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

A Heat–Viscoelastic Structure Interaction Model with Neumann and Dirichlet Boundary Control at... 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 $${\mathcal {A}}$$ A is the free dynamics operator, and $${\mathcal {B}}_N$$ B N is the (unbounded) control operator in the case of Neumann control acting at the interface, it is shown that $${\mathcal {A}}^{-\frac{1}{2}}{\mathcal {B}}_N$$ A - 1 2 B N is a bounded operator from the interface measured in the $$\mathbf{L}^2$$ 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 $$(-{\mathcal {A}})^{1/2}$$ ( - A ) 1 / 2 , i.e., $${\mathcal {D}}((-{\mathcal {A}})^{1/2})$$ D ( ( - A ) 1 / 2 ) , where $${\mathcal {A}}$$ A has highly coupled boundary conditions at the interface. To this end, here we prove that $${\mathcal {D}}((-{\mathcal {A}})^{\frac{1}{2}})\equiv {\mathcal {D}}((-{\mathcal {A}}^*)^{\frac{1}{2}})\equiv V$$ D ( ( - A ) 1 2 ) ≡ D ( ( - A ∗ ) 1 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 $$\rightarrow $$ → 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 $${\mathcal {B}}_D$$ B D is the (unbounded) control operator in the case of Dirichlet control acting at the interface, it is then shown here that $${\mathcal {A}}^{-1}{\mathcal {B}}_D$$ A - 1 B D is a bounded operator from the interface measured this time in the $$\mathbf{H}^{\frac{1}{2}}$$ H 1 2 -norm to the energy space. Similar consequences follow. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-016-9348-2
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2016

References

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