[00520] Controllability of Generalized Fractional Dynamical Systems

Session Time & Room : 5B (Aug.25, 10:40-12:20) @A207

Type : Contributed Talk

Abstract : In this paper necessary and sufficient conditions are established for the controllability of
linear fractional dynamical system of the form
\begin{eqnarray}
^CD^{\alpha,\rho}_{0^+}x(t)&=& Ax(t)+Bu(t), \ \ t\in J=[0,T]\\
x(0)&=&x_0
\end{eqnarray}
where $0<\alpha<1,\rho>0, \rho\neq 1$ and $x\in R^n$ is the state vector, $u\in R^m$ is the control vector, $x_0\in R^n$
and $A$ is an $n\times n$ matrix and $B$ is an $n\times m$ matrix.
Here the generalized fractional derivative is taken as
\begin{eqnarray*}
^CD^{\alpha,\rho}_{0^+}x(t)=\frac{\rho^{\alpha}}{\Gamma(1-\alpha)}
\int_0^t \frac{1}{(t^{\rho}-s^{\rho})^{\alpha}}x^{\prime}(s)ds
\end{eqnarray*}
Further sufficient conditions are obtained for the following nonlinear fractional system
\begin{eqnarray}
^CD^{\alpha,\rho}_{0^+}x(t)&=& Ax(t)+Bu(t)+f(t,x(t)), \\
x(0)&=&x_0
\end{eqnarray}
where the function $f:J\times R^n\to R^n$ is continuous.
The results for linear systems are obtained by using the Mittag-Leffler function and the Grammian matrix. Controllability of
nonlinear fractional system is established by means of Schauder's fixed point theorem. Examples are provided to illustrate the results.