arXiv subjects

Robert Vrabel

Publications and source records attributed to Robert Vrabel.

22 records · Page 2Linked to original sources

arXiv2017

Singularly perturbed linear Neumann problem with the characteristic roots on the imaginary axis. A non-resonant case

In this note we are dealing with the problem of existence and asymptotic behavior of solutions for the non-resonant singularly perturbed linear Neumann boundary value problem \begin{eqnarray*} εy"+ky=f(t),\quad k>0,\quad 0<ε<<1,\quad t\in\langle a,b\rangle \end{eqnarray*} \begin{equation*} y'(a)=0,\quad y'(b)=0. \end{equation*} Our approach is based on the analysis of an integral equation equivalent to this problem.

math.CA↗

arXiv2017

On the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems

A new systematic approach to the construction of approximate solutions to a class of nonlinear singularly perturbed feedback control systems using the boundary layer functions especially with regard to the possible occurrence of the boundary layers is proposed. For example, problems with feedback control, such as the steady-states of the thermostats, where the controllers add or remove heat, depending upon the temperature registered in another place of the heated bar, can be interpreted with a second-order ordinary differential equation subject to a nonlocal three--point boundary condition. The $O(ε)$ accurate approximation of behavior of these nonlinear systems in terms of the exponentially small boundary layer functions is given. At the end of this paper, we formulate the unsolved controllability problem for nonlinear systems.

math.OC↗

arXiv2017

Frequency control of singularly perturbed forced Duffing's oscillator

We analyze the dynamics of the forced singularly perturbed differential equation of Duffing's type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative. We give some generalizations of results obtained recently by B.S. Wu, W.P. Sun and C.W. Lim, Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations, Journal of Sound and Vibration, Volume 307, Issues 3-5, (2007), pp. 953-960. The new method for an analysis of the nonlinear oscillations which is based on the dynamic change of coordinates is proposed.

math.DS↗

arXiv2017

Generalization of the Matrix Determinant Lemma and its application to the controllability of single input control systems

Linear control theory provides a rich source of inspiration and motivation for development in the matrix theory. Accordingly, in this paper, a generalization of Matrix Determinant Lemma to the finite sum of outer products of column vectors is derived and an alternative proof of one of the fundamental results in modern control theory of the linear time--invariant systems $\dot x=Ax+Bu,$ $y=Cx$ is given, namely that the state controllability is unaffected by state feedback, and even more specifically, that for the controllability matrices $\mathcal{C}$ of the single input open and closed loops the equality $\det\left(\mathcal{C}_{(A,B,C)}\right)$ $=\det\left(\mathcal{C}_{(A-BK,B,C)}\right)$ holds.

math.OC↗