1,459 research outputs found
Exponential decay properties of a mathematical model for a certain fluid-structure interaction
In this work, we derive a result of exponential stability for a coupled
system of partial differential equations (PDEs) which governs a certain
fluid-structure interaction. In particular, a three-dimensional Stokes flow
interacts across a boundary interface with a two-dimensional mechanical plate
equation. In the case that the PDE plate component is rotational inertia-free,
one will have that solutions of this fluid-structure PDE system exhibit an
exponential rate of decay. By way of proving this decay, an estimate is
obtained for the resolvent of the associated semigroup generator, an estimate
which is uniform for frequency domain values along the imaginary axis.
Subsequently, we proceed to discuss relevant point control and boundary control
scenarios for this fluid-structure PDE model, with an ultimate view to optimal
control studies on both finite and infinite horizon. (Because of said
exponential stability result, optimal control of the PDE on time interval
becomes a reasonable problem for contemplation.)Comment: 15 pages, 1 figure; submitte
Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound
We consider an initial-boundary-value problem for a thermoelastic Kirchhoff &
Love plate, thermally insulated and simply supported on the boundary,
incorporating rotational inertia and a quasilinear hypoelastic response, while
the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law
giving rise to a 'second sound' effect. We study the local well-posedness of
the resulting quasilinear mixed-order hyperbolic system in a suitable solution
class of smooth functions mapping into Sobolev -spaces. Exploiting the
sole source of energy dissipation entering the system through the hyperbolic
heat flux moment, provided the initial data are small in a lower topology
(basic energy level corresponding to weak solutions), we prove a nonlinear
stabilizability estimate furnishing global existence & uniqueness and
exponential decay of classical solutions.Comment: 46 page
On Stability of Hyperbolic Thermoelastic Reissner-Mindlin-Timoshenko Plates
In the present article, we consider a thermoelastic plate of
Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising
from Cattaneo's law. In the absense of any additional mechanical dissipations,
the system is often not even strongly stable unless restricted to the
rotationally symmetric case, etc. We present a well-posedness result for the
linear problem under general mixed boundary conditions for the elastic and
thermal parts. For the case of a clamped, thermally isolated plate, we show an
exponential energy decay rate under a full damping for all elastic variables.
Restricting the problem to the rotationally symmetric case, we further prove
that a single frictional damping merely for the bending compoment is sufficient
for exponential stability. To this end, we construct a Lyapunov functional
incorporating the Bogovski\u{i} operator for irrotational vector fields which
we discuss in the appendix.Comment: 27 page
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