**5. Conclusions**

For a linear time-invariant control system in a Hilbert space with bounded operator coefficients, we examined the problem of arbitrary assignment of the upper Bohl exponent by means of linear state feedback with a time-varying linear bounded gain operator function. We have proved that the property of exact controllability of the open-loop system is sufficient for arbitrary assignability of the upper Bohl exponent of the closed-loop system. We plan to extend these results to systems without necessarily bounded operator *A* but generating a *C*0-continuous semigroup. We plan to prove similar results for systems with dynamic output feedback. Further development of these results may be their extension to systems with periodic coefficients and with arbitrary time-varying non-periodic

coefficients, to systems in general Banach spaces, or to systems with discrete time. We expect to apply the results to specific systems, for example, to systems with delays, considering them as abstract systems of differential equations in an infinite-dimensional space.

**Author Contributions:** Conceptualization, V.Z.; methodology, V.Z.; formal analysis, V.Z. and M.Z.; investigation, V.Z. and M.Z.; writing–original draft preparation, V.Z. and M.Z.; writing–review and editing, V.Z. and M.Z.; visualization, V.Z. and M.Z.; supervision, V.Z.; project administration, V.Z.; funding acquisition, V.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Ministry of Science and Higher Education of the Russian Federation in the framework of state assignment No. 075-00232-20-01, Project 0827-2020-0010 "Development of the theory and methods of control and stabilization of dynamical systems".

**Conflicts of Interest:** The authors declare no conflict of interest.
