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

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• 作者：Roberto Triggiani
• 刊名：Applied Mathematics and Optimization
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：73
• 期：3
• 页码：571-594
• 全文大小：612 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Systems Theory and Control
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  
• 出版者：Springer New York
• ISSN：1432-0606
• 卷排序：73

文摘

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 \({\mathcal {A}}\) is the free dynamics operator, and \({\mathcal {B}}_N\) is the (unbounded) control operator in the case of Neumann control acting at the interface, it is shown that \({\mathcal {A}}^{-\frac{1}{2}}{\mathcal {B}}_N\) is a bounded operator from the interface measured in the \(\mathbf{L}^2\)-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 \((-{\mathcal {A}})^{1/2}\), i.e., \({\mathcal {D}}((-{\mathcal {A}})^{1/2})\), where \({\mathcal {A}}\) has highly coupled boundary conditions at the interface. To this end, here we prove that \({\mathcal {D}}((-{\mathcal {A}})^{\frac{1}{2}})\equiv {\mathcal {D}}((-{\mathcal {A}}^*)^{\frac{1}{2}})\equiv V\), 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 \(\rightarrow \) 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 interface. On the other hand, if \({\mathcal {B}}_D\) is the (unbounded) control operator in the case of Dirichlet control acting at the interface, it is then shown here that \({\mathcal {A}}^{-1}{\mathcal {B}}_D\) is a bounded operator from the interface measured this time in the \(\mathbf{H}^{\frac{1}{2}}\)-norm to the energy space. Similar consequences follow.KeywordsHeat-structure interactionOptimal boundary regularityHeat-structure interaction systems