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operators and completely integrable nonlinear lattices
 Mathematical Surveys and Monographs
, 2000
"... to post this online edition! This version is for personal use only! If you like this book and want to support the idea of online versions, please consider buying this book: ..."
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Cited by 149 (43 self)
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to post this online edition! This version is for personal use only! If you like this book and want to support the idea of online versions, please consider buying this book:
the H ∞ calculus and sums of closed operators
 LeM] [Mar] [McI] [PT] [She] [Ste] [ST] [Wei] C. Le
"... Abstract. We develop a very general operatorvalued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ calculus. Using this we prove theorem of DoreVenni type on sums of commuting sect ..."
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Cited by 27 (4 self)
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Abstract. We develop a very general operatorvalued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ calculus. Using this we prove theorem of DoreVenni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the Rboundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting. 1.
Well Posedness For Damped Second Order Systems With Unbounded Input Operators
 DIFFERENTIAL AND INTEGRAL EQUATIONS
, 1995
"... We consider damped second order in time systems such as those arising in structures with piezoceramic actuators and sensors. These systems are naturally formulated as abstract second order systems with unbounded nonhomogeneous term. Existence, uniqueness and continuous dependence of solutions in a ..."
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Cited by 24 (17 self)
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We consider damped second order in time systems such as those arising in structures with piezoceramic actuators and sensors. These systems are naturally formulated as abstract second order systems with unbounded nonhomogeneous term. Existence, uniqueness and continuous dependence of solutions in a weak or variational setting are given. A semigroup formulation is presented and conditions under which the variational solutions and semigroup solutions are the same are discussed.
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Evolutionary Semigroups and Dichotomy of Linear SkewProduct Flows on Locally Compact Spaces with Banach Fibers
 J. DIFF. EQNS
, 1994
"... We study evolutionary semigroups generated by a strongly continuous semicocycle over a locally compact metric space acting on Banach bers. This setting simultaneously covers evolutionary semigroups arising from nonautonomuous abstract Cauchy problems and C 0 semigroups, and linear skewproduct ..."
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Cited by 16 (7 self)
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We study evolutionary semigroups generated by a strongly continuous semicocycle over a locally compact metric space acting on Banach bers. This setting simultaneously covers evolutionary semigroups arising from nonautonomuous abstract Cauchy problems and C 0 semigroups, and linear skewproduct flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skewproduct flow. To this end a Banach algebra of weighted composition operators is studied. The results are applied in the study of: "roughness" of the dichotomy, dichotomy and solutions of nonhomogeneous equations, Green's function for a linear skewproduct flow, "pointwise" dichotomy versus "global" dichotomy, and evolutionary semigroups along trajectories of the flow.
Sufficient Conditions For Exponential Stability And Dichotomy Of Evolution Equations
 Forum Math
, 1998
"... . We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u 0 (t) = A(t)u(t) () on a Banach space X . Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform const ..."
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Cited by 13 (7 self)
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. We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u 0 (t) = A(t)u(t) () on a Banach space X . Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform constants and A(\Delta) has a sufficiently small Holder constant, then () has exponential dichotomy. We further study robustness of exponential dichotomy under time dependent unbounded Miyaderatype perturbations. Our main tool is a characterization of exponential dichotomy of evolution families by means of the spectra of the socalled evolution semigroup on C 0 (R; X) or L 1 (R; X). 1. Introduction and preliminaries Exponential dichotomy is one of the fundamental asymptotic properties of solutions of the linear Cauchy problem (CP ) ae d dt u(t) = A(t)u(t); t ? s; u(s) = x in a Banach space X. It also plays an important role in the investigation of qualitative properties of nonlinear evolut...
Approximation in LQR Problems for Infinite Dimensional Systems With Unbounded Input Operators
, 1990
"... We present a variational framework based on sesquilinear forms for Galerkin approximation techniques for state feedback control in problems governed by infinite dimensional dynamical systems. Both parabolic and second order in time, hyperbolic partial differential equations with unbounded input and ..."
