The paper is devoted to obtaining the necessary and sufficient conditions of the solvability of weakly perturbed boundary-value problems for the nonlinear operator-differential Riccati equation in the Hilbert space on the interval and whole line with parameter ?. We find the solution of the given boundary value problem which for ε = 0 turns in one of the solutions of generating boundary value problem. Solution of the generating problem is constructed with the using generalized operator in analytical form. Iterative process for finding of solutions of weakly nonlinear equation with quadratic error is constructed.
Riccati equation plays an important role in the theory of optimal control, physics and many others applications. It should be emphasized that such equations are very often used in the games theory and calculus of variations. It should be noted here that in general, many papers are devoted to obtaining the conditions of the solvability in the regular case. We noted such papers as [
There are many papers where the matrix Riccati equations and operator-differential Riccati equations were investigated. As a rule, such equations were investigated in the regular case where the given problem had a unique solution. In the nonregular case, such equation was investigated (in the periodic case) in the work of Boichuk O. A. and Krivosheya S. A. [
In the present paper, using the technique of generalized inverse operators, we derive a criterion for the solvability of the given problem, generating problem and analyzing the structure of the solution set. We construct the iterative process for finding the solutions of weakly nonlinear problem which is the modification of Newton method and converges with quadratic error.
The article consists of three parts.
The first part of the given paper is devoted to the statement of the problem and denotations.
The second part is devoted to obtaining the necessary and sufficient conditions of the existence of bounded solutions of generating boundary value problem.
The last part is devoted to obtaining the necessary and sufficient conditions of the existence of solutions of weakly nonlinear Riccati operator-differential equation.
In this paper, Riccati equation was investigated in the critical case in the Hilbert space. And we obtain the full theorem of the solvability in linear case.
We consider the following boundary value problem
where
Consider the case when the differential equation is defined on the finite interval
where
respectively. Obviously, that
Using the operator
in the form
where M is an arbitrary operator and
The main statement of this part is the following theorem.
Theorem 1. Consider the boundary value problem (8), (9).
1) There are exist solutions of the boundary value problem (8), (9) if and only if the following condition is true
where
is generalized Green’s operator and
2) There are exist generalized solutions of boundary value problem (8), (9) if and only if
where
where
is generalized Green’s operator.
3) There are exist the quasisolutions if and only if
with the same generalized invertible.
Sketch of the proof: Substituting in the boundary condition we will have
and then we obtain the following operator equation
Using the notion of generalized invertible operator [
1) If
If the condition (16) is satisfied then the set of the solutions of the Equation (15) has the following form:
for any linear and bounded operator C. Then the family of solutions of the boundary value problem (8), (9) has the form
where
is generalized Green’s operator.
2) If
if and only if the following relation is hold
where
where
is generalized Green’s operator.
3) If the
with the same generalized invertible (see also the paper [
Now we consider the boundary value problem (1), (2). We find the solution
Theorem 2. (necessary condition) Let the boundary value problem (1), (2) has the solution
Proof. Suppose that the boundary value problem (1), (2) has solution
where
Such as condition (10) is true, then the condition of solvability (21) we can rewrite in the following form
Dividing by
or in the form
From this condition we obtain the theorem 2.
Now we obtain the sufficient condition of the solvability of boundary value problem (8), (9). We make the change of the variable
The family of solutions of the Equation (23) has the following form
under condition (21)
Substituting in this expression (25) and using the Equation (20), we have
Then we can rewrite this expression in the following form of the operator equation
where
and
If the following condition
is true then the equation (30) has the solution
Under condition (32), we can prove that boundary value problem (23), (24) have solutions. In a such way, we prove the following theorem.
Theorem 3. (sufficient condition) Under condition (32) boundary value problem (23), (24) is solvable. Solution of the given boundary problem can be found with using the following converging iterative process
with zero initial data.
Proof.
Proof of this theorem uses the modification of the fixed point theorem and is performed as well as the proof of the theorem 3 from the paper [
Example 1.
Considering the following boundary value problem with the matrix-valued in l2 functions,
nonhomogenous part has the following form
and conditions on infinity
In this case
Operator
with
In this case, the operator
From the condition (39) we obtain that
In the such way, we have
(46)
The unperturbed problem has the following form
Consider the following problem with the matrix,
nonhomogenous part has the following form
In this case
where
where
Here are
O. O.Pokutnyi, (2015) Boundary Value Problem for an Operator-Differential Riccati Equation in the Hilbert Space on the Interval. Advances in Pure Mathematics,05,865-873. doi: 10.4236/apm.2015.514081