# Fardigola L. V.

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

↓ 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.

### Controllability problems for the string equation

Fardigola L. V., Khalina K. S.

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.

### 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.

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.

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

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.

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

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