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Cited by 13 (6 self)
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We present a variational framework based on sesquilinear forms for Galerkin approximation techniques for state feedback control in problems governed by infinite dimensional dynamical systems. Both parabolic and second order in time, hyperbolic partial differential equations with unbounded input and unbounded observation operators are included as special cases of our treatment. 1 Introduction In this paper we discuss the linear quadratic regulator (LQR) problem for a class of (essentially parabolic) unbounded input or boundary control problems. A variational framework using sesquilinear forms is developed to treat Dirichlet and Neuman boundary control problems for parabolic equations and strongly damped elastic systems. Using such a framework, convergence of Galerkin approximations to solutions of Riccati equations is also established. The boundary control problem for parabolic systems has been studied extensively over the last two decades, inspired by the monograph of J.L. Lions [21] ...
Exponential Stability, Exponential Expansiveness, and Exponential Dichotomy of Evolution Equations on the HalfLine
"... For an evolution family on the halfline U = (U(t; s)) ts0 of bounded linear operators on a Banach space X we introduce operators G 0 ; GX and I X on certain spaces of X valued continuous functions connected with the integral equation u(t) = U(t; s)u(s) + R t s U(t; )f()d: We characterize exponen ..."
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Cited by 12 (5 self)
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For an evolution family on the halfline U = (U(t; s)) ts0 of bounded linear operators on a Banach space X we introduce operators G 0 ; GX and I X on certain spaces of X valued continuous functions connected with the integral equation u(t) = U(t; s)u(s) + R t s U(t; )f()d: We characterize exponential stability, exponential expansiveness and exponential dichotomy of U by properties of G 0 ; GX and I X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively. 1991 Mathematics Subject Classification: Primary: 34G10, 47D06, Secondary: 47H20 Keywords and Phrases: evolution equation, evolution family, evolution semigroup, exponential stability, exponential expansiveness, exponential dichotomy, integral equation. Introduction We consider the nonautonomous linear evolution equation d dt u(t) = A(t)u(t); t 2 J = R+ or R; (NCP ) This work was done while the first author was visiting the Department of Math...
Analytic semigroups: applications to inverse problems for flexible structures
 IN DIFFERENTIAL EQUATIONS WITH APPLICATIONS
, 1991
"... We present new convergence and stability results for least squares inverse problems involving systems described by analytic semigroups. The practical importance of these results is demonstrated by application to several examples from problems of estimation of material parameters in flexible structur ..."
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Cited by 10 (9 self)
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We present new convergence and stability results for least squares inverse problems involving systems described by analytic semigroups. The practical importance of these results is demonstrated by application to several examples from problems of estimation of material parameters in flexible structures using accelerometer data.
Robust stability of linear evolution operators on Banach spaces
, 1994
"... In this paper we introduce a stability radius for a wide class of linear infinitedimensional timevarying systems under structured timevarying perturbations. A framework is presented which allows the same degree of unboundedness in the perturbations as in the generator of the nominal model. Keyw ..."
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Cited by 7 (0 self)
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In this paper we introduce a stability radius for a wide class of linear infinitedimensional timevarying systems under structured timevarying perturbations. A framework is presented which allows the same degree of unboundedness in the perturbations as in the generator of the nominal model. Keywords: infinite dimensional systems, timevarying, evolution operators, Cauchy problem, robust stability, structured perturbations Notation X; X;X;U i ; Y i ; U; Y Banach spaces over K = R or K = C with norms k \Delta k X etc. L(X; Y ) (resp. L(X)) Banach space of bounded linear operators from X to Y (resp. on X) provided with the operator norm k \Delta kL(X;Y ) (resp. k \Delta kL(X) ) U(X) fL 2 L(X); L invertible in L(X)g, the set of invertible bounded linear operators on X provided with the operator norm C(s; t; X) The set of Xvalued continuous functions on [s; t]; s t 1 PC(R+ ; L(X; Y )) The set of piecewise continuous L(X; Y )valued operator functions on R+ = [0; 1) PC b (R + ;...