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

Lu Q, Zuazua E (2016)

Publication Type: Journal article

Publication year: 2016

Journal

Mathematics of Control Signals and Systems Springer Verlag (Germany)

Book Volume: 28

Article Number: 10

Journal Issue: 2

DOI: 10.1007/s00498-016-0162-9

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.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Sichuan University (SCU) / 四川大学 CN China (CN) Universidad Autónoma de Madrid (UAM) ES Spain (ES)

How to cite

APA:

Lu, Q., & Zuazua, E. (2016). On the lack of controllability of fractional in time ODE and PDE. Mathematics of Control Signals and Systems, 28(2). https://dx.doi.org/10.1007/s00498-016-0162-9

MLA:

Lu, Qi, and Enrique Zuazua. "On the lack of controllability of fractional in time ODE and PDE." Mathematics of Control Signals and Systems 28.2 (2016).

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