Thermoelastic plate model with a control term in the thermal equation is considered. The main result in this paper is that with thermal control, locally distributed within the interior and square integrable in time and space, any finite energy solution can be driven to zero at the control time T.
IntroductionIn this paper, we investigate the null controllability of thermoelastic plates when the control (heat source) acts in the thermal equation. In general, these models consist of an elastic motion equation and a heat equation, which are coupled in such a way that the energy transfer between them is taken into account.The plate, we consider here, is derived in the light of [18]. Transverse shear effects are neglected (Euler-Bernoulli model), and the plate is hinged on its edge. In addition to internal and external heat source, the temperature dynamics are driven by internal frictional forces caused by the motion of the plate. The latter connection is expressed by the second law of thermodynamics for irreversible processes, which relates the entropy to the elastic strains. Accounting for thermal effects, we assume that the heat flux law involves only the temperature gradient by the Fourier law.Let Ω be a bounded, open, connected subset of R 2 , with a C ∞ boundary and ω any open subset of Ω. Let T > 0 and setWe consider a model which describes the small vibrations of a homogeneous, elastically and thermally isotropic Kirchhoff plate, under the influence of a control function f ∈ L 2 ((0,T) × ω). In absence of exterior forces, and with
SUMMARYWe show that the solution of a semilinear transmission problem between an elastic and a thermoelastic material, decays exponentially to zero. That is, denoting by E(t) the sum of the ÿrst, second and third order energy associated with the system, we show that there exist positive constants C and satisfyingMoreover, the existence of absorbing sets is achieved in the non-homogeneous case.
This work is focused on the dissipative system
$$
\begin{cases}
\partial_{tt}u+\partial_{xxxx}u
+\partial_{xx}\theta-\big(\beta+\|\partial_x u\|_{L^2(0,1)}^2\big)\partial_{xx}u=f\\
\noalign{\vskip.7mm}
\partial_{t} \theta -\partial_{xx}\theta -\partial_{xxt} u= g
\end{cases}
$$
describing the dynamics
of an extensible thermoelastic beam, where
the dissipation is entirely contributed
by the second equation ruling the evolution of $\theta$.
Under natural boundary conditions, we prove the existence
of bounded absorbing sets.
When the external sources $f$ and $g$ are time-independent,
the related semigroup of solutions is shown to possess the global
attractor of optimal regularity for all parameters $\beta\in\mathbb{R}$.
The same result holds true when the first equation
is replaced by
$$
\partial_{tt} u-\gamma\partial_{xxtt} u+\partial_{xxxx}u
+\partial_{xx}\theta-\big(\beta+\|\partial_x u\|_{L^2(0,1)}^2\big)\partial_{xx}u=f
$$
with $\gamma>0$. In both cases, the solutions on the attractor are strong solutions
We study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations. We show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution. In the case of the lack of exponential decay, we find the polynomial rate of decay of the solution. Some examples are given.
a b s t r a c tIn this work we formulate a nonlinear mathematical model for the thermoelastic beam assuming the Fourier heat conduction law. Boundary conditions for the temperature are imposed on the ending cross sections of the beam. A careful analysis of the resulting steady states is addressed and the dependence of the Euler buckling load on the beam mean temperature, besides the applied axial load, is also discussed. Finally, under some simplifying assumptions, we deduce the model for the bending of an extensible thermoelastic beam with fixed ends. The behavior of the resulting dissipative system accounts for both the elongation of the beam and the Fourier heat conduction. The nonlinear term enters the motion equation, only, while the dissipation is entirely contributed by the heat equation, ruling the thermal evolution.
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