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Sentinelles pour les systèmes distribués. À données incomplètes. (French) Zbl 0759.93043

Recherches en Mathématiques Appliquées. 21. Paris etc.: Masson. xii, 227 p. (1992).
It is the purpose of the book to present a method, called by the author “the method of sentinels”, by which information on unknown data in a system of evolution equations can be obtained. A typical example for the problems considered is the second order parabolic equation \[ u_ t+Au=f(u)+g\quad\text{for } 0<t<T,\;x\in\Omega, \tag{1a} \] with initial and boundary conditions \[ u=h\quad\text{for } t=0,\;x\in\Omega, \tag{1b} \]
\[ u=0\quad\text{for } 0<t<T,\;x\in\partial\Omega. \tag{1c} \] Here, the known quantities are the second order elliptic operator \(A\), the real valued function \(f\), the open set \(\Omega\subset R^ n\) and the constant \(T>0\), while the functions \(g\) and \(h\) are only incompletely known. Also known, by observation, are the values of the solution \(u\) in some sub- cylinder \((t_ 0,t_ 1)\times U\) of \((0,T)\times\overline\Omega\).
The most common method of taking advantage of this knowledge of \(u\) when trying to derive information on \(g\) and \(h\) is probably the method of least squares. In contrast to this method, where \(g\) and \(h\) play the same role, the method of sentinels tries to obtain information on \(g\) without information on \(h\) (and vice versa).
Sentinels have been introduced by the author in two papers [C. R. Acad. Sci., Paris, Ser. I, 307, 819-823 (1988; Zbl 0664.93041) and ibid. 307, 865-870 (1988; Zbl 0681.93005)]. In the present book, by working out a lot of examples, it is shown that the method can be applied to a wide range of (deterministic) problems. Roughly speaking, at present the method is applicable to a given problem provided the state \(u\) depends in a differentiable way on the unknown quantities (i.e. for (1): on \(g\) and \(h)\).
The book consists of six chapters: Chapts. 1-4 deal with dissipative problems. For the problems treated in Chapt. 1, (1) with \(U\) open in \(\Omega\) is a typical example. In Chapt. 3, unknown data appear also in the boundary conditions, while in Chapt. 4 the observation set \(U\) is assumed to lie on the boundary, and \((t_ 0,t_ 1)\) is allowed to shrink to a single point. In Chapt. 2, the quality of the information obtained by sentinels is investigated (e.g. the question: How many sentinels are necessary to have complete information on the unknown quantities?). In Chapt. 5, general evolution systems are considered (e.g. hyperbolic systems and Petrovsky type systems), while the last chapter deals with some new situations: incompletely known domains \(\Omega\) or operators \(A\). The book contains no index.

MSC:

93C20 Control/observation systems governed by partial differential equations
35R30 Inverse problems for PDEs
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
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