On the null controllability of heat equation with memory

Null Controllability of Linear and Semilinear Nonlocal Heat Equations with an Additive Integral Kernel

“…Finally, observe that hypothesis (H) implies that the kernel K has to vanish exponentially as t goes to 0 + and to T − . Nevertheless, following the classical approach of [5] (see also [17]), it is possible to remove the decay assumption at t = 0, but this at the price of losing any information on the controllability cost. In fact, in this case we shall argue by a fixed point procedure, implying that we do not have a constructive method to build the control.…”

“…It combines energy estimates and the fact that β ≤ α in Q (see, e.g., [5]). Furthermore, using (3.22) and the classical approach presented in several works ( [5,8,9,17]), it is possible to obtain the following result. Proposition 3.2.…”

Controllability Properties for Equations with Memory of Fractional Type

“…Approximate controllability for a heat equation perturbed by a memory term t 0 H(t− s)u(s) ds when the relaxation kernel has compact support is proved via Carleman inequalities in [14]. A positive result on controllability to the target zero for this class of systems when H(t) has very special properties is proved in [22] via Carleman estimates. Controllability of systems with fractional derivatives (in time) have been advocated in particular by M. Yamamoto, and we cite the paper [5] where approximate controllability is proved for the system (1) with boundary controls, in the case that N (t) is the Dirac delta and K(t) = (1/Γ(1 − α))t −α (see also [12,24]).…”

Lack of null controllability of viscoelastic flows

“…[10][11][12][13][14]; for hyperbolic PDEs with memory controllability is achieved by perturbation of the elastic case. Parabolic problems appear to be more subtle, since the incorporation of memory effects into Carleman inequalities is a nontrivial problem; we refer to [8,24] for recent work showing both negative and positive results under certain situations. However, in a viscoelastic fluid the future evolution is not controlled to zero simply by controlling the velocity, and the focus of this paper is control of the full state of the system.…”