# On the lack of controllability of fractional in time ODE and PDE

## Abstract

This paper is devoted to the analysis of the problem of controllability of fractional (in time) ordinary and partial differential equations (ODE/PDE). The fractional time derivative introduces some memory effects on the system that need to be taken into account when defining the notion of control. In fact, in contrast with the classical ODE and PDE control theory, when driving these systems to rest, one is required not only to control the value of the state at the final time, but also the memory accumulated by the long-tail effects that the fractional derivative introduces. As a consequence, the notion of null controllability to equilibrium needs to take into account both the state and the memory term. The existing literature so far is only concerned with the problem of partial controllability in which the state is controlled, but the behavior of the memory term is ignored. In the present paper, we consider the full controllability problem and show that, due to the memory effects, even at the ODE level, controllability cannot be achieved in finite time. This negative result holds even for finite-dimensional systems in which the control is of full dimension. Consequently, the same negative results hold also for fractional PDE, regardless of their parabolic or hyperbolic nature. This negative result exhibits a completely opposite behavior with respect to the existing literature on classical ODE and PDE control where sharp sufficient conditions for null controllability are well known.

## Keywords

Fractional in time ODE PDE Partial controllability Null controllability Observability## Notes

### Acknowledgments

This work is supported by the Advanced Grant FP7-246775 NUMERIWAVES of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the BERC 2014-2017 program of the Basque Government, the MTM2011-29306-C02-00 and SEV-2013-0323 Grants of the MINECO and a Humboldt Award at the University of Erlangen-Nuremberg, and the NSF of China under Grant 11471231, and the Fundamental Research Funds for the Central Universities in China under Grant 2015SCU04A02. This work was initiated while the authors were visiting the CIMI-Toulouse in the context of the activities of the Excellence Chair on ”PDE, Control and Numerics”. The authors acknowledge the CIMI for the hospitality and support and D. Matignon for fruitful discussions.

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