Representation of solutions for linear fractional systems with pure delay and multiple delays

2021 ◽  
Author(s):  
Ahmed M. Elshenhab ◽  
Xing Tao Wang
Keyword(s):  
Multiple Delays ◽  
Fractional Systems ◽  
Representation Of Solutions ◽  
Pure Delay

scholarly journals REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS AND NONPERMUTABLE VARIABLE COEFFICIENTS

2020 ◽  
Vol 25 (2) ◽  
pp. 303-322
Author(s):  
Michal Pospíšil
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Multiple Delays ◽  
Matrix Coefficients ◽  
Representation Of Solutions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Delay Differential

Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications.Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.

scholarly journals Controllability criteria for time–delay fractional systems with a retarded state

2016 ◽  
Vol 26 (3) ◽  
pp. 521-531 ◽  
Author(s):  
Beata Sikora
Keyword(s):  
Time Delay ◽  
Sufficient Conditions ◽  
The State ◽  
Multiple Delays ◽  
Fractional Systems ◽  
Numerical Examples ◽  
Relative Controllability ◽  
The Laplace Transform ◽  
Necessary And Sufficient ◽  
Theoretical Results

Abstract The paper is concerned with time-delay linear fractional systems with multiple delays in the state. A formula for the solution of the discussed systems is presented and derived using the Laplace transform. Definitions of relative controllability with and without constraints for linear fractional systems with delays in the state are formulated. Relative controllability, both with and without constraints imposed on control values, is discussed. Various types of necessary and sufficient conditions for relative controllability and relative U-controllability are established and proved. Numerical examples illustrate the obtained theoretical results.

scholarly journals Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 71 ◽  
Author(s):  
Olga Tsekhan
Keyword(s):  
Sufficient Conditions ◽  
Singularly Perturbed ◽  
Degenerate System ◽  
Multiple Delays ◽  
Complete Controllability ◽  
Singularly Perturbed System ◽  
System With Delay ◽  
Perturbed System ◽  
Time Invariant ◽  
Controllability Conditions

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

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