The wave equation with a boundary control at both endpoints on an arbitrary time interval (Q2703876)

The author returns to the most basic wave equation, that is the vibration of a string, of length \(\ell\) given by \(u_{tt}- u_{xx}= 0\), on a fixed time interval \([0,T]\), with the initial conditions \(u(x,0)= \varphi(x)\in W^2_2(x)\), \(u_t(x, 0)= \psi(x)\in W^1_2(x)\), and also alternate finite conditions at time \(T: u(x, T)= \varphi_1(x)\in W^2_2(x)\), \(u_t(x, T)= \psi_1(x)\in W^1_2(x)\) and to the problem of boundary control of the vibrating string. Observe that the velocity of wave propagation along the string is silently assumed to be equal to one. Also, one observes that membership in the specified Sobolev spaces is a necessary condition for existence of solutions, simply because it assert the physical requirement that the energy of the system is finite.NEWLINENEWLINENEWLINEThis class of problems has been considered first by \textit{D. L. Russell} in 1965 [MRC Technical report \#566] and in later publications, \textit{J. L. Lions} [SIAM Rev. 30, 1-68 (1988; Zbl 0644.49028)] and by \textit{A. G. Butkovskij} in his 1985 monograph [Phase portraits of controlled dynamical systems (1985; Zbl 0579.93002)], and by other authors, who established existence of weak solutions in an appropriate Hilbert space. However, the present paper gives the most complete and detailed solution of the boundary control for the vibrating string with more explicit assumptions about the admissible classes of boundary control. The problem posed here is finding the explicit form of the boundary controls \(\mu(t)= u(0,t)\in W^2_2[0,T]\), and \(\nu(t)= u(\ell, t)\in W^2_2[0, T]\), matching a set of initial-boundary conditions assuring existence and uniqueness of solutions. The first problem assigns matching conditions at time \(t= 0\), \(\mu(0)= \varphi(0)\), \(\nu(0)= \varphi(\ell)\), \(\mu'(0)= \psi(0)\), \(\nu'(0)= \psi(\ell)\), while the second problem assigns similar conditions at time \(t= T\). The third problem concerns the solutions to the homogeneous vibrating string equations. Since the vibrating shape of the string advances at unit velocity, one can guess the relations between \(\ell\) and \(T\), which can be classified into separate cases \(\ell\leq T\), with the specific case \(\ell< T\leq 2\ell\), and \(\ell> T\). Nonexistence of solutions, existence without uniqueness, and existence and uniqueness are proved for the problems I, II and III listed above.NEWLINENEWLINENEWLINEThe paper reviewed here presents competent and careful analysis of the posed boundary control problems, ending in explicit formulas. In the opinion of the reviewer, not much is left to be said about these problems, that has not been solved by the author.