Dorota Bors

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.

Global Solvability of Hammerstein Equations with Applications to BVP Involving Fractional Laplacian

Some sufficient conditions for the nonlinear integral operator of the Hammerstein type to be a diffeomorphism defined on a certain Sobolev space are formulated. The main result assures the invertibility of the Hammerstein operator and in consequence the global solvability of the nonlinear Hammerstein equations. The applications of the result to nonlinear Dirichlet BVP involving the fractional Laplacian and to some specific Hammerstein equation are presented.

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.

Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian

We consider a class of partial differential equations with the fractional Laplacian and the homogeneous Dirichlet boundary data. Some sufficient condition under which the solutions of the equations considered depend continuously on parameters is stated. The application of the results to some optimal control problem is presented. The methods applied in the paper make use of the variational structure of the problem.

Systems described by Volterra type integral operators

In the paper some sufficient condition for the nonlinear integral operator of the Volterra type to be a diffeomorphism defined on the space of absolutely continuous functions are formulated. The proof relies on consideration of the linearized equation together with Palais-Smale condition, thus a combination of topological and variational methods is used. The applications of the result to the control systems with feedback and to the specific nonlinear Volterra equation are presented.

On the controllability to the interval of the system governed by a hyperbolic equation

Purpose – The purpose of this paper is to present the sufficient condition for the controllability to the interval of the system defined by a nonlinear hyperbolic equation.Design/methodology/approach – The proof of the main theorem is based on the Miranda theorem which is a generalization of the Bolzano‐Weierstrass theorem.Findings – In the course of the work, it was found that under some assumptions a system is controllable to the interval with the aid of controls from the oblique interval spanned by the given controls. In the end of the paper, an example illustrating the proposed method is derived.Originality/value – This paper illustrates a procedure for finding a control which steers the system defined by a nonlinear hyperbolic equation to a given state.

Controllability of one-dimensional and two-dimensional systems

In the paper sufficient conditions for the controllability to the interval of two-dimensional continuous Roesser system and one-dimensional continuous system are proved. The idea of proof is based on applying a lemma which is a multidimensional version of Bolzano-Poincare theorem.

On Mayer problem for systems governed by second-order ODE

In this article, we consider an optimal control problem of Mayer type. We relax the standard growth condition imposed on the right-hand side of the equation of dynamics and provide an alternative approach to classical reasoning. Namely, we state some sufficient conditions for the existence of optimal solution for the aforementioned problem. An example illustrating controllability of an object with variable mass is also presented.

On some nonlinear second order control systems

In this paper, we consider control systems governed by second order differential equations. Sufficient conditions for the existence of solutions and the corresponding stability criteria are proved. Moreover, conditions for the existence of optimal processes are presented. The systems under investigation have natural technical and physical interpretation. The method and results of the paper may be generalised to the continuous 2D systems.