Computational methods for the fast boundary stabilization of flexible structures. Part 1: The case of beams

An efficient active control strategy for flexible systems [V. Komornik, Rapid boundary stabilization of linear distributed systems, SIAM J. Control Optim. 35 (5) (1997) 1591–1613] is thoroughly investigated from the numerical point of view. A non-standard computational framework is proved to be relevant both for simulation and control synthesis. The proposed formulation proves necessary in so far as a standard numerical approach is shown to fail. The observed properties of the state feedback law confirm the theory as far as beams are concerned. In particular, one can achieve an arbitrarily large decay rate of some weak norm of the system. Moreover, the control law compares favourably with the integral force feedback in terms of efficiency. Finally, smoothing procedures are introduced to decrease the control spill-over associated with the possible lack of compatibility between boundary control and initial conditions. This purely artificial spill-over is proved to result from oversimplification of the control process modelling.