**Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback**

#### **Vasilii Zaitsev \*,† and Inna Kim †**

Laboratory of Mathematical Control Theory, Udmurt State University, 426034 Izhevsk, Russia; kimingeral@gmail.com

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

† These authors contributed equally to this work.

Received: 6 May 2020; Accepted: 22 May 2020; Published: 24 May 2020

**Abstract:** We consider a control system defined by a linear time-varying differential equation of *n*-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefficients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin's theorem on sufficient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufficient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefficients by linear time-invariant static output feedback. Illustrative examples are considered.

**Keywords:** linear differential equation; exponential stability; linear output feedback; stabilization; uncertain system

**MSC:** 34D20; 93C05; 93D15; 93D23
