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

  • L. V. Fardigola

Abstract

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.
Published
25.08.2018
How to Cite
FardigolaL. V. “Transformation Operators in Controllability Problems for the Degenerate Wave Equation With Variable Coefficients”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, no. 8, Aug. 2018, pp. 1128-42, http://umj.imath.kiev.ua/index.php/umj/article/view/1623.
Section
Research articles