Examples of S—expansions of Lie Algebras

In this note, we give examples of S—expansions of Lie algebras of finite and infinite dimension. For the finite dimensional case, we expand all real three-dimensional Lie algebras. In the case of infinite dimension, we perform contractions obtaining new Lie algebras of infinite dimension.

Infinite Geraghty type extensions and their applications on integral equations

In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018), (Fisher in Math. Mag. 48(4):223–225, 1975), and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). We also prove a similar Geraghty type construction for Fisher (Math. Mag. 48(4):223–225, 1975) in an infinite dimension using similar techniques as in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018) and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). As an application, we ensure the existence of solutions for infinite dimensional Fredholm integral equation and Uryshon type integral equation.

Nonlinear small-gain theorems for input-to-state stability of infinite interconnections

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.

Ornstein–Uhlenbeck semigroups in infinite dimension

This is a survey paper about Ornstein–Uhlenbeck semigroups in infinite dimension and their generators. We start from the classical Ornstein–Uhlenbeck semigroup on Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for Ornstein–Uhlenbeck semigroups on Banach spaces. This article is part of the theme issue ‘Semigroup applications everywhere’.

Theoretical observers for infinite dimensional skew-symmetric systems

The observer construction has a main importance in the control theory and its applications for the systems of infinite dimension. Even if the system’ state has an infinite dimension, its estimation is given using some physical measures of finite dimensions. Considering unbounded boundary observations operators and assuming that the exact observability property holds, we build some Luenberger like observers which assure the exponential stability of the error system under some regularity conditions.