NEW REMARK ON THE PROPERTIES OF STABLE INTERNAL CONTROL SYSTEMS OF A CERTAIN ORDER OF LINEAR DIFFERENTIAL EQUATIONS

Main Article Content

EBIENDELE EBOSELE PETER
AGWELI ALIU MOMOH

Abstract

The objective of this paper is to investigate and give sufficient and necessary conditions that guarantee the properties of stability of internal control systems for a certain order of Linear Differential Equations. This corresponds to globally exponentially stable equilibrium (point or position) of a system with input and output properties. We state and prove the classical Liapunov Theorem which allows us to reduce the stability analysis to an algebraic problem (computation of the eigenvalues of the matrix). We also introduce the quadratic Lyapunov functions and Lyapunov matrix equations that enhances the results of the defined objective. The Routh – Hurwitz criterion was given without proof. Since the results obtained in this study is different from the results and approach’s obtained in the literature, which implies that the results of this study are essentially new.

Keywords:
Properties of stable internal control, system, Lyapunov functions, linear differential equations

Article Details

How to Cite
PETER, E. E., & MOMOH, A. A. (2020). NEW REMARK ON THE PROPERTIES OF STABLE INTERNAL CONTROL SYSTEMS OF A CERTAIN ORDER OF LINEAR DIFFERENTIAL EQUATIONS. Journal of Basic and Applied Research International, 26(6), 1-7. Retrieved from https://ikprress.org/index.php/JOBARI/article/view/5251
Section
Original Research Article

References

Bacciotti A, Rosier L. Liapunov functions and Stability in control theory. Springer, Berlin; 2005.

Bellen A, Sennaro M. Numerical methods for delay Differential equations. Oxford University Press; 2013.

Bellman R, Cooke K. Differential – Difference equations. Academic Press, New York; 1963.

Breda D, Maset S, Vermiglio R. Stability of linear differential equations, Anumerical Approach with MATLAB. Springer Briefs in Electrical and Computer Engineering; 2015.

Bernstein DS. Matrix mathematics: Theory facts and formulas with application to Linear system theory. Princeton University Press, Princeton; 2005.

Brockett R. The early days of geometric nonlinear control. Automatic. 2014;50(9): 2203–2224.

Bravyi EL. Solvability of the Cauchy problem for higher- order linear functional Differential Equations. 2012;48(4):465–476.

Brockett RW. Finite dimensional linear system. Wiley, New York; 1970.

Conti R. Linear differential equations and control. Academic Press, London; 1976.

Ebiendele EP, Agweli AM. On the bounded and unique solvability of the boundary value problem of the equation in the space of scalar functions with absolute continuous derivative of the (N-1) order and its isotonic green operator for a certain class of linear functional differential equations Asian Journal of Mathematics and Computer Research. 2020;26(4):251–263.

Ebiendele P. New criterion that guarantee sufficient conditions for globally asymptotically stable periodic solutions of nonlinear differential equations with delay. Journal of Advances in mathematics and computer sciences. 2019;31(5):1 – 10.

[Article no. JAMCS 9171]

Ebiendele EP, Nosakhare UF. On the stability and unique state T-periodic solution for regular perturbation system for certain class of ordinary differential equations. Journal of Basic and Applied Research International. 2019;25(1):53–61.

Ebiendele EP, Nosakhare UF. On the uniquely solvability of Cauchy problem and dependences of parameters for a certain class of linear functional differential equations; 2018.

Ebiendele PE. On the Stable and Unstable state of a certain class of delay Differential Equations. Archives of Applied Science Research. 2017;9(3):35-40.

Ebiendele EP, Asuelinmen O. On uniform boundedness and stability of solutions for Predator – Prey model for class of Delay Nonlinear Differential Equations. Asian Journal of mathematics and computer Research. 2018;25(4):238–248.

Gyo ̈ri I, Hartung F. On numerical approximation using Differential Equations with Piecewise – constant arguments. Periodica mathematica Hungarica. 2008;56(1):55-69.

Terrel WJ. Stability and Stabilization: An introduction. Princeton press, Princeton; 2009