Boundary Value Problem for an Operator-Differential Riccati Equation in the Hilbert Space on the Interval

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 ε. 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.


Introduction
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 [1]- [12] where considering equation was investigated in the finitedimensional and in the infinite-dimensional cases.It should be noted that this equation was investigated as in the operator and matrix form or as in the operator-differential and differential form (see the bibliography).
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. [12].In the paper [4], the discrete Riccati equation was investigated.In the paper of Pronkin [8], the question about quasiperiodic solutions of the matrix Riccati equation with coefficients which are Arnold functions is investigated.Dissertation of Christian Wyss [1] is also devoted to the perturbation theory for Hamiltonian operator matrices and Riccati equations in the Hilbert space.
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.

Statement of the Problem
We consider the following boundary value problem where ( ) ,  H ; or in another words these operators are the ways in the space of linear and bounded operators ( ) parameter.We find the solution ( )

Solutions on the Finite Interval
Consider the case when the differential equation is defined on the finite interval [ ] 0;T or on the infinite inter- val.Let us the operator t K τ whose action on the operator-valued function where ( ) U t , ( ) V t are evolution operators [13], of the following operator-differential equations respectively.Obviously, that ( ) Using the operator t K τ , we write out the general solution of the nonperturbed problem ( ) 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 under this condition the family of solutions have the following form 2) There are exist generalized solutions of boundary value problem ( 8), ( 9) if and only if where Q + is strong generalized invertible operator.Then the family of solutions of the Equation ( 8) has the form where 3) There are exist the quasisolutions if and only if and in this case we have the solutions in the following form 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 [14] [15], we have the following variants for the equation: 15) has the solution if and only if the following condition is true 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 we have the generalized solutions in the following form if and only if the following relation is hold where Q + is strong generalized invertible operator [15].Then the family of solutions of the boundary value problems ( 8), ( 9) has the form where we have the quasisolutions in the form ( with the same generalized invertible (see also the paper [16]).

Necessary Condition
Now we consider the boundary value problem (1), (2).We find the solution ( ) where Such as condition (10) is true, then the condition of solvability (21) we can rewrite in the following form Dividing by ε and passing to the limit when ε tends to zero we obtain From this condition we obtain the theorem 2.

Sufficient Condition of the Solvability
Now we obtain the sufficient condition of the solvability of boundary value problem ( 8), (9).We make the change of the variable ( ) Then we obtain the following boundary value problem The family of solutions of the Equation ( 23) has the following form 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 ( ) ( )  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 [17].
Considering the following boundary value problem with the matrix-valued in l 2 functions, 0 1 0 0 0 0 0 nonhomogenous part has the following form ( ) and conditions on infinity In this case Operator × *-star matrix which has the following form In this case, the operator From the condition (39) we obtain that The unperturbed problem has the following form Consider the following problem with the matrix, ( ) nonhomogenous part has the following form In this case

Z t ε
of the boundary value problem (1), (2) which for 0 ε = turns in one of the solutions ( ) 0 Z t of boundary value problem (8), (9).Now we obtain the necessary condition of the existence of such solutions.Theorem 2. (necessary condition) Let the boundary value problem (1), (2) has the solution ( ) , Z t ε which for 0 ε = turns in one of the solution Then the operator 0 C satisfies the following operator equation for generating operators 20)Proof.Suppose that the boundary value problem (1), (2) has solution ( ) with C 0 .By virtue of the theorem 1 thefollowing condition of solvability is true 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), ( the block of the Fibonacci matrix B.