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On the observability of time-discrete conservative linear systems. (English) Zbl 1143.65044

Summary: We consider various time discretization schemes of abstract conservative evolution equations of the form \(\dot z= Az\), where \(A\) is a skew-adjoint operator. We analyze the problem of observability through an operator \(B\). More precisely, we assume that the pair \((A,B)\) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by N. Burq and M. Zworski [J. Am. Math. Soc. 17, No. 2, 443–471 (2004; Zbl 1050.35058)]. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations.

MSC:

65J10 Numerical solutions to equations with linear operators
93B07 Observability
46N40 Applications of functional analysis in numerical analysis
47N40 Applications of operator theory in numerical analysis
35Q53 KdV equations (Korteweg-de Vries equations)
93C55 Discrete-time control/observation systems

Citations:

Zbl 1050.35058
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References:

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