Thermoelastic Plate Equations Research Articles

Controllability of block diagonal systems and applications

In this paper we present necessary a sufficient conditions for the exact and approximate controllability of Block-Diagonal Control Systems representing a broad class of linear evolution equations in Hilbert spaces. We apply these results to the controlled nD wave equation, heat equation and thermoelastic plate equation.

Local Energy Decay Estimate of Solutions to the Thermoelastic Plate Equations in Two- and Three-Dimensional Exterior Domains

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations: u_{tt} + ∆^2u + ∆θ = 0 \text{ and } θ_t − ∆θ − ∆u_t = 0\quad \text{in } Ω × (0, ∞), subject to Dirichlet boundary conditions: u|_Γ = D_ν u|_Γ = θ|_Γ = 0 and initial conditions (u,u_t,θ)|_{t=0} = (u_0, v_0, θ_0) . Here Ω is an exterior domain (domain with bounded complement) in ℝ^n with n = 2 or n= 3 , the boundary Γ of which is assumed to be a C^4 -hypersurface.

On the L p analytic semigroup associated with the linear thermoelastic plate equations in the half-space

The paper is concerned with linear thermoelastic plate equations in the half-space R + n = { x = ( x 1 , … , x n ) ❘ x n > 0 } : u tt + Δ 2 u + Δ θ = 0 and θ t - Δ θ - Δ u t = 0 R + n × ( 0 , ∞ ) , subject to the boundary condition: u | x n = 0 = D n u | x n = 0 = θ | x n = 0 = 0 and initial condition: ( u , D t u , θ ) | t = 0 = ( u 0 , v 0 , θ 0 ) ∈ H p = W p , D 2 × L p × L p , where W p , D 2 = { u ∈ W p 2 ❘ u | x n = 0 = D n u | x n = 0 = 0 } . We show that for any p ∈ ( 1 , ∞ ) , the associated semigroup { T ( t ) } t ≥ 0 is analytic in the underlying space H p . Moreover, a solution ( u , θ ) satisfies the estimates: ∥ ∇ j ( ∇ 2 u ( · , t ) , u t ( · , t ) , θ ( · , t ) ) ∥ L q ( R + n ) ≤ C p,q t - j2 - n2 ( 1p - 1q ) ∥ ( ∇2 u0 , v0 , θ0 ) ∥ Lp ( R +n ) (t>0) for j = 0 , 1 , 2 provided that 1 < p ≤ q ≤ ∞ when j = 0 , 1 and that 1 < p ≤ q < ∞ when j = 2 , where ∇ j stands for space gradient of order j .

Initial-boundary value problems for a class of nonlinear thermoelastic plate equations

This paper studies initial-boundary value problems for a class of nonlinear thermoelastic plate equations. Under some certain initial data and boundary conditions, it obtains an existence and uniqueness theorem of global weak solutions of the nonlinear thermoelstic plate equations, by means of the Galerkin method. Moreover, it also proves the existence of strong and classical solutions.

The transmission problem to thermoelastic plate of hyperbolic type

In this paper, we consider the thermoelastic plate equations with localized thermal dissipation of memory type, proposed by Gurtin & Pipkin (1968, Arch. Ration. Mech. Anal., 31, 113-126). We will show that the solution of the corresponding model decays exponentially as time goes to infinity, provided the relaxation function decays exponentially. The main difference between the current model and other thermoelastic systems is that the whole system is of hyperbolic type and the 'dissipation' is weaker (indefinite) than that given by the Fourier law for the heat flux.

