Exponential stability for linear evolutionary equations

Abstract

We give an approach to exponential stability within the framework of evolutionary equations due to Picard [Math. Methods Appl. Sci. 32(14) (2009), 1768-1803]. We derive sufficient conditions for exponential stability in terms of the material law operator which is defined via an analytic and bounded operator-valued function and give an estimate for the expected decay rate. The results are illustrated by three examples: differential-algebraic equations, partial differential equations with finite delay and parabolic integro-differential equations.S. Trostorff / Exponential stability for linear evolutionary equations orem 2.4 of this article). Moreover, the question of causality, which can be seen as a characterizing property for evolutionary processes, was addressed, which leads to additional constraints on the operator M, namely that M is defined via the Fourier-Laplace transformation of an analytic and bounded function M : B C (r, r) → L(H) (for more details see Section 2). Especially, the analyticity of M is crucial for the causality, due to the correlation of supports of L 2 -functions and the analyticity of their Laplace transforms by the Paley-Wiener theorem (see [21, Theorem 19.2]). Later on, these results were generalized to the case of A being a maximal monotone relation in [22,24].In this work we give sufficient criteria for the exponential stability of the evolutionary problem in terms of the function M . The study of stability issues for differential equations, which goes back to Lyapunov (see [14] for a survey), has become a very active field of research for many decades and there exist numerous works dealing with this topic. We just like to mention some standard approaches to exponential stability. The first strategy goes back to Lyapunov. The aim is to find a suitable function (a so-called Lyapunov function) yielding a certain differential inequality which allows to derive statements about the asymptotic behavior of solutions of the differential equation. A second approach, which applies to linear differential equations, is based on the theory of semigroups. In this framework, different criteria for exponential stability were derived in terms of the semigroup or its generator, e.g. Datko's lemma ([8] or [9, p. 300]), Gearharts theorem ([11] or [9, p. 302]) or the Spectral Mapping Theorem (see [9, p. 302, Theorem 1.10]). A third approach uses the Fourier or the Laplace transform of a solution to derive statements of their asymptotics. These methods are sometimes referred to as Frequency Domain Methods. In our framework it seems to be appropriate to employ the last approach, since, by the definition of M, methods of vector-valued complex analysis are at hand through the Fourier-Laplace transformation. Note that, due to the general structure of evolutionary equations, the results apply to a broad class of differential equations, such as differential-algebraic equations, equations with memory effects or integro-differential equations, where semigroup methods may be difficult to apply.The article is s...

Optimal energy decay in a one-dimensional wave-heat system with infinite heat part

“…Waurick showed in [11] that as N → ∞, the sequence of solutions (u N ) N ∈N converges weakly in L 2 loc ([0, 1] × R) to u, the solution to the limit equation 1 2 ∂ 2 t u(ξ, t) + ∂ t u(ξ, t) + 1 2 u(ξ, t) − 2∂ 2 ξ u(ξ, t) = f (ξ, t) + ∂ t f (ξ, t), t ∈ R, subject to zero initial conditions and Neumann boundary on both ends. Furthermore, he showed that this limit admitted exponentially stable solutions in the sense of [10]. However, when the elliptic part, u N (ξ, t) − ∂ 2 ξ u N (ξ, t) = f (ξ, t), is replaced with the corresponding parabolic part, ∂ t u N (ξ, t) − ∂ 2 ξ u N (ξ, t) = f (ξ, t), the limit equation becomes 1 2 ∂ 2 t u(ξ, t) + ∂ t u(ξ, t) − 2∂ 2 ξ u(ξ, t) = f (ξ, t) + ∂ t f (ξ, t), t ∈ R, subject again to zero initial conditions and Neumann boundary.…”

A Note on Exponential Stability for Evolutionary Equations.

“…However, we provide an alternative proof, allowing to drop one constraint on the operator M imposed in [4]. We emphasize that due to the abstract structure of (1), our results apply to a broad class of differential equations, such as differential-algebraic equations, integro-differential equations or delay differential equations (see [4] for several applications). Throughout, let H be a complex Hilbert space with inner product ·|· and induced norm | · |.…”

“…[3]). We want to address the exponential stability of problems of the form (1), where A is a linear maximal monotone (or m-accretive) operator on some Hilbert space H and give sufficient criteria for the exponential stability in terms of the operator M. This was already done by the author in [4]. However, we provide an alternative proof, allowing to drop one constraint on the operator M imposed in [4].…”

“…We want to address the exponential stability of problems of the form (1), where A is a linear maximal monotone (or m-accretive) operator on some Hilbert space H and give sufficient criteria for the exponential stability in terms of the operator M. This was already done by the author in [4]. However, we provide an alternative proof, allowing to drop one constraint on the operator M imposed in [4]. We emphasize that due to the abstract structure of (1), our results apply to a broad class of differential equations, such as differential-algebraic equations, integro-differential equations or delay differential equations (see [4] for several applications).…”

Homogenisation and the weak operator topology

“…Note that M(∂ t,µ ) is realised as a densely defined operator acting from L 2 µ (R; K) to L 2 µ (R; H) for all µ > ν. By[27, Lemma 3.6], for f ∈ dom(M(∂ t,µ ))∩dom(M(∂ t,η )) we have M(∂ t,µ )f = M(∂ t,ν )f.…”