Relative Exact Controllability of Fractional Linear Systems with Delays in Control [1]

In the paper linear, fractional, continuous time, infinite-dimensional, stationary dynamical control systems with multiple variable point delays in admissible control described by 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 exact controllability in a given finite time interval are formulated and proved. The main result of the paper is to show, that global relative exact controllability of fractional linear systems with different types of delays in admissible control is equivalent to invertibility of suitably defined relative exact controllability operator. In the proof of main result, methods taken from the theory of linear bounded operators in Hilbert spaces are used. Some remarks and comments on the existing controllability results for linear fractional dynamical system with delays are also presented.