ABSTRACT
The abstract model considered in this paper is motivated (at least) by an entire class of boundary control problems for partial differential equations defined on bounded domain. This consists of wave-like or plate-like equations (second order in time), with a strong internal damping, which is modeled by a high-order operator. It is precisely this latter term that is responsible for changing the character of the equation and for inducing parabolic-like properties to the resulting dynamics. In the case of concrete p.d.e. problems with homogeneous boundary conditions, or in the abstract case which encompasses these p.d.e. problems, these equations have been thoroughly studied in the Hilbert space framework in [C-T.1]-[C-T.4], and later in the Banach space framework in [F-0.1], to which we refer for further information. In the case where the control action is exercised in a (suitable) boundary condition, these resulting mixed problems give rise to model (1.3.10), or (1.3.11), where the presence of the operator B \—the novelty of these models-is an intrinsic property. For brevity, we shall confine our application of the abstract theory to two canonical examples, where the damping operator (in the case of homogeneous boundary conditions) is a (possibly, fractional) power of the principal (elastic) space operator. More general cases, such as e.g., [T.l, Section 3.3] could likewise be treated, where now the damping operator is only comparable (in the technical sense of positive self-adjoint operators in [C-T.1]-[C-T.3]) to a (fractional) power of the elastic operator.
