We consider elliptic and parabolic equations with infinitely many variables. We prove some results of existence, uniqueness and regularity of solutions.

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator $K$. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un $C_0$-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case $\parallel K\parallel \ge 1$. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the...

To any bounded analytic semigroup on Hilbert space or on $L^p$-space, one may associate natural ’square functions’. In this survey paper, we review old and recent results on these square functions, as well as some extensions to various classes of Banach spaces, including noncommutative $L^p$-spaces, Banach lattices, and their subspaces. We give some applications to $H^{\infty }$ functional calculus, similarity problems, multiplier theory, and control theory.

We study stability and integrability of linear non-autonomous evolutionary Cauchy-problem $$\left(\mathrm{P}\right)\left\{\begin{array}{c}\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)t\text{-a.e.}\text{on}[0,\tau ],u\left(0\right)=0,\hfill \end{array}\right.$$ where $A:[0,\tau ]\to ℒ(X,D)$ is a bounded and strongly measurable function and $X$, $D$ are Banach spaces such that $D\underset{d}{\to }\hookrightarrow X$. Our main concern is to characterize $L^p$-maximal regularity and to give an explicit approximation of the problem (P).

This paper is concerned with systems with control whose state evolution is described by linear skew-product semiflows. The connection between uniform exponential stability of a linear skew-product semiflow and the stabilizability of the associated system is presented. The relationship between the concepts of exact controllability and complete stabilizability of general control systems is studied. Some results due to Clark, Latushkin, Montgomery-Smith, Randolph, Megan, Zabczyk and Przyluski are generalized....

In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight $(1-\mu )$ is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a C0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert...

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 ${W_t^H}_{t\in [0,T]}$ 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) $d{X}_t=A{X}_{t}dt+Bd{W}_t^H$ (t∈ [0,T]), $X_0=0$ almost surely, where A is the generator of a $C_0$-semigroup ${S\left(t\right)}_{t\ge 0}$ of bounded linear operators on...

Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $ℒ(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac{1}{2}$ we show that a function $\Phi :(0,T)\to ℒ(H,E)$ 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...

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...