$L_p$ theory for the linear thermoelastic plate equations in bounded and exterior domains

The paper is concerned with linear thermoelastic plate equations in a domain $\Omega$: $$ u_{tt} + \Delta^2u+ \Delta \theta = 0 \ \text{ and }\ \theta_t - \Delta\theta - \Delta u_t = 0 \quad\text{in $\Omega\times(0, \infty)$}, $$ subject to the Dirichlet boundary condition $u|_\Gamma = D_\nu u|_\Gamma = \theta|_\Gamma = 0$ and initial condition $(u, u_t, \theta)|_{t=0} = (u_0, v_0, \theta_0) \in W^2_{p, D}(\Omega)\times L_p\times L_p$. Here, $\Omega$ is a bounded or exterior domain in ${{\mathbb R}}^n$ ($n\geq2$). We assume that the boundary $\Gamma$ of $\Omega$ is a $C^4$ hypersurface and we define $W^2_{p, D}$ by the formula $W^2_{p, D} = \{ u \in W^2_p : u|_\Gamma = D_\nu u|_\Gamma = 0\}$. We show that, for any $p \in (1, \infty)$, the associated semigroup $\{T(t)\}_{t\geq0}$ is analytic. Moreover, if $\Omega$ is bounded, then $\{T(t)\}_{t\geq0}$ is exponentially stable.

BoundedH∞-calculus for pseudo-differential Douglis-Nirenberg systems of mild regularity

We consider pseudo-differential Douglis–Nirenberg systems on ℝn with components belonging to the standard Hörmander class S1,δ *(ℝn × Rn ), 0 ≤ δ < 1. Parameter-ellipticity with respect to a subsector Λ ⊂ ℂ is introduced and shown to imply the existence of a bounded H∞-calculus in suitable scales of Sobolev, Besov, and Hölder spaces. We also admit non pseudo-differential perturbations. Applications concern systems with coefficients of mild Hölder regularity and the generalized thermoelastic plate equations (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

AbstractThis paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=Dνu|=θ|=0 and initial condition: (u, ut, θ)|t=0=(u0, v0, θ0). Here, Ω is a bounded domain in ℝn(n≧2). We assume that the boundary ∂Ω of Ω is a C4 hypersurface. We obtain an Lp–Lq maximal regularity theorem. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations

The paper is concerned with the existence of almost periodic solutions to the so-called semilinear thermoelastic plate systems. For that, the strategy consists of seeing these systems as a particular case of the semilinear parabolic evolution equations (*) x ′ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) , t ∈ R , where A ( t ) for t ∈ R is a family of sectorial linear operators on a Banach space X satisfying the so-called Acquistapace–Terreni conditions, and f is a function defined on a real interpolation space X α for α ∈ ( 0 , 1 ) . Under some reasonable assumptions it will be shown that (*) has a unique almost periodic solution. We then make use of the previous result to obtain the existence and uniqueness of an almost periodic solution to the thermoelastic plate systems.

Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations

We prove the existence of a compact, finite dimensional, global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a nonlinear system of thermoelastic plate equations, without any mechanical (viscous or structural) dissipation in the plate component. The plate dynamics is modelled following Berger's approach; we investigate both cases when rotational inertia is included into the model and when it is not. A major part in the proof is played by an estimate--known as stabilizability estimate--which shows that the difference of any two trajectories can be exponentially stabilized to zero, modulo a compact perturbation. In particular, this inequality yields bounds for the attractor's fractal dimension which are independent of two key parameters, namely $\gamma$ and $\kappa$, the former related to the presence of rotational inertia in the plate model and the latter to the coupling terms. Finally, we show the upper semi-continuity of the attractor with respect to these parameters.

Control-theoretic properties of structural acoustic models with thermal effects, I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.

A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation

A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation

Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations

In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions u tt+Δ 2u+αΔθ=f 1(t,u,θ),t⩾0, x∈Ω, θ t−βΔθ−αΔu t=f 2(t,u,θ),t⩾0, x∈Ω, θ=u=Δu=0,t⩾0, x∈∂Ω, where α≠0, β>0, Ω is a sufficiently regular bounded domain in R N ( N⩾1) and f e 1,f e 2 : R×L 2(Ω) 2→L 2(Ω) define by f e ( t, u, θ)( x)= f( t, u( x), θ( x)), x∈Ω, are continuous and locally Lipschitz functions. First, we prove that the linear system ( f 1= f 2=0) generates an analytic strongly continuous semigroup which decays exponentially to zero. Second, under some additional condition we prove that the nonlinear system has a bounded solution which is exponentially stable, and for a large class of functions f 1, f 2 this bounded solution is almost periodic. Finally, we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution.

