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Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

Zdzisław Brzeźniak, Jan van Neerven (2000)

Studia Mathematica

Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process W t H t [ 0 , T ] with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) d X t = A X t d t + B d W t H (t∈ [0,T]), X 0 = 0 almost surely, where A is the generator of a C 0 -semigroup S ( t ) t 0 of bounded linear operators on...

Stochastic evolution equations driven by Liouville fractional Brownian motion

Zdzisław Brzeźniak, Jan van Neerven, Donna Salopek (2012)

Czechoslovak Mathematical Journal

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ( H , E ) -valued functions with respect to H -cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1 . For 0 < β < 1 2 we show that a function Φ : ( 0 , T ) ( H , E ) is stochastically integrable with respect to an H -cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H -cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...

Stochastic integration of functions with values in a Banach space

J. M. A. M. van Neerven, L. Weis (2005)

Studia Mathematica

Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process W H ( t ) t [ 0 , T ] . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is...

Strong stabilization of controlled vibrating systems

Jean-François Couchouron (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with feedback stabilization of second order equations of the form ytt + A0y + u (t) B0y (t) = 0, t ∈ [0, +∞[, where A0 is a densely defined positive selfadjoint linear operator on a real Hilbert space H, with compact inverse and B0 is a linear map in diagonal form. It is proved here that the classical sufficient ad-condition of Jurdjevic-Quinn and Ball-Slemrod with the feedback control u = ⟨yt, B0y⟩H implies the strong stabilization. This result is derived from a general compactness...

Strong stabilization of controlled vibrating systems

Jean-François Couchouron (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with feedback stabilization of second order equations of the form ytt + A0y + u (t) B0y (t) = 0, t ∈ [0, +∞[, where A0 is a densely defined positive selfadjoint linear operator on a real Hilbert space H, with compact inverse and B0 is a linear map in diagonal form. It is proved here that the classical sufficient ad-condition of Jurdjevic-Quinn and Ball-Slemrod with the feedback control u = ⟨yt, B0y⟩H implies the strong stabilization. This result is derived from a general compactness theorem...

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

Let Y be a Banach space and let S L p be a subspace of an L p space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to S ( Y ) L p ( Y ) . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and e - t B is a positive contraction on L p for any...

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