Equation (21), is relevant to control
that defined by the formula
(28)
(29)
The proof of the theorem for the general case is given in [17] . Control
,
defined by formula (27) (control with the minimum norm) is found by Kalman R.Е. [19] [20] .
From (26), (27) as follows
where
(30)
(31)
The solution of problem 5 follows from the following theorem.
Theorem 6: For controllability of the solution of the linear integrodifferential Equation (2) under conditions (3), (4) it is necessary and sufficient to satisfy the following conditions:
1) set
defined by formula (26) was positively defined.
2) identical equations are satisfied
(32)
where
defined by formulas(30), (31).
Proof. Let
is the solution to the controllability problem for the system (2)-(4), where
,
,
,
. Function
is the solution for the control system problem (18)-(20), where
,
,
,
, set
.
So
,
, it is necessary and sufficient that the set
and the identical equations are satisfied (32). The connection between the solutions of the control problem (2)-(4) and (18)-(20) is defined by equations (32). From the controllability of the system (18)-(20) follows the controllability of the system (2)-(4). The theorem is proved.
As follows from identities (32), the desired control,
the control
is determined from the solution of the optimization problem
(33)
with the following conditions
(34)
where
The solution to the optimization problem (33), (34) is given in part 2, where
,
,
,
.
The solution of problems 6, 7 follows from the following theorem.
Theorem 7. Let the set
, the value
, where
is the solution of optimization problem (33)-(34). then:
1) the pair
translates the trajectory of the system (2)-(4) from any starting point
in the time
to any desired end state
in the time
;
2) solution of the controllability problem for the system (2)-(4) function
The proof of the theorem follows from Theorems 5, 6.
Example 2. Consider a controllable process
(35)
In vector form, the system (35) will be written as
(36)
where
1) Necessary and sufficient conditions of controllability. A linear controlled system has the form (see (18)-(20)).
(37)
where the sets are
Using the initial point, we find the following vectors and matrixes:
The matrix
is positively determined. Consequently, there exists a control.
, where
Function
,
where
then
Now the necessary and sufficient controllability conditions for the system (37) will be written as
2) Construction of a solution to the controllability problem. The control
is defined as a solution of optimization problem: there is need to find a minimum of
The Freche function derivative
at any point
. Minimizing sequence
where
if
.
7. Conclusions
As a result of scientific research, the following results were obtained:
· solvability and construction of the Fredholm integral equation of the first kind are reduced to solving an extremal problem in a Hilbert space;
· necessary and sufficient conditions for the existence of a solution of the Fredholm integral equation of the first kind are obtained;
· algorithms for constructing minimizing sequences have been developed, their convergence to solutions of Fredholm integral equations of the first kind has been proven for cases with and without restrictions on the desired solution;
· a constructive theory of solvability and construction of a solution to a linear integrodifferential equation with distributed control delay was created;
· the scientific novelty of the results obtained lies in the fact that a new method has been created for studying of the Fredholm integral equation of the first kind based on convex analysis and the theory of extremal problems in Hilbert space;
· the practical value of the results obtained lies in the fact that a constructive algorithm has been created for solving the Fredholm integral equation of the first kind and an integrodifferential equation with distributed control, easily implemented by modern means of computational mathematics and computer technology.