Optimal Energy Decay for the Wave-Heat System on a Rectangular Domain

Batty, C. J. K.; Paunonen, Lassi; Seifert, David

doi:10.1137/18m1195796

Cited by 22 publications

(10 citation statements)

References 18 publications

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“…Meanwhile, assume that Cw = ∇w in Ω, Obviously, (A, C) is exactly observable. The stability of wave equation in rectangular domain with locally viscous damping was ever considered by [4] and [23], where they obtained that under u(x, t) = αw t (x, t), the system can be stabilized polynomially, and the optimal decay rate is given as follows:…”

Section: Wave Equation Without Gcc: Rectangular Domainmentioning

confidence: 99%

Slow decay and turnpike for infinite-horizon hyperbolic LQ problems

Han¹,

Zuazua²

2021

Preprint

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This paper is devoted to analysing the explicit slow decay rate and turnpike in the infinitehorizon linear quadratic optimal control problems for hyperbolic systems. Assume that some weak observability or controllability are satisfied, by which, the lower and upper bounds of the corresponding algebraic Riccati operator are estimated, respectively. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to the LQ optimal control problems constraint to networks of one-dimensional wave equations and also some multi-dimensional ones with local controls which lack of GCC(Geometric Control Condition).

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Section: Wave Equation Without Gcc: Rectangular Domainmentioning

confidence: 99%

Slow decay and turnpike for infinite-horizon hyperbolic LQ problems

Han¹,

Zuazua²

2021

Preprint

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“…Motivated by the fundamental works [10,17] and the subsequent ones [3,6,7,11,19], the frequency domain approach has become a very useful tool to analyze the long-time behaviors (including exponential/polynomial/logarithmic stability) of many autonomous systems in infinite dimensions. This approach leads to not only a relatively concise and elementary argument, but also an optimal decay rate estimate for the problem under consideration at least for some relatively simple geometrical configurations, such as one-dimensional or rectangular domains (e.g., [4,5,13,15]). Meanwhile, compared to the arguments based on micro-local analysis, this approach requires considerably weaker smoothness conditions on the data (including particularly the boundaries of the space domains in which the systems evolve).…”

Section: Define a Linear Unbounded Operatormentioning

confidence: 99%

Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system

Rao¹,

Zhang²

2021

CPAA

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<p style='text-indent:20px;'>We consider the asymptotic behavior of a linear model arising in fluid-structure interactions. The system is formed by a heat equation and a wave equation in two distinct domains, which are coupled by atransmission condition along the interface of the domains. By means of the frequency domain approach, we establish some decay rates for the whole system. Our results also showthat the decay of the fluid-structure interaction depends not only on the transmission of the damping from the heat equation to the wave equation, but also on the location of the damping for the wave equation.</p>

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“…(11) Now, in our first result, we provide a semigroup wellposedness for A : D(A) ⊂ H → H. This is given in the following theorem: 10)- (11), generates a C 0 -semigroup of contractions. Consequently, the solution 2)-( 5), or equivalently (9), is given by…”

Section: The Thick Wave-thin Wave-heat Generatormentioning

confidence: 99%

“…Such a strong stability can be seen as a measure of the "strength" of the coupling condition. For the classical heat-wave system (without the 2-D wave equation on the interface) this question is by now rather well understood and precise decay rates are well known (see [3,9] and references within.) (We should emphasize that the high-frequency oscillations in the structure are not efficiently dissipated and therefore there is no exponential decay of the energy.…”

Section: Introduction 1motivation and Literaturementioning

confidence: 99%

Wellposedness, Spectral Analysis and Asymptotic Stability of a Multilayered Heat-Wave-Wave System

Avalos¹,

Geredeli²,

Muha³

2019

Preprint

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In this work we consider a multilayered heat-wave system where a 3-D heat equation is coupled with a 3-D wave equation via a 2-D interface whose dynamics is described by a 2-D wave equation. This system can be viewed as a simplification of a certain fluid-structure interaction (FSI) PDE model where the structure is of composite-type; namely it consists of a "thin" layer and a "thick" layer. We associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay.Our first result is semigroup well-posedness for the (FSI) PDE dynamics. Utilizing here a Lumer-Phillips approach, we show that the fluid-structure system generates a C 0 -semigroup on a chosen finite energy space of data. As our second result, we prove that the solution to the (FSI) dynamics generated by the C 0 -semigroup tends asymptotically to the zero state for all initial data. That is, the semigroup of the (FSI) system is strongly stable. For this stability work, we analyze the spectrum of the generator A and show that the spectrum of A does not intersect the imaginary axis.

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Optimal Energy Decay for the Wave-Heat System on a Rectangular Domain

Cited by 22 publications

References 18 publications

Slow decay and turnpike for infinite-horizon hyperbolic LQ problems

Slow decay and turnpike for infinite-horizon hyperbolic LQ problems

Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system

Wellposedness, Spectral Analysis and Asymptotic Stability of a Multilayered Heat-Wave-Wave System

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