Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers

Lasiecka, Irena; Triggiani, Roberto

doi:10.3934/cpaa.2016001

Cited by 14 publications

(18 citation statements)

References 35 publications

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“…to be interpreted as in (1.7) (LHS) and (2.10a), whose domain is recalled below in (1.18a-b) from [27]. Instead, B N is defined by (2.14c) in the following sense.…”

Section: Model With Neumann Control At the Interfacementioning

confidence: 98%

“…The present paper is a natural successor of [27,36]. It deals with the same heatstructure interaction model where the structure has the distinctive feature that it possesses a viscoelastic ('strong') damping term.…”

Section: Introductionmentioning

confidence: 96%

“…Naturally, the key property established in [27]-that the free dynamics generates a s.c. (C 0 −) semigroup which, moreover, is analytic on the natural energy space (Theorem 2.1 below)-is the motivating starting point of the present investigation to be critically employed in our analysis. As in [27], we shall focus on the setting where the structure is immersed in the fluid leaving to other efforts the study of the fluid within a structure (such as blood running inside arteries [9]). Prior Fig.…”

Section: Introductionmentioning

confidence: 99%

“…It deals with the same heatstructure interaction model where the structure has the distinctive feature that it possesses a viscoelastic ('strong') damping term. The basic difference is that [27] studied the free dynamics (boundary/interface homogeneous case) with no forcing term at the interface between the two media, while the focus of the present paper is a study of the non-homogeneous, boundary-driven dynamics, where a control or forcing term g acts in the 'Neumann' interface conditions, see (1.1e), or in the 'Dirichlet' interface conditions, see (5.1d). Naturally, the key property established in [27]-that the free dynamics generates a s.c. (C 0 −) semigroup which, moreover, is analytic on the natural energy space (Theorem 2.1 below)-is the motivating starting point of the present investigation to be critically employed in our analysis.…”

Section: Introductionmentioning

confidence: 99%

“…The basic difference is that [27] studied the free dynamics (boundary/interface homogeneous case) with no forcing term at the interface between the two media, while the focus of the present paper is a study of the non-homogeneous, boundary-driven dynamics, where a control or forcing term g acts in the 'Neumann' interface conditions, see (1.1e), or in the 'Dirichlet' interface conditions, see (5.1d). Naturally, the key property established in [27]-that the free dynamics generates a s.c. (C 0 −) semigroup which, moreover, is analytic on the natural energy space (Theorem 2.1 below)-is the motivating starting point of the present investigation to be critically employed in our analysis. As in [27], we shall focus on the setting where the structure is immersed in the fluid leaving to other efforts the study of the fluid within a structure (such as blood running inside arteries [9]).…”

Section: Introductionmentioning

confidence: 99%

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A Heat–Viscoelastic Structure Interaction Model with Neumann and Dirichlet Boundary Control at the Interface: Optimal Regularity, Control Theoretic Implications

Triggiani

2016

Appl Math Optim

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We consider a heat-structure interaction model where the structure is subject to viscoelastic (strong) damping, and where a (boundary) control acts at the interface between the two media in Neumann-type or Dirichlet-type conditions. For the boundary (interface) homogeneous case, the free dynamics generates a s.c. contraction semigroup which, moreover, is analytic on the energy space and exponentially stable here Lasiecka et al. (Pure Appl Anal, to appear). If A is the free dynamics operator, and B N is the (unbounded) control operator in the case of Neumann control acting at the interface, it is shown that A − 1 2 B N is a bounded operator from the interface measured in the L 2 -norm to the energy space. We reduce this problem to finding a characterization, or at least a suitable subspace, of the domain of the square root (−A) 1/2 , i.e., D((−A) 1/2 ), where A has highly coupled boundary conditions at the interface. To this end, here we prove that D((−A)2 ) ≡ V , with the space V explicitly characterized also in terms of the surviving boundary conditions. Thus, this physical model provides an example of a matrix-valued operator with highly coupled boundary conditions at the interface, where the so called Kato problem has a positive answer. (The well-known sufficient condition Lions in J Math Soc JAPAN 14(2):233-241, 1962, Theorem 6.1 is not applicable.) After this result, some critical consequences follow for the model under study. We explicitly note here three of them: (i) optimal (parabolic-type) boundary → interior regularity with boundary control at the interface; (ii) optimal boundary control theory for the corresponding quadratic cost problem; (iii) min-max game theory problem with control/disturbance acting at the In memory of A. V. Balakrishnan: long-time friend, mentor, collaborator. B Roberto Triggiani 123 572 Appl Math Optim (2016) 73:571-594interface. On the other hand, if B D is the (unbounded) control operator in the case of Dirichlet control acting at the interface, it is then shown here that A −1 B D is a bounded operator from the interface measured this time in the H 1 2 -norm to the energy space. Similar consequences follow.

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“…to be interpreted as in (1.7) (LHS) and (2.10a), whose domain is recalled below in (1.18a-b) from [27]. Instead, B N is defined by (2.14c) in the following sense.…”

Section: Model With Neumann Control At the Interfacementioning

confidence: 98%

Section: Introductionmentioning

confidence: 96%