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Sur les sentinelles des systèmes distribués. Le cas des conditions initiales incomplètes. (On the sentinels of distributed systems. The situation where initial conditions are incomplete). (French) Zbl 0664.93041

The author presents a new concept in the theory of control of distributed parameter systems: the sentinels. Sentinels are functionals of the state trajectory which are as close as possible to means and which are insensitive to unknown data of the problem. Here is treated the case where initial conditions are incomplete. Other examples are presented in a following note [ibid., No.17, 865-870 (1988)].
More precisely, let the state y be given by \(y'+{\mathcal A}y=\bar Q\) in \(\Omega\) \(\times (0,T)\) with the initial condition: \(y(0)=\bar y^ 0+\tau \hat y^ 0\) where \(\tau\) is small. Let \(\omega\) be an open set \(\omega\) \(\subset \Omega\) and \(h_ 0\geq 0\) function in \(\omega\) \(\times [0,t)\) such that: \[ \iint_{\omega \times (0,T)}h_ 0 dx dt=1; \] a sentinel is defined by: \[ {\mathcal S}(\tau;\hat y^ 0;w)=\iint_{\omega \times (0,T)}(h_ 0+w)y dx dt, \] in which w is determined in such a way that: \[ (d/d\tau){\mathcal S}(\tau;\hat y_ 0;w)|_{\tau =0}=0\quad for\quad all\quad \hat y_ 0 \] and that \(\| w\|_{L^ 2(\omega \times (0,T))}\) be minimum. It turns out that this is a problem of exact controllability either standard or new and it can be solved by the HUM method developed by the author.
In a more recent paper, the definition has been modified in order that the sentinels remain sensitive to some input of the system [Proc. 5th Symp. Control Distrib. Parameter Syst. (Perpignan 1989)].
Reviewer: J.Henry

MSC:

93C20 Control/observation systems governed by partial differential equations
49J20 Existence theories for optimal control problems involving partial differential equations
93B35 Sensitivity (robustness)
49K40 Sensitivity, stability, well-posedness
93B05 Controllability
93B03 Attainable sets, reachability
35B37 PDE in connection with control problems (MSC2000)
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