Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H ∞ (X) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.
We extend the ν-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions.
Abstract. We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in R n , n ≤ 3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is shown that finite energy solutions decay exponentially to zero with the rate depending only on the (finite energy) size of initial conditions. The proofs are based on methods of weak compactness along with nonlocal partial differential operator multipliers which supply the sought after "recovery" inequalities. Regularity of solutions is also discussed by exploiting the underlying analyticity of the linearized semigroup along with a related maximal parabolic regularity [1,16,44].
Mathematics Subject Classification (2000). Primary 74F05; Secondary 35B30, 35B40, 74H40. Keywords. Quasilinear thermoelastic plates, existence of weak solutions, uniform decays of finite energy solutions.
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