Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems

We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterwards, we formulate the notion of regional controllability for fractional systems with control delays and give some of their important properties. Our main method consists in defining an attainable set, which allow us to prove exact and weak controllability. Moreover, main results include not only those of controllability but also a powerful Hilbert uniqueness method that allow us to solve the minimum energy optimal control control problem. Precisely, an explicit control is obtained that drives the system from an initial given state to a desired regional state with minimum energy. Examples are given to illustrate the obtained theoretical results.