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

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 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.2010 Mathematics Subject Classification. 30E05, 35L05, 35R11, 45K05, 93B05, 93B60, 93C05.

Control and numerical approximation of fractional diffusion equations

“…Nevertheless, in recent years, several results have been obtained on the controllability properties of hyperbolic (wave) and dispersive (Schrödinger) models involving the fractional Laplacian. The interested reader may refer, for instance, to [12,18,92]. As for the numerical approximation of wave-type models, this issue is known to be quite delicate.…”

“…In recent years, this class of models has got the attention of the control community (see for instance [17,33,66] or [18] for models involving the fractional Laplacian). In particular, it has been observed that these models can be cast as coupled PDE-ODE systems, in which the ODE component introduces non-propagation effects similar to those produced by a low-order (s ≤ 1/2) fractional Laplacian.…”

“…In addition to that, control problems in the non-local setting have been largely considered in recent years. An incomplete bibliography at this respect includes [5,12,14,15,18,21,33,43,66,69,70,91,92,93].…”

Multiplicity of solutions for fractional $q(.)$-Laplacian equations

“…The research on fractional Laplace operators and their applications is very attractive and extended. In the last decade, many authors from different fields of the pure and applied mathematics have considered PDE models involving the fractional Laplacian and addressed many relevant questions such as existence, uniqueness and regularity of solutions [5,10,13,14,15,17,31,32,34,39,40,41,47,48,49,50], spectral properties [22,23,28], or even more applied issues, for example control problems [8,9,11,12,42,43,44,45] or the description of several phenomena arising in finance and quantum mechanics [4,14,27].…”

Internal control for a non-local Schrödinger equation involving the fractional Laplace operator

“…Concerning wave-like models, the approximate controllability from the exterior of fractional wave equations has been proved in [35,57], while [6] treats the null-controllability of a one-dimensional fractional wave equation with memory. Nonetheless, as far as the author knows, there are currently no controllability results for the fractional Schrödinger equation.…”