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Analytical and numerical methods for the stability analysis of linear fractional delay differential equations. (English) Zbl 1250.65099

An asymptotic stability analysis of linear scalar delay-differential equations of fractional order $q\in \left(0,1\right)$ of type:

${}^{c}{D}^{q}y\left(t\right)=Ay\left(t\right)+By\left(t-\tau \right),\phantom{\rule{3.33333pt}{0ex}}t>0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

with the initial condition $y\left(t\right)=\phi \left(t\right)$, $t\in \left[-\tau ,0\right]$, where $\tau >0$ is a fixed time delay, $\phi \in {L}^{\infty }\left(\left[-\tau ,0\right],ℝ\right)$ is a given initial data and ${}^{c}{D}^{q}f$ is the Caputo fractional-order derivative of order $q$ defined by ${}^{c}{D}^{q}f\left(t\right)\equiv {\Gamma }{\left(1-q\right)}^{-1}\phantom{\rule{0.277778em}{0ex}}{\int }_{0}^{t}{\left(t-s\right)}^{-q}{f}^{\text{'}}\left(s\right)ds·$ After introducing the standard definition of asymptotic stability i.e., that the solution of (1) tends to zero as $t\to +\infty$ and a brief review of the well-known asymptotic stability results for $q\to 1$, the authors derive some necessary algebraic conditions on the parameters $A,B,\tau ,q$ for the asymptotic stability of the null solution. These results are obtained by combining analytical tools like Laplace transform with numerical approximation of integrals. Some stability regions in the space of parameter values are presented.

##### MSC:
 65L07 Numerical investigation of stability of solutions of ODE 65L03 Functional-differential equations (numerical methods) 34K20 Stability theory of functional-differential equations 34K28 Numerical approximation of solutions of functional-differential equations 65L20 Stability and convergence of numerical methods for ODE 34A08 Fractional differential equations 34K06 Linear functional-differential equations