393 research outputs found

    A functional analytic approach towards nonlinear dissipative well-posed systems

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    The aim of this paper is to develop a functional analytic approach towards nonlinear systems. For linear systems this is well known and the resulting class of well-posed and regular linear systems is well studied. Our approach is based on the theory of nonlinear semigroup and we explain it by means of an example, namely equations of quasi-hyperbolic type

    Root locii for systems defined on Hilbert spaces

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    The root locus is an important tool for analysing the stability and time constants of linear finite-dimensional systems as a parameter, often the gain, is varied. However, many systems are modelled by partial differential equations or delay equations. These systems evolve on an infinite-dimensional space and their transfer functions are not rational. In this paper a rigorous definition of the root locus for infinite-dimensional systems is given and it is shown that the root locus is well-defined for a large class of infinite-dimensional systems. As for finite-dimensional systems, any limit point of a branch of the root locus is a zero. However, the asymptotic behaviour can be quite different from that for finite-dimensional systems. This point is illustrated with a number of examples. It is shown that the familiar pole-zero interlacing property for collocated systems with a Hermitian state matrix extends to infinite-dimensional systems with self-adjoint generator. This interlacing property is also shown to hold for collocated systems with a skew-adjoint generator

    C0C_0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

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    Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. The main result of this paper is a simple test for C0C_0-semigroup generation in terms of the boundary conditions. The result is illustrated with several examples

    Variational principles for self-adjoint operator functions arising from second-order systems

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    Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form z¨(t),y+d[z˙(t),y]+a0[z(t),y]=0. \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0. Here a0\mathfrak{a}_0 and d\mathfrak{d} are densely defined, symmetric and positive sesquilinear forms on a Hilbert space HH. We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix A\mathcal{A}, the forms t(λ)[x,y]:=λ2x,y+λd[x,y]+a0[x,y], \mathfrak{t}(\lambda)[x,y] := \lambda^2\langle x,y\rangle + \lambda\mathfrak{d}[x,y] + \mathfrak{a}_0[x,y], where λC\lambda\in \mathbb C and x,yx,y are in the domain of the form a0\mathfrak{a}_0, and a corresponding operator family T(λ)T(\lambda). Using form methods we define a generalized Rayleigh functional and characterize the eigenvalues above the essential spectrum of A\mathcal{A} by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.Comment: to appear in Operators and Matrice

    On continuity of solutions for parabolic control systems and input-to-state stability

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    We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.Comment: 19 pages, final version of preprint, Prop. 6 and Thm 7 have been generalised to arbitrary Banach spaces, the assumption of boundedness of the semigroup in Thm 10 could be droppe

    Analyticity and Riesz basis property of semigroups associated to damped vibrations

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    Second order equations of the form z+A0z+Dz=0z'' + A_0 z + D z'=0 in an abstract Hilbert space are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix AA associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of AA in the phase space
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