Notions Of Admissibility Research Articles

Resolvent conditions for the control of parabolic equations

Since the seminal work of Russell and Weiss in 1994, resolvent conditions for various notions of admissibility, observability and controllability, and for various notions of linear evolution equations have been investigated intensively, sometimes under the name of infinite-dimensional Hautus test. This paper sets out resolvent conditions for null-controllability in arbitrary time: necessary for general semigroups, sufficient for analytic normal semigroups. For a positive self-adjoint operator A, it gives a sufficient condition for the null-controllability of the semigroup generated by −A which is only logarithmically stronger than the usual condition for the unitary group generated by iA. This condition is sharp when the observation operator is bounded. The proof combines the so-called “control transmutation method” and a new version of the “direct Lebeau–Robbiano strategy”. The improvement of this strategy also yields interior null-controllability of new logarithmic anomalous diffusions.

Reducibility of domain representations and Cantor–Weihrauch domain representations

We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations.A representation reduces to another if its representing map factors through the other representation. Reductions form a pre-order on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, non-triviality. Equivalent representations represent the same set of functions on the represented space.Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch's TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE.To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility.

Visco-elastic relaxation with a van der waals type stress

Contemporary science models certain phase transitions as conservation laws with a non-monotone stress-strain relation. The system of conservation laws are often hyperbolic except in a region in phase space, where the system is elliptic or of composite type. It is possible to prove existence and uniqueness of solutions for specific models which exhibit such change of type, but notions of admissibility and the connections between the mathematical theory and engineering experiments are not fully understood. In this paper we model phase transition by a visco-elastic relaxation law, and describe numerical experiments on this system. In particular we examine special initial data which, in the context of the mixed system, gives rise to multiple solutions.