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Transformation operators in the problems of controllability for the degenerate wave equation with variable coefficients. (English. Ukrainian original) Zbl 1454.93110

Ukr. Math. J. 70, No. 8, 1300-1318 (2019); translation from Ukr. Mat. Zh. 70, No. 8, 1128-1142 (2018).
Summary: We study a control system \(w_{tt} = \dfrac{1}{\rho}(kw_x)_x + \gamma w\), \(w(0,t) = u(t)\), \(x \in (0,l)\), \(t \in (0,T)\), 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 for free time.

MSC:

93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
35L10 Second-order hyperbolic equations
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