Abstract
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 \(\left( \begin{gathered} F_1 {\text{ 0 }}F_2 \hfill \\ F_3 {\text{ }}--F_1^* {\text{ }}F_5 \hfill \\ - F_5^* {\text{ }}F_2^* {\text{ }} - F_4 \hfill \\ \end{gathered} \right)\), 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.
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Kurina, G.A. Invertibility of an Operator Appearing in the Control Theory for Linear Systems. Mathematical Notes 70, 206–212 (2001). https://doi.org/10.1023/A:1010254808822
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DOI: https://doi.org/10.1023/A:1010254808822