A deconvolution problem related to a singular system
The authors derive a recursive algorithm for solving the deconvolution problem \[ y(t)= \int^t_0 H(t-s) u(s)\,ds,\quad t> 0, \] assuming that the impulse response function \(H\) is generated by the singular system \[ {d\over dt} (My)+ Ly= Bu,\quad My(0)= 0. \] Here, \(L\) and \(M\) are closed linear operators on a Hilbert space, with \(L\) possessing a bounded inverse. A change of variables (given by \(Ly= x\)) is used to formulate the identification problem in a \(C_0\)-semigroup framework. The resulting algorithm is in the spirit of the one introduced in \textit{Yu. S. Osipov} and \textit{A. V. Kryazhimskii} [Inverse problems for ordinary differential equations: dynamical solutions (Gordon and Breach, Amsterdam) (1995; Zbl 0884.34015)]; see also the recent papers by \textit{V. Maksimov} and \textit{L. Pandolfi} [J. Optimization Theory Appl. 103, No. 2, 401--420 (1999; Zbl 0956.49020)] and by the authors [SIAM J. Control Optimization 40, No. 5, 1384--1405 (2002; Zbl 1018.93016)] on related algorithms for distributed and finite-dimensional systems, respectively.
scientific article; zbMATH DE number 2081782
scientific article