Invertibility of an Operator Appearing in the Control Theory for Linear Systems

We give sufficient conditions for the existence of a bounded inverse operator for a linear operator appearing in the theory of optimal control of linear systems in Hilbert space and having a matrix representation of the form
$$ \begin {pmatrix} F_1&0&F_2 \\F_3&-F_1^*&F_5 \\-F_5^*&F_2^*&-F_4 \end{pmatrix} , $$
, where $F3$, $F4$ are nonnegative self-adjoint operators. The invertibility of the operator under study is used to prove the unique solvability of a certain two-point boundary-value problem that arises from conditions for optimal control.