Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. (English) Zbl 0643.93050

Datko, Richard

SIAM J. Control Optimization 26, No. 3, 697-713 (1988).

The paper under review can be viewed as a continuation of a previous paper by the author together with J. Lagnese and M. Polis [ibid. 24, 152-156 (1986; Zbl 0592.93047)]. Here it is shown that the previous result is by no means a singular example of destabilizing effect of arbitrary small decays in the feedback control. The author provides, as a one-dimensional example, the cantilevered Euler-Bernoulli beam with the canonical feedback through the bending moments at the free end, giving uniform exponential stability. It is shown that an arbitrary small delay in the feedback may lead to exponential instability of the system. A similar result is given for a rectangular membrane. A more complete picutre is obtained for distributed control stabilization problems, where destabilization by delay in the control appears to be a common feature. A final example (string) explicates the additional difficulties one meets in boundary control problems.

Reviewer: G.Leugering

MSC:

93D15	Stabilization of systems by feedback
93C20	Control/observation systems governed by partial differential equations
35L10	Second-order hyperbolic equations
74H45	Vibrations in dynamical problems in solid mechanics
93B35	Sensitivity (robustness)

Keywords:

cantilevered Euler-Bernoulli beam; delay in the feedback; exponential instability; distributed control stabilization; delay in the control

Citations:

Zbl 0592.93047