On a thermoelastic plate equation in an exterior domain

On a thermoelastic plate equation in an exterior domain

On a thermoelastic plate equation in an exterior domain

AbstractObtained are the existence of solutions and the local energy decay of a linear thermoelastic plate equation in a 3 dim. exterior domain. The thermoplate equation is formulated as a Sobolev equation in the abstract framework. Our proof of the existence theorem is based on an argument due to Goldstein (Semigroups of Linear Operators and Applications. Oxford University Press: New York, 1985). To obtain the local energy decay, we use the commutation method in order to treat the high‐frequency part and a precise expansion of the resolvent operator obtained by constructing the parametrix in order to treat the low‐frequency. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Boundary stabilizability of nonlinear structural acoustic models with thermal effects on the interface

A three-dimensional structural acoustic model is considered. This model consists of a wave equation defined on a 3-dimensional bounded domain Ω coupled with a thermoelastic plate equation defined on Γ 0 – a flat surface of the boundary ∂Ω. The main issue studied here is that of uniform stabilizability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the `minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ 0, suffices for the stabilization of the entire structure.

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain {omega} coupled with a thermoelastic plate equation defined on {gamma}{sub 0}-a flat surface of the boundary {omega}. Thus, the coupling between the wave and the plate takes place on the interface {gamma}{sub 0}. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the 'minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to {gamma}{sub 0}, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural dampingmore » most often considered in the literature; (iii) there is no mechanical damping placed on the interface {gamma}{sub 0}; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary partial {omega} are subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides 'optimal' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin.« less

A sharp trace result on a thermo-elastic plate equation with coupled hinged/Neumann boundary conditions

We consider a thermo-elastic plate equation with rotational forces [Lagnese.1]and with coupled hinged mechanical/Neumann thermal boundary conditions (B.C.).We give a sharp result on the Neumann trace of the mechanical velocity, which is '$\frac{1}{2}$' sharper in the space variable than the result than one would obtain by a formal application of trace theory on the optimal interior regularity. Two proofs byenergy methods are given: one which reduces the analysis to sharp wave equation'sregularity theory; and one which analyzes directly the corresponding Kirchoff elasticequation. Important implications of this result are noted.

Finite Element Compensators for Thermo-Elastic Systems with Boundary Control and Point Observation

We consider a thermo-elastic plate equation on a bounded domain Ω, subject to a boundary control as a ‘bending moment’, and with point observation in the interior of Ω The free dynamics generates a s.c. analytic semigroup on the natural energy space. Both control operator and observation operator are fully unbounded, with a combined degree of unboundedness > 1, the super-critical case. We construct an approximation theory, based on FEM, which leads to a finite-dimensional dynamic compensator, possessing all the expected desirable properties, once it is inserted into the original continuous system. In particular, it preserves the same margin of stability of the continuous problem, except, perhaps, for an arbitrary ε>0.

Analyticity, and lack thereof, of thermo-elastic semigroups

We consider (linear) thermo-elastic plate equations under five sets of canonical B.C., including two cases where the mechanical and the thermal variables are coupled on the boundary. The challenging so-called free B.C. case of [Lag], [A-L] is included. The main results are as follows. If rotational forces are not accounted for, then the resulting s.c. contraction semigroup is, moreover, analytic on the natural (energy) space under all such canonical B.C. By contrast, if rotational forces are accounted for, then the corresponding s.c. contraction semigroup has a structural property that makes it more akin to a s.c. group (at least in the mechanical part); a fortiori, it is neither compact, nor differentiable, nor uniformly continuous for all t > 0. Analyticity of the s.c. thermo-elastic semigroup, particularly in the difficult case of free B.C., has been an open problem for some time in specialized circles. Similarly, a general description of the cases where analyticity fails has been the object of inquiries.