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Dynamical distributed controller for the synchronization problem of integer and fractional order partial differential equation systems

In this work we present a methodology to design dynamical distributed controllers for the synchronization problem of systems governed by partial differential equations of integer and fractional order. We consider fractional systems whose space derivatives are of integer order and time derivatives are of fractional commensurate order. The methodology is based on finding canonical forms by means of a change of variable, such that a distributed controller can be designed in a natural way in the form of a chain of integrators. To study the stability of the integer order closed loop system, we propose to use the spectral and semi-group theory for infinite dimensional Hilbert spaces. On the other hand, the stability criterion previously established by Matignon is used for the stability analysis of the fractional order case. Additionally, we tackle the synchronization problem of multiple systems, which is reduced to a problem of generalized multi-synchronization. To illustrate the effectiveness of the proposed methodology, examples and numerical results are given.