Stanislaw Walczak

Application of 2D systems to investigation of a process of gas filtration

In the paper, a process of gas filtration described by the 2D system with controls is considered. Sufficient conditions for the existence of optimal process are proved.

On helly's theorem for functions of several variables and its applications to variational problems ∗

In this paper prove Hellys principle of choice for functions of two variables. This theroem is applied to the investigation of some variational problems. An application is given to a gas absorption problem

On the Diffeomorphisms Between Banach and Hilbert Spaces

Abstract In this paper, we give some sufficient conditions for f : X → H to be a diffeomorphism, where X is a Banach space and H is a Hilbert space. The proof of the result is based on the mountain pass theorem. Using this result, in the final part of the paper, we prove an existence theorem for some class of integro-differential equations.

On the existence of a solution for some distributed optimal control hyperbolic system

We consider a Bolza problem governed by a linear time-varying Darboux-Goursat system and a nonlinear cost functional, without the assumption of the convexity of an integrand with respect to the state variable. We prove a theorem on the existence of an optimal process in the classes of absolutely continuous trajectories of two variables and measurable controls with values in a fixed compact and convex set.

Fractional Sobolev Spaces via Riemann-Liouville Derivatives

Using Riemann-Liouville derivatives, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones. Next, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and compactness of some imbeddings. An application to boundary value problems is given as well.

On some properties of Goursat–Darboux systems with distributed and boundary controls

In this paper, a Goursat–Darboux control system is considered. In the first part, the bang-bang principle and some approximation results concerning the piecewise constant controls, for a linear system with distributed and boundary controls, are proved. In the second part, an approximation result concerning the bang-bang controls, for a special case of nonlinear (in a state) system with distributed controls, is derived.

Nonlinear elliptic systems with variable boundary data

In this paper, a boundary value problem for elliptic systems is considered. Some sufficient conditions under which the solutions of this problem continuously depend on boundary data are proved.

A fractional imbedding theorem

In this paper we give an answer for the following question: when

Compactness of fractional imbeddings

AC_{a + }^\alpha \subset AC_{a + }^\beta ,