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Zbl 0891.93016
Liu, Kangsheng
Locally distributed control and damping for the conservative systems. (English)
[J] SIAM J. Control Optimization 35, No.5, 1574-1590 (1997). ISSN 0363-0129; ISSN 1095-7138/e

The paper deals with the controllability and exponential stabilizability for the control system $y' = Ay +Bu$ where $A$ generates a $C_0$-semigroup on a Hilbert space $H$ and $B\in L(U,H)$, U being a Hilbert space. When $A^{\ast} = -A$, the equivalence between controllability, exponential stabilizability, an observability inequality and a frequency domain condition is presented. A related result for boundary control systems has been given by {\it V. Komornik} [SIAM J. Control Optim. 35, No. 5, 1591-1613 (1997)]. When $A$ generates a group and the system is exactly controllable, it is shown that every $y_0 \in D(A)$ can be transferred to the origin by smooth controls. A frequency domain characterization for exactly controllable second order elastic systems is then given. The above results are applied to the study of the control and damping of the wave, Schrödinger and Petrovsky equations. Precisely, one obtains a sufficient condition for the subregion on which the application of the control (resp. damping) gives exact controllability (resp. the exponential decay property) on a not necessarily smooth region. Finally, a relationship between the controllability of the three control systems mentioned above is established.
[O.Cârjá (Iaşi)]

MSC 2000:: *93B05 Controllability
93D15 Stabilization of systems by feedback
35B40 Asymptotic behavior of solutions of PDE
35B37 PDE in connection with control problems
93C25 Control systems in abstract spaces

Keywords: conservative PDE; controllability; damping; exponential stabilizability; frequency domain condition; Petrovsky equations

Cited in: Zbl 1117.34054 Zbl 0959.93024

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