Adaptive Control: Stability, Convergence, and Robustness

Abstract

The most significant contribution to the recorded ground displacement during an earthquake comes from surface waves propagating along the ground.A consistent mathematical theory describing such surface wave propagation, then, is at the heart of seismology.Since the Earth can be approximated by a layered, elastic half-space that is infinitely deep, this problem can be handled as a special waveguide problem.The theory of open waveguides is of exceptional importance to this effort.Thus a book describing waveguide theory applied to geology really is quite useful and quite logical.The Earth is a highly dispersive system.A temporally short impulse in the source region reaches a distant observer as a disturbance of long duration.The dispersion, which affects surface waves, also is a typical waveguide phenomenon.For this reason, it is not surprising that surface waves are suited to treatment using the theory of waveguides.Such a treatment is presented in great mathematical detail in the book under review.Some general remarks are made about waveguides with discontinuities in the Introduction.In Chap.2, Malischewsky considers boundary conditions and their seismological interpretation, and concludes with differential operators in closed domains and in open domains.The real mathematical ingenuity begins with Chap.3, "The Undisturbed Waveguide."After demonstrating that waveguides exhibit an eigenspectrum, Malischewsky shows that the formalism of quantum mechanics is appropriate for their description.The list of topics covered is suggestive of a quantum mechanics text: 3.1.The mode conception .... 3.2.2.The eigenspectrum .... 3.2.3.The eigenfunctions .... 3.2.4.Orthogonality and normalization .... and 3.3.6.The radiation conditions.