Controllability of Fractional Linear Systems with Delays in Control

In the chapter linear, fractional, continuous time, finite-dimensional, dynamical control systems with multiple variable point delays and distributed delay in admissible control described by linear ordinary differential state equations are considered. Using notations, theorems and methods taken directly from functional analysis and linear controllability theory, necessary and sufficient conditions for global relative controllability in a given finite time interval are formulated and proved. The main result of the chapter is to show, that global relative controllability of fractional linear systems with different types of delays in admissible control is equivalent to non-singularity of a suitably defined relative controllability matrix. In the proofs of the main results, methods and concepts taken from the theory of linear bounded operators in Hilbert spaces are used.  Applying a relative controllability matrix for relative controllable systems steering admissible control is proposed, which steers the fractional system from a given initial complete state to the desired final relative state. Some remarks and comments on the existing controllability results for linear fractional dynamical system with delays are also presented.