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

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 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.

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

“…The non-homogeneous case. As already noted, a companion publication [40] deals with the 'boundary' (interface) non-homogeneous problem, whereby the homogeneous boundary condition (B.C.) in (1e), respectively (1d), will be replaced by the following boundary non-homogeneous conditions, respectively: either ∂(w + w t ) ∂ν = ∂u ∂ν + g; or else u = w t + g on (0, T ] × Γ s .…”

“…Here, g is a control function of Neumann-type, respectively, of Dirichlet-type. Writing each problem in the abstract formẋ = Ax + B N g (Neumann case) oṙ x = Ax + B D g (Dirichlet case), with explicit operators B N and B D , reference [40] shows two main preliminary results: (1) that A − 1 2 B N is a bounded operator from functions at the interface measured in the L 2 -norm to the energy space H 0 ; (2) that A −1 B D is a bounded operator from functions at the interface measured this time in the H 1 2 -norm to the energy space H 0 . In the much more challenging Neumann case, the proof that…”

“…It is a first study which is preliminarily focused on the boundary (interface) homogeneous problem because of space constraints. It is followed [40] by the ultimate case of interest, where the heat-structure PDE system is subject to a boundary control acting at the interface between the two media: the heat component and the structure component. The control may be either of Neumann-type or of Dirichlet-type, depending on the specific interface condition on which it acts, see (4) below.…”

“…The control may be either of Neumann-type or of Dirichlet-type, depending on the specific interface condition on which it acts, see (4) below. For this boundary non-homogeneous case, we study in [40] the corresponding (optimal) boundary → interior regularity, from which solution to corresponding optimal control problems,…”

Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay

“…Such configuration belongs therefore now to parabolic-parabolic coupling. This and much more is established in [28], [35], [36], [37] initially in the case of a heat-viscoelastic wave interaction model. A natural generalization to the case where the heat is replaced by a fluid (with pressure) followed in [38].…”