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

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. Preliminary Results

**Theorem**

**2.**

**Proof.**

## 3. Time-Invariant Stabilization by Static State Feedback

**Definition**

**1.**

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

## 4. Time-Invariant Stabilization by Static Output Feedback

**Definition**

**2.**

**Theorem**

**4.**

**Theorem**

**5.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Aeyels, D.; Peuteman, J. Uniform asymptotic stability of linear time-varying systems. In Open Problems in Mathematical Systems and Control Theory; Blondel, V., Sontag, E.D., Vidyasagar, M., Willems, J.C., Eds.; Springer: London, UK, 1999; pp. 1–5. [Google Scholar] [CrossRef]
- Ilchmann, A.; Owens, D.H.; Prätzel-Wolters, D. Sufficient conditions for stability of linear time-varying systems. Syst. Control Lett.
**1987**, 9, 157–163. [Google Scholar] [CrossRef][Green Version] - Bylov, B.F.; Vinograd, R.E.; Grobman, D.M.; Nemytskii, V.V. Theory of Lyapunov Exponents; Nauka: Moscow, Russia, 1966. [Google Scholar]
- Demidovich, B.P. Lectures on the Mathematical Stability Theory; Nauka: Moscow, Russia, 1967. [Google Scholar]
- Zhu, J.J. A necessary and sufficient stability criterion for linear time-varying systems. In Proceedings of the 28th Southeastern Symposium on System Theory, Baton Rouge, Louisiana, USA, 31 March–2 April 1996; pp. 115–119. [Google Scholar] [CrossRef]
- Levin, A.Y. Absolute nonoscillatory stability and related questions. St. Petersburg Math. J.
**1993**, 4, 149–161. [Google Scholar] - Ragusa, M.A. Necessary and sufficient condition for a VMO function. Appl. Math. Comput.
**2012**, 218, 11952–11958. [Google Scholar] [CrossRef] - Zhou, B. On asymptotic stability of linear time-varying systems. Automatica
**2016**, 68, 266–276. [Google Scholar] [CrossRef] - Wan, J.-M. Explicit solution and stability of linear time-varying differential state space systems. Int. J. Control Autom. Syst.
**2017**, 15, 1553–1560. [Google Scholar] [CrossRef] - Vrabel, R. A note on uniform exponential stability of linear periodic time-varying systems. IEEE Trans. Autom. Control
**2020**, 65, 1647–1651. [Google Scholar] [CrossRef][Green Version] - Zhou, B.; Tian, Y.; Lam, J. On construction of Lyapunov functions for scalar linear time-varying systems. Syst. Control Lett.
**2020**, 135, 104591. [Google Scholar] [CrossRef] - Kharitonov, V.L. The asymptotic stability of the equilibrium state of a family of systems of linear differential equations. Differ. Uravn.
**1978**, 14, 2086–2088. [Google Scholar] - Petersen, I.R. A stabilization algorithm for a class of uncertain linear systems. Syst. Control Lett.
**1987**, 8, 351–357. [Google Scholar] [CrossRef] - Zhou, K.; Khargonekar, P.P. Robust stabilization of linear systems with norm-bounded time-varying uncertainty. Syst. Control Lett.
**1988**, 10, 17–20. [Google Scholar] [CrossRef] - Khargonekar, P.P.; Petersen, I.R.; Zhou, K. Robust stabilization of uncertain linear systems: Quadratic stabilizability and H
^{∞}control theory. IEEE Trans. Autom. Control**1990**, 35, 356–361. [Google Scholar] [CrossRef] - Xie, L.; de Souza, C.E. Robust H
_{∞}control for linear systems with norm-bounded time-varying uncertainty. IEEE Trans. Autom. Control**1992**, 37, 1188–1191. [Google Scholar] [CrossRef] - Zhabko, A.P.; Kharitonov, V.L. Necessary and sufficient conditions for the stability of a linear family of polynomials. Autom. Remote Control
**1994**, 55, 1496–1503. [Google Scholar] - Kharitonov, V.L. Robust stability analysis of time delay systems: A survey. Annu. Rev. Control
**1999**, 23, 185–196. [Google Scholar] [CrossRef] - Sadabadi, M.S.; Peaucelle, D. From static output feedback to structured robust static output feedback: A survey. Annu. Rev. Control
**2016**, 42, 11–26. [Google Scholar] [CrossRef][Green Version] - Blanchini, F.; Colaneri, P. Uncertain systems: Time-varying versus time-invariant uncertainties. In Uncertainty in Complex Networked Systems. Systems and Control: Foundations and Applications; Başar, T., Ed.; Birkhäuser: Cham, Switzerland, 2018. [Google Scholar] [CrossRef]
- Carniato, L.A.; Carniato, A.A.; Teixeira, M.C.M.; Cardim, R.; Mainardi Junior, E.I.; Assunção, E. Output control of continuous-time uncertain switched linear systems via switched static output feedback. Int. J. Control
