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154
Jacobi operators and completely integrable nonlinear lattices
 MATHEMATICAL SURVEYS AND MONOGRAPHS
, 2000
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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 85 (12 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.
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 42 (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...
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 38 (6 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...
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 35 (20 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.
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 22 (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.
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 18 (8 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] ...
Receding horizon optimal control for infinite dimensional systems
, 2002
"... The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is veried provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as qu ..."
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Cited by 17 (3 self)
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The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is veried provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as quadratic functional. Applications to the NavierStokes equations, semilinear wave equations and reaction diffusion systems are given.
A solution to the problem of Lpmaximal regularity
"... Abstract. We give a negative solution to the problem of the Lpmaximal regularity on various classes of Banach spaces including Lqspaces with 1 < q 6 = 2 < +∞. 1. ..."
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Cited by 17 (2 self)
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Abstract. We give a negative solution to the problem of the Lpmaximal regularity on various classes of Banach spaces including Lqspaces with 1 < q 6 = 2 < +∞. 1.