Abstract
Nonlinear boundary stabilization for semilinear Euler-Bernoulli equations is considered. The main new feature of the problem is that the nonlinear stabilization is achieved via boundary moments only and the decay rates are established in the natural energy space associated with the model. This is in contrast with previous literature, where the control of moments and displacements was necessary for stabilization and, moreover, the obtained decay rates were obtained in the sense of distributions.
The need for stability results achieved via moments only and expressed in terms of the topology associated with natural finite energy function is dictated by several applications arising in the area of control of “smart materials” where PZT technology is applied. On the mathematical side, the said stability results are obtained with a help of microlocal analysis which is necessary to establish several new “trace” estimates which do not follow from standard PDE/trace theory. These trace estimates are, in turn, critical in obtaining the main “stabilizability” inequality, reconstructing the energy from the measurements of the moments only.
This research was supported by NSF Grant DMS-9504822 and the Army Research Office Grant DAAH04-96-1-0059.
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Ji, G., Lasiecka, I. (1998). Energy Decay in H 2 × L 2 for Semilinear Plates with Nonlinear Boundary Dissipation Acting via Moments Only. In: Optimal Control. Applied Optimization, vol 15. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6095-8_11
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DOI: https://doi.org/10.1007/978-1-4757-6095-8_11
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