Ukrainian Mathematical Journal

, Volume 70, Issue 8, pp 1300–1318 | Cite as

Transformation Operators in the Problems of Controllability for the Degenerate Wave Equation with Variable Coefficients

  • L. V. Fardigola

We study a control system \( {w}_{tt}=\frac{1}{\rho }{\left(k{w}_x\right)}_x+\gamma w,w\left(0,t\right)=u(t),x\in \left(0,l\right),t\in \left(0,T\right), \) in special modified spaces of the Sobolev type. Here, ρ, k, and γ are given functions on [0, l) , u 𝜖 L(0,T) is a control, and T > 0 is a constant. The functions ρ and k are positive on [0, l) and may tend to zero or to infinity as xl. The growth of distributions from these spaces is determined by the growth of ρ and k as xl. Applying the method of transformation operators, we establish necessary and sufficient conditions for the L-controllability and approximate L-controllability at a given time and for free time.


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Authors and Affiliations

  • L. V. Fardigola
    • 1
  1. 1.Verkin Institute for Low Temperature Physics and EngineeringUkrainian National Academy of SciencesKharkivUkraine

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