**2018**, 93, 1127–1146. [Google Scholar] [CrossRef][Green Version] - Gu, D.-K.; Liu, G.-P.; Duan, G.-R. Robust stability of uncertain second-order linear time-varying systems. J. Frankl. Instit.
**2019**, 356, 9881–9906. [Google Scholar] [CrossRef] - Barmish, B.R. Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. J. Optim. Theory Appl.
**1985**, 46, 399–408. [Google Scholar] [CrossRef] - Xie, L.; Shishkin, S.; Fu, M. Piecewise Lyapunov functions for robust stability of linear time-varying systems. Syst. Control Lett.
**1997**, 31, 165–171. [Google Scholar] [CrossRef] - Ramos, D.C.W.; Peres, P.L.D. An LMI approach to compute robust stability domains for uncertain linear systems. In Proceedings of the 2001 American Control Conference, Arlington, VA, USA, 25–27 June 2001. [Google Scholar] [CrossRef]
- Montagner, V.F.; Peres, P.L.D. A new LMI condition for the robust stability of linear time varying systems. In Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, USA, 9–12 December 2003. [Google Scholar] [CrossRef]
- Bliman, P.A. A convex approach to robust stability for linear systems with uncertain scalar parameters. SIAM J. Control Optim.
**2004**, 42, 2016–2042. [Google Scholar] [CrossRef][Green Version] - Geromel, J.C.; Colaneri, P. Robust stability of time varying polytopic systems. Syst. Control Lett.
**2006**, 55, 81–85. [Google Scholar] [CrossRef] - Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A. Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions. Automatica
**2007**, 43, 309–316. [Google Scholar] [CrossRef] - Hu, T.; Blanchini, F. Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions. Automatica
**2010**, 46, 190–196. [Google Scholar] [CrossRef] - Chesi, G. Sufficient and necessary LMI conditions for robust stability of rationally time-varying uncertain systems. IEEE Trans. Autom. Control
**2013**, 58, 1546–1551. [Google Scholar] [CrossRef][Green Version] - Gritli, H.; Belghith, S. New LMI conditions for static output feedback control of continuous-time linear systems with parametric uncertainties. In Proceedings of the 2018 European Control Conference (ECC), Limassol, Cyprus, 12–15 June 2018. [Google Scholar] [CrossRef]
- Gritli, H.; Belghith, S.; Zemouche, A. LMI-based design of robust static output feedback controller for uncertain linear continuous systems. In Proceedings of the 2019 International Conference on Advanced Systems and Emergent Technologies (IC_ASET), Hammamet, Tunisia, 19–22 March 2019. [Google Scholar] [CrossRef]
- Gelig, A.H.; Zuber, I.E. Invariant stabilization of classes of uncertain systems with delays. Autom. Remote Control
**2011**, 72, 1941–1950. [Google Scholar] [CrossRef] - Zakharenkov, M.; Zuber, I.; Gelig, A. Stabilization of new classes of uncertain systems. IFAC-PapersOnLine
**2015**, 48, 1024–1027. [Google Scholar] [CrossRef] - Gelig, A.H.; Zuber, I.E.; Zakharenkov, M.S. New classes of stabilizable uncertain systems. Autom. Remote Control
**2016**, 77, 1768–1780. [Google Scholar] [CrossRef] - Gelig, A.K.; Zuber, I.E. Multidimensional output stabilization of a certain class of uncertain systems. Autom. Remote Control
**2018**, 79, 1545–1557. [Google Scholar] [CrossRef] - Zaitsev, V.A. Modal control of a linear differential equation with incomplete feedback. Differ. Equ.
**2003**, 39, 145–148. [Google Scholar] [CrossRef]

**Figure 1.**Graphs of the solutions to (63).

**Figure 2.**Graphs of the solutions to (84).

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**MDPI and ACS Style**

Zaitsev, V.; Kim, I. Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback. *Mathematics* **2020**, *8*, 853.
https://doi.org/10.3390/math8050853

**AMA Style**

Zaitsev V, Kim I. Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback. *Mathematics*. 2020; 8(5):853.
https://doi.org/10.3390/math8050853

**Chicago/Turabian Style**

Zaitsev, Vasilii, and Inna Kim. 2020. "Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback" *Mathematics* 8, no. 5: 853.
https://doi.org/10.3390/math8050853