2019
Том 71
№ 6

All Issues

Fardigola L. V.

Articles: 5
Article (Ukrainian)

Transformation operators in controllability problems for the degenerate wave equation with variable coefficients

Fardigola L. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1128-1142

We study the control system $w_{tt} = \cfrac1{\rho } (kw_x) x + \gamma w,\; w(0, t) = u(t),\; x \in (0, l), t \in (0, T)$, in special modified spaces of the Sobolev type. Here, $\rho , k,$ and \gamma are given functions on $[0, l)$; $u \in L^{\infty} (0, T)$ is a control, and $T > 0$ is a constant. The functions $\rho$ and $k$ are positive on $[0, l)$ and may tend to zero or to infinity as $x \rightarrow l$. The growth of distributions from these spaces is determined by the growth of $\rho$ and $k$ as $x \rightarrow l$. Applying the method of transformation operators, we establish necessary and sufficient conditions for the $L^{\infty}$ -controllability and approximate $L^{\infty}$ -controllability at a given time and at a free time.

Article (Ukrainian)

Controllability problems for the string equation

Fardigola L. V., Khalina K. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 939–952

For the string equation controlled by boundary conditions, we establish necessary and sufficient conditions for 0-and ε-controllability. The controls that solve such problems are found in explicit form. Moreover, using the Markov trigonometric moment problem, we construct bangbang controls that solve the problem of ε-controllability.

Brief Communications (Ukrainian)

On the Possibility of Stabilization of Evolution Systems of Partial Differential Equations on $ℝ^n × [0, + ∞)$Using One-Dimensional Feedback Controls

Fardigola L. V., Sheveleva Yu. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1289-1296

We establish conditions for the stabilizability of evolution systems of partial differential equations on $ℝ^n × [0, + ∞)$ by one-dimensional feedback controls. To prove these conditions, we use the Fourier-transform method. We obtain estimates for semialgebraic functions on semialgebraic sets by using the Tarski–Seidenberg theorem and its corollaries. We also give examples of stabilizable and nonstabilizable systems.

Article (Russian)

Nonlocal two-point boundary-value problems in a layer with differential operators in the boundary condition

Fardigola L. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1122–1128

We obtain criteria of well-posedness and strong well-posedness (smoothing of solutions as compared with given functions) of boundary-value problems for linear partial differential evolution equations in an infinite layer. The boundary condition is nonlocal and gives a relation between the values of the unknown function and its derivatives with respect to spatial coordinates on shifts of connected components of the boundary of the layer inside the layer.

Article (Ukrainian)

Well-posed problems in a layer with differential operators in boundary conditions

Fardigola L. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1083–1090