Invertibility of an operator arising in the theory of the control of linear systems.
Sufficient conditions for the existence of a bounded inverse operator for a linear operator in Hilbert space having a matrix representation of the form \[ F = \left( \begin{matrix} F_1 & 0 & F_2 \\ F_3 & -F_1^* & F_5 \\ -F_5^* & F_2^* & F_4 \end{matrix}\right) \] are obtained, where \(F_3, F_4 \) are nonnegative self-adjoint operators. It is shown that such operators appear, in particular, in the study of problems of optimal control of linear systems unresolved for the derivatives. The conditions obtained are not necessary for the invertibility of an arbitrary operator \(F\) of such form. A simple operator confirming this proposition is given. The invertibility conditions are used to prove the unique solvability of a certain two-point boundary-value problem that arises from conditions for optimal control.
scientific article; zbMATH DE number 1927743