Controllability of Fractional Linear Systems with Delays

The main purpose of this paper is to study controllability of linear continuous-time fractional dynamical systems containing both lumped constant delay in state variables and distributed delays in admissible controls. Necessary and sufficient conditions for relative controllability in finite time interval are formulated and proved using theory of linear bounded operators, solution properties of fractional differential equations and results taken directly from linear matrix algebra. The main result of the paper is to show, that global relative controllability of fractional linear systems with different types of delays is equivalent to non-singularity of suitably defined relative controllability matrix.