CONTROLLABILITY OF FREDHOLM’S INTEGRO-DIFFERENTIAL EQUATIONS WITH BY A DEGENERATE KERNEL IN HILBERT SPACES

Abstract

The work examines integro-differential equations Fredholm with a degenerate kernel with Hilbert control spaces.  The need to study these equations is related to numerous ones applications of integro- differential equations in mathematics, physics, technology, economy and other fields. Complexity the study of integro-differential equations is connected with the fact that the integral-differential operator is not solvable everywhere. There are different approaches to the solution of not everywhere solvable linear operator equations: weak perturbation of the right-hand side of this equation with further application of the Vishyk-Lyusternyk method, introduction to system of impulse action, control, etc. The problem of obtaining coefficient conditions of solvability and analytical presentation of general solutions of integro-differential equations is a rather difficult problem, so frequent solutions will suffice are obtained by numerical methods. In this connection, Fredholm’s integro-differential equations with degenerate kernel and control in Hilbert spaces no were investigated. Therefore, the task of establishing conditions is urgent controllability, construction of general solutions in an analytical form and corresponding general controls of integro-differential equations with a degenerate kernel in abstract Hilbert spaces. As an intermediate result in the work using the results of pseudoinversion of integral operators in Hilbert spaces the solvability criterion and the form of general solutions are established integro-differential equations without control in the abstract Hilbert spaces. To establish the controllability criterion is not solvable everywhere integro-differential equations with Hilbert control spaces, the general theory of research is not applied everywhere solvable operator equations. At the same time, they are used significantly orthoprojectors, pseudo-inverse operators to normally solvable ones operators in Hilbert spaces. With the use of orthoprojectors, pseudo-inverse operators and pseudoinversion of integraloperators, a criterion is obtained solutions and the general form of solutions of integro-differential equations with a degenerate kernel with control y Hilbert spaces. An image of the general appearance is obtained control under which these solutions exist.