We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, ...provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the B- functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the B-calculus thus making the paper essentially self-contained.
We prove a bi-sublinear embedding for semigroups generated by non-smooth complex-coefficient elliptic operators in divergence form and for certain mutually dual pairs of Orlicz-space norms. This ...generalizes a result by Carbonaro and Dragičević from power functions to more general Young functions that still behave like powers. To achieve this, we generalize a Bellman function constructed by Nazarov and Treil.
We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for ...triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.
We are concerned with asymptotic behaviours of solutions for linear wave equations with frictional damping only on Wentzell boundary, but without any interior damping. Making some elaborate and ...subtle analysis of an associated auxiliary system, we obtain an ideal estimate of the resolvent of the generator of the system along the imaginary axis. This enables us to prove that the energies of the system decay polynomially. Our energy stability result presents a solution, in the linear case, to the problem proposed by Cavalcanti et al. (2007) 8, which was put forward as a “hard problem” due to the lack of interior damping.
Let T denote a positive operator with spectral radius 1 on, say, an L^p-space. A classical result in infinite dimensional Perron–Frobenius theory says that, if T is irreducible and power bounded, ...then its peripheral point spectrum is either empty or a subgroup of the unit circle.
In this note we show that the analogous assertion for the entire peripheral spectrum fails. More precisely, for every finite union U of finite subgroups of the unit circle we construct an irreducible stochastic operator on \ell ^1 whose peripheral spectrum equals U.
We also give a similar construction for the C_0-semigroup case.
We investigate a special class of nonlinear infinite dimensional systems. These are obtained by subtracting a nonlinear maximal monotone (possibly multi-valued) operator ℳ from the semigroup ...generator of a scattering passive linear system. While the linear system may have unbounded linear damping (for instance, boundary damping) which is only densely defined, the nonlinear damping operator ℳ is assumed to be defined on the whole state space. We show that this new class of nonlinear infinite dimensional systems is well-posed and incrementally scattering passive. Our approach uses the theory of maximal monotone operators and the Crandall–Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax–Phillips type nonlinear semigroup that represents the whole system.
The approximate controllability of a parabolic integro-differential equation system has already been shown for the case where the memory kernel is a constant. However, when this condition on the ...memory kernel and considering instead only the weak constraint of boundedness, the problem becomes more complex and it is less easy to prove the approximate controllability of the system. In this study, based on the operator semigroup method, a detailed proof of the approximate controllability of the system is given for the case where the memory kernel is nonconstant and bounded. In addition, three numerical simulation studies are performed to verify the theoretical proof.
We study a class of integro-differential equations in Hilbert spaces which fits the Coleman–Gurtin model of heat conduction with memory. Well-posedness of the equation is established and sufficient ...conditions for exponential stability of the equation are given. This is done by using the semigroup approach. An illustrative example is given and the obtained theoretical results are verified by numerical simulations.
We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration ...of the water along that boundary, as a multiple of a scalar input function u, times a given function h of the height along the active boundary. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ζ̇. We prove that for suitable functions h, there exists a bounded feedback functional F such that the feedback u=Fz renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1+t)−16. Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill et al. (2019).
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