Boundary control of spherically symmetric oscillations of a three-dimensional ball (Q2783072)

The author has displayed in the past a unique talent for looking at problems that are regarded as routine and discovering a new viewpoint, thus arriving at new and more general results. Here he continues developing a theory based on his studies of the vibration of a string. He considers oscillations of the three-dimensional ball described by the equations \(u_{tt}- u_{rr}- 2u_r/r= 0\), with \(u(r,0)= \varphi(r)\), \(u_t(r,0)= \psi(r)\), \(u(R,t)= \mu(t)\), \(0\leq r\leq R\), \(0\leq t\leq T\). Problems with these kinds of boundary and initial conditions are called ``mixed problems''. The main problem consists of finding the minimal time interval \([0,T]\) sufficient for attaining the prescribed final boundary state \(u(r,T)\), \(u_t(r,T)\). In analogy to his treatment of the vibrating string, he defines three types of mixed problems. As in d'Alambert's solution, the generalized solutions represent the superposition of two ``waves''. The minimal time duration for guiding the system to the required state by means of boundary control while insisting on bounded energy is derived.NEWLINENEWLINENEWLINEThe reviewer has minor objections to some terminology of the translator. He would prefer to see the original term used by the author: ``A three-dimensional ball'', to the translator's ``3-ball'', and the expression ``an arbitrarily given state'' to the translator's ``arbitrarily present state'', etc.NEWLINENEWLINENEWLINEFor the most part, the article has been translated correctly, and so this criticism of the translation is not serious.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].