Null-controllability properties of a fractional wave equation with a memory term

Umberto Biccari; Mahamadi Warma

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arXiv · 1901.10194

Null-controllability properties of a fractional wave equation with a memory term

Umberto Biccari·Mahamadi Warma

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Abstract

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset $\omega(t)$ which is moving with a constant velocity $c\in\mathbb{R}$, the main result of the paper states that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.

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Umberto Biccari, Mahamadi Warma. 2019-01-29. Null-controllability properties of a fractional wave equation with a memory term. https://arxiv.org/abs/1901.10194

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