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

• Published in: Applied mathematics & optimization Vol. 73; no. 3; pp. 571 - 594
• Main Author: Triggiani, Roberto
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2016; Springer Nature B.V
• Subjects: Boundaries; Boundary conditions; Boundary control; Boundary layer; Calculus of Variations and Optimal Control; Optimization; Control; Control theory; Dirichlet problem; Energy measurement; Game theory; Heat transfer; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Operators; Optimization; Partial differential equations; Simulation; Studies; Systems Theory; Theoretical; Viscoelasticity; Heat-structure interaction; Optimal boundary regularity; Heat-structure interaction systems
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-016-9348-2

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 ) 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 → 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 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.