On the approximate calculation of time-minimal distributed controls of vibrations by finite element approximations. (English) Zbl 0769.93015
Summary: This paper is concerned with distributed null-control of vibrations governed by an abstract wave equation. Based on a method for the exact computation of minimum \(L^ 2\)-norm controls for given time intervals and time-minimal controls which are bounded with respect to the \(L^ 2\)- norm, an approximation method is developed which is based on Galerkin’s method and convergence results are derived.
MSC:
| 93B05 | Controllability |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
References:
| [1] | Collatz L (1949) Eigenwertaufgaben mit technischen Anwendungen. Akademische Verlagsgesellschaft Geest und Portig K-G: Leipzig. · Zbl 0035.17504 |
| [2] | Fattorini HO (1977) The time optimal problem for distributed control of systems governed by the wave equation. In: Aziz AK, Wingate JW, Balas, MJ (eds), Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York, pp 151-175. |
| [3] | Krabs W (1985) On time-minimal distributed control of vibrating systems governed by an abstract wave equation. Appl Math Optim 13:137-149. · Zbl 0569.49016 · doi:10.1007/BF01442203 |
| [4] | Lamp U (1987) Zur Berechnung zeitminimaler verteilter Steuerungen von schwingenden Systemen unter Verwendung der Finite Element Methode. Dissertation, Darmstadt. · Zbl 0681.73054 |
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