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 ${W_H\left(t\right)}_{t\in [0,T]}$. 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...

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...

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...

Let Y be a Banach space and let $S\subset L_p$ 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 $S\left(Y\right)\subset L_p\left(Y\right)$. 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 $e^{-tB}$ is a positive contraction on $L_p$ for any...

We show that the domain of the Ornstein-Uhlenbeck operator on $L^p$$({ℝ}^N,\mu dx...$

We establish existence of mild solutions for the semilinear first order functional abstract Cauchy problem and we prove that the set of mild solutions of this problem is connected in the space of continuous functions.

We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method is investigated.

Si considera, in uno spazio di Hilbert $H$ l'operatore lineare ${\mathfrak{M}}_0\phi =1/2\text{Tr}\left[D^2\phi \right]+〈x,AD\phi 〉-〈DU\left(x\right),D\phi 〉$, dove $A$ è un operatore negative autoaggiunto e $U$ è un potenziale che soddisfa a opportune condizioni di integrabilità. Si dimostra con un metodo analitico che ${\mathfrak{M}}_0$ è essenzialmente autoaggiunto in uno spazio $L^2\left(H,\nu \right)$ e si caratterizza il dominio della sua chiusura $\mathfrak{M}$ come sottospazio di $W^{2,2}\left(H,\nu \right)$. Si studia inoltre la «spectral gap property» del semigruppo generato da $\mathfrak{M}$.

We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution $u$ which is bounded with values in $C^{2+\theta }\left(\overline{\mathrm{\Omega }}\right)$(0 < < 1), with${\partial }_{t}u$ bounded with values in $C^{\theta }\left(\overline{\mathrm{\Omega }}\right)$.