**About the Editor**

**Dumitru Baleanu** is a professor at the Institute of Space Sciences, Magurele-Bucharest, Romania, and a visiting staff member at the Department of Mathematics, Cankaya University, Ankara, Turkey. Dumitru Baleanu go<sup>t</sup> his Ph.D. from the Institute of Atomic Physics in 1996. His fields of interest include fractional dynamics and its applications, fractional differential equations and their applications, discrete mathematics, image processing, bio-informatics, mathematical biology, mathematical physics, soliton theory, Lie symmetry, dynamic systems on time scales, computational complexity, the wavelet method and its applications, quantization of systems with constraints, the Hamilton–Jacobi formalism, and geometries admitting generic and non-generic symmetries. Dumitru Baleanu is co-author of 15 books published by Springer, Elsevier, and World Scientific. His H index is 60, and he is a highly cited researcher in Mathematics and Engineering. Dumitru Baleanu won the 2019 Obada Prize. This prize recognizes and encourages innovative and interdisciplinary research that cuts across traditional boundaries and paradigms. It aims to foster universal values of excellence, creativity, justice, democracy, and progress, and to promote the scientific, technological, and humanistic achievements that advance and improve our world.

In addition, together with G.C. Wu, L.G. Zeng, X.C. Shi, and F. Wu, Dumitru Baleanu is the coauthor of Chinese Patent No ZL 2014 1 0033835.7 regarding chaotic maps and their important role in information encryption.

## **Preface to "Advances in Differential and Difference Equations with Applications 2020"**

Differential and difference equations are extreme representations of complex dynamical systems. During the last few decades, the theory of fractional differentiation has been successfully applied to the study of anomalous social and physical behaviors, where scaling power law of fractional order appears universal as an empirical description of such complex phenomena. Recently, the difference counterpart of fractional calculus has started to be intensively used for a better characterization of some real-world phenomena. Systems of delay differential equations have started to occupy a place of central importance in various areas of science, particularly in biological areas.

This book presents some19 original results regarding the theory and application of differential and difference equations which can be successfully used in dealing with real-world problems in various branches of science and engineering.

> **Dumitru Baleanu** *Editor*

## **On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback**

#### **Vasilii Zaitsev \*,† and Marina Zhuravleva †**

Laboratory of Mathematical Control Theory, Udmurt State University, Izhevsk 426034, Russia; mrnzo@yandex.ru

**\*** Correspondence: verba@udm.ru

† These authors contributed equally to this work.

Received: 18 May 2020; Accepted: 14 June 2020; Published: 17 June 2020

**Abstract:** We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

**Keywords:** linear control system; Hilbert space; state feedback control; exact controllability; upper Bohl exponent

**MSC:** 34D08; 34A35; 93C05
