Null-controllability properties of a fractional wave equation with a memory term
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2020-06
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American Institute of Mathematical Sciences
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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 ω(t) which is moving with a constant velocity c ∈ 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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Biccari, U., & Warma, M. (2020). Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations and Control Theory, 9(2), 399-430. https://doi.org/10.3934/EECT.2020011