4 research outputs found
Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities
In this paper, by means of the Riesz basis approach, we study the stability
of a weakly damped system of two second order evolution equations coupled
through the velocities. If the fractional order damping becomes viscous and the
waves propagate with equal speeds, we prove exponential stability of the system
and, otherwise, we establish an optimal polynomial decay rate. Finally, we
provide some illustrative examples
New global Carleman estimates and null controllability for forward/backward semi-linear parabolic SPDEs
In this paper, we study the null controllability for some linear and
semi-linear parabolic SPDEs involving both the state and the gradient of the
state. To start with, an improved global Carleman estimate for linear forward
(resp. backward) parabolic SPDEs with general random coefficients and
-valued source terms is derived. Based on this, we further develop a new
global Carleman estimate for linear forward (resp. backward) parabolic SPDEs
with -valued source terms, which enables us to deal with the global
null controllability for linear backward (resp. forward) parabolic SPDEs with
gradient terms. As byproduct, a special energy-type estimate for the controlled
system that explicitly depends on the parameters and the weighted
function is obtained. Furthermore, by employing a fixed-point
argument, we extend the previous linear controllability results to some
semi-linear backward (resp. forward) parabolic SPDEs