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## [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.
• Classification : 93B05, 34A08, Controllability, Fractional Dynamical Systems
• Format : Talk at Waseda University
• Author(s) :
• Balachandran Krishnan (Department of Mathematics, Bharathiar University, Coimbatore-641046)