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13
Wellposedness and asymptotic behaviour of nonautonomous linear evolution equations
 A. Lorenzi, B. Ruf (Eds.): “Evolution Equations, Semigroups and Functional Analysis,” Birkhäuser
, 2002
"... We review results on the existence and the long term behaviour of nonautonomous linear evolution equations. Emphasis is put on recent results on the asymptotic behaviour using a semigroup approach. ..."
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Cited by 6 (4 self)
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We review results on the existence and the long term behaviour of nonautonomous linear evolution equations. Emphasis is put on recent results on the asymptotic behaviour using a semigroup approach.
A sufficient condition for exponential dichotomy of parabolic evolution equations
 Weis (Eds.), "Evolution Equations and their Applications in Physical and Life Sciences (Proceedings Bad Herrenalb
, 1998
"... In this note we establish the exponential dichotomy of the abstract Cauchy problem (CP) ..."
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Cited by 4 (4 self)
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In this note we establish the exponential dichotomy of the abstract Cauchy problem (CP)
Asymptotic behaviour of parabolic nonautonomous evolution equations
, 2002
"... The long term behaviour of autonomous linear Cauchy problems on a Banach space X has been studied systematically and quite successfully by means of spectral theory ..."
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Cited by 4 (1 self)
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The long term behaviour of autonomous linear Cauchy problems on a Banach space X has been studied systematically and quite successfully by means of spectral theory
A Characteristic Equation For NonAutonomous Partial Functional Differential Equations
, 2000
"... . We characterize the exponential dichotomy of nonautonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time periodic case. From this we deduce robustness results. We further study the almost periodic ..."
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Cited by 3 (3 self)
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. We characterize the exponential dichotomy of nonautonomous partial functional differential equations by means of a spectral condition extending known characteristic equations for the autonomous or time periodic case. From this we deduce robustness results. We further study the almost periodicity of solutions to the inhomogeneous equation. Our approach is based on the spectral theory of evolution semigroups. 1. Introduction For the autonomous partial functional differential equation u(t) = Au(t) + Lu t ; t 0; u(t) = OE(t); \Gammar t 0; (1.1) there is a well developed semigroup approach; in particular, a powerful spectral theory is available. Here we assume that A generates a C 0 semigroup V (\Delta) on a Banach space X. Further, r 0, OE 2 E := C([\Gammar; 0]; X); L 2 L(E; X), and we let u t () := u(t + ) for 2 [\Gammar; 0], t 0, and u : [\Gammar; 1) ! X. Then the operator AL := d d ; D(AL ) := fOE 2 C 1 ([\Gammar; 0]; X) : OE(0) 2 D(A); OE 0 (0) = AOE(0) + ...
Parabolic Evolution Equations With Asymptotically Autonomous Delay
 Report No.2, Fachbereich Mathematik und Informatik, Universitat
, 2001
"... . We study retarded parabolic nonautonomous evolution equations whose coefficients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the nonautonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation ten ..."
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Cited by 3 (3 self)
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. We study retarded parabolic nonautonomous evolution equations whose coefficients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the nonautonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation tends to the stationary solution at infinity. We use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation. 1. Introduction In the present paper we continue the investigation of the longterm behaviour of asymptotically autonomous evolution equations begun in [30]. There we studied the Cauchy problem u(t) = A(t)u(t) + f(t); t ? s 0; u(s) = x; (1.1) on a Banach space X assuming that the linear operators A(t), t 0, are sectorial of the same type and satisfy the `AcquistapaceTerreni' condition (see (P) below) and that there exists another sectorial operator A such that lim t!1 R(w; A(t...
Feedbacks For NonAutonomous Regular Linear
 SIAM J. Control Optim
, 2001
"... We introduce nonautonomous wellposed and (absolutely) regular linear systems as quadrupels consisting of an evolution family and output, input and input output maps subject to natural hypotheses. In the spirit of G. Weiss' work these maps are represented in terms of admissible observation a ..."
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Cited by 2 (2 self)
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We introduce nonautonomous wellposed and (absolutely) regular linear systems as quadrupels consisting of an evolution family and output, input and input output maps subject to natural hypotheses. In the spirit of G. Weiss' work these maps are represented in terms of admissible observation and control operators (the latter in an approximative sense) in the time domain. In this setting the closedloop system exists for a canonical class of `admissible' feedbacks, and it inherits the absolute regularity and other properties of the given system. In particular, one can iterate feedbacks. We study a second order parabolic partial differential equation in nondivergence form with point control and observation in space dimension n 3.
ADMISSIBILITY AND EXPONENTIAL DICHOTOMY OF EVOLUTIONARY PROCESSES ON HALFLINE
"... Abstract. In the present paper we give a new way to characterize the exponential dichotomy of evolutionary processes in terms of ”Perrontype” theorems, without the socalled evolution semigroup. Also, there are obtained another proofs of some results gives by Van Minh, Räbiger and Schnaubelt. 1. ..."
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Abstract. In the present paper we give a new way to characterize the exponential dichotomy of evolutionary processes in terms of ”Perrontype” theorems, without the socalled evolution semigroup. Also, there are obtained another proofs of some results gives by Van Minh, Räbiger and Schnaubelt. 1.
Exponential Dichotomy Of Cocycles, Evolution Semigroups, And Translation Algebras
, 1998
"... . We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space \Theta acting on a Banach space X . Our main tool is the associated evolution semigroup on C 0 (\Theta; X). We prove that the cocycle has exponentia ..."
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. We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space \Theta acting on a Banach space X . Our main tool is the associated evolution semigroup on C 0 (\Theta; X). We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mapping/annular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 (Z; X). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, exponential dichot...
EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS TO SOME CLASSES OF NONAUTONOMOUS HIGHERORDER DIFFERENTIAL EQUATIONS
"... Abstract. In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper. 1. ..."
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Abstract. In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper. 1.
C. Chilarescu – A. Pogan – C. Preda ∗ A CHARACTERIZATION OF THE EXPONENTIAL STABILITY OF EVOLUTIONARY PROCESSES IN TERMS OF THE
"... Abstract. A characterization of the exponential stability of evolutionary rocesses in terms of the admissibility of some pairs of spaces, is given. The method of ”test functions ” from a very large class of spaces is used. Thus are obtained generalizations of some results given by N. van Minh, F. Ra ..."
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Abstract. A characterization of the exponential stability of evolutionary rocesses in terms of the admissibility of some pairs of spaces, is given. The method of ”test functions ” from a very large class of spaces is used. Thus are obtained generalizations of some results given by N. van Minh, F. Rabiger and R. Schnaubelt. 1.