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Wellposedness and asymptotic behaviour of nonautonomous linear evolution equations, Evolution equations, semigroups and functional
, 2002
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THE REGULARITY OF THE WAVE EQUATION WITH PARTIAL DIRICHLET CONTROL AND COLOCATED OBSERVATION∗
"... Abstract. In this paper we analyze a multidimensional controlled wave equation on a bounded domain, subject to partial Dirichlet control and colocated observation. By means of a partial Fourier transform, it is shown that the system is wellposed and regular in the sense of D. Salamon and G. Weiss. ..."
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Abstract. In this paper we analyze a multidimensional controlled wave equation on a bounded domain, subject to partial Dirichlet control and colocated observation. By means of a partial Fourier transform, it is shown that the system is wellposed and regular in the sense of D. Salamon and G. Weiss. The corresponding feedthrough operator is found to be the identity operator on the input space.
FEEDBACK THEORY FOR NONAUTONOMOUS REGULAR LINEAR SYSTEMS WITH INPUT AND STATE DELAYS
"... Abstract. We show that the class of regular non–autonomous systems is invariant under perturbation by time–varying state and input delays. In particular, we give explicit formulas of the resulting input, output, and input–output maps. This result is used to solve the feedback problem for the delayed ..."
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Abstract. We show that the class of regular non–autonomous systems is invariant under perturbation by time–varying state and input delays. In particular, we give explicit formulas of the resulting input, output, and input–output maps. This result is used to solve the feedback problem for the delayed system. The relationship between the open and the closed loop system is investigated. Our results are applied to a parabolic boundary control problem with input and state delays. 1.
Two classes of passive timevarying wellposed linear systems
"... Abstract. We investigate two classes of timevarying wellposed linear systems. Starting from a timeinvariant scatteringpassive system, each of the timevarying systems is constructed by introducing a timedependent inner product on the state space and modifying some of the generating operators. ..."
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Abstract. We investigate two classes of timevarying wellposed linear systems. Starting from a timeinvariant scatteringpassive system, each of the timevarying systems is constructed by introducing a timedependent inner product on the state space and modifying some of the generating operators. These classes of linear systems are motivated by physical examples such as the electromagnetic field around a moving object. To prove the wellposedness of these systems, we use the LaxPhillips semigroup induced by a wellposed linear system, as in scattering theory. Key words. Wellposed linear system, operator semigroup, linear timevarying system, LaxPhillips semigroup, scattering passive system.
REGULARIZATION AND FREQUENCY–DOMAIN STABILITY OF WELL–POSED SYSTEMS
"... Abstract. We study linear control systems with unbounded control and observation operators using certain regularization techniques. This allows us to introduce a modification of the transfer function for the system also if the input and output operators are not admissible in the usual sense. The mo ..."
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Abstract. We study linear control systems with unbounded control and observation operators using certain regularization techniques. This allows us to introduce a modification of the transfer function for the system also if the input and output operators are not admissible in the usual sense. The modified transfer function is utilized to show exponential stability of sufficiently smooth solutions for the internal system under appropriate admissibility conditions on the system operators and appropriately modified stabilizability and detectability assumptions. If the internal system satisfies additional regularity properties, then we even obtain its uniform exponential stability. 1.