Opuscula Mathematica
Loading...
ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2013
Volume
Vol. 33
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 33 (2013)
Projects
Pages
Articles
A note on Blasius type boundary value problems
(2013) Andrzejczak, Grzegorz; Nockowska-Rosiak, Magdalena Ewa; Przeradzki, Bogdan
The existence and uniqueness of a solution to a generalized Blasius equation with asymptotic boundary conditions is proved. A new numerical approximation method is proposed.
On vertex b-critical trees
(2013) Blidia, Mostafa; Eschouf, Noureddine Ikhlef; Maffray, Frédéric
A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that has neighbors of all other colors. The b-chromatic number of a graph $G$ is the largest $k$ such that $G$ admits a b-coloring with $k$ colors. A graph $G$ is b-critical if the removal of any vertex of $G$ decreases the b-chromatic number. We prove various properties of b-critical trees. In particular, we characterize b-critical trees.
The lq-controller synthesis problem for infinite-dimensional systems in factor form
(2013) Grabowski, Piotr
The general lq-problem with infinite time horizon for well-posed infinite-dimensional systems has been investigated by George Weiss and Martin Weiss and by Olof Staffans with a complement by Kalle Mikkola and Olof Staffans. Our aim in this paper is to present a solution of a general lq-optimal controller synthesis problem for infinite-dimensional systems in factor form. The systems in factor form are an alternative to additive models, used in the theory of well-posed systems, which rely on leading the analysis exclusively within the basic state space. As a result of applying the simplified analysis in terms of the factor systems and an another derivation technique, we obtain an equivalent, however, astonishingly not the same formulae expressing the optimal controller in the time-domain and the method of spectral factorization. The results are illustrated by two examples of the construction of both the optimal control and optimal controller for some standard lq-problems met in literature: a control problem for a class of boundary controlled hyperbolic equations initiated by Chapelon and Xu, to which we give full solution and an example of the synthesis of the optimal control/controller for the standard lq-problem with infinite-time horizon met in the problem of improving a river water quality by artificial aeration, proposed by Zołopa and the author.
Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type
(2013) Âkovlev, Sergej I.; Âkovleva, Valentina
It is shown that any $\mu \in \mathbb{C}$ is an infinite multiplicity eigenvalue of the Steklov smoothing operator $S_h$ acting on the space $L^1_{loc}(\mathbb{R})$. For $\mu \neq 0$ the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.
The maximum principle for viscosity solutions of elliptic differential functional equations
(2013) Karpowicz, Adrian
This paper is devoted to the study of the maximum principle for the elliptic equation with a deviated argument. We will consider viscosity solutions of this equation.