Atlas home || Conferences | Abstracts | about Atlas
Host: University of Colorado
Sponsor: American Mathematical Society, National Science Foundation
Homepage: http://www.ams.org/meetings/src-gulliver.html
Email: wsd@ams.org
Organizers: Robert D. Gulliver (University of Minnesota, Minneapolis) (co-chair), Walter Littman (University of Minnesota, Minneapolis) (co-chair), Roberto Triggiani (University of Virginia) (co-chair)
Deadline for abstracts: March 03, 1999
Description:
The proposed conference seeks to explore the infusion of differential geometric
methods into the analysis of control theory problems for partial differential equations
(P.D.E.s). Very recent research supports the expectation that Bochner techniques in
differential geometry, when brought to bear on the classes of P.D.E.s modelling and
control problems discussed below, will yield significant mathematical advances. These
include:
(a) Intrinsic, coordinate-free models of (nonlinear) shells equations, more suitable for mathematical investigation than present, exceedingly complicated, coordinate-based models, mostly derived in the mechanical literature.
(b) A priori direct (trace regularity) and reverse (continuous observability) inequalities for mixed problems for second order hyperbolic P.D.E.s, Maxwell equations, plate-like equations, Schrodinger P.D.E.s (Petrowski-type) etc., defined on a multidimensional Euclidean domain, with emphasis on the variable coefficient case. In the case of dissipative systems, reverse inequalities, which are generally more challenging to achieve, yield energy decay (stabilization) results.
(c) Establishment of direct and reverse a priori inequalities for highly coupled systems of P.D.E.s arising in modern technological applications, such as: shell models and thermoelastic models defined on 2-dimensional surface-like domains; structurally acoustic models defined on acoustic 3-dimensional chambers with curved walls, possibly subject to thermoelastic effects; etc.
As to (a), the role of Riemann geometry is expected to be paramount in capturing geometric features of general shells, both static and dynamic; to express, in intrinsic form, the correct boundary conditions; and to establish the required estimates.
As to (b), very recent research has indicated that Riemann geometric methods can profitably be used to complement and extend known analysis-based methods of proving the a priori reverse inequalities in the general case of variable coefficients. Riemann geometric methods appear to bring a few advantages: (i) they essentially reduce the analysis to the constant coefficient principal part case, where strategies are well understood; (ii) they ultimately provide easier-to-verify conditions, with a distinct geometric flavor involving notions such as convexity in the Riemann metric and gaussian curvature; (iii) they require only a finite, natural degree of smoothness, rather than high smoothness as in pseudo-differential analysis.
As to (c), the overall system may consist either of two P.D.E.s of the same type (hyperbolic/hyperbolic coupling), or else of different type (hyperbolic/parabolic coupling), possibly defined on different contiguous domains, and with strong, possibly boundary, coupling. For example the elastic wall of an acoustic chamber may be subject to high internal damping, whereby the original plate equation becomes parabolic-like. This is a vastly open research topic for basic P.D.E.s theory in general, and for control and optimization theory in particular. Because of their original description on curved domains (manifolds), these problems appear particularly well suited for differential geometric methods to supplement analytic approaches, at the level of both modeling and analysis, including notions such as operators on manifolds, forms, gaussian curvature of the domain, etc. Here the difficulties are compounded over those described in point a) above for a single shell, since the shell may be just one component of a composite, highly coupled system.
The proposed exploratory conference will show once more that mathematical research knows no boundaries between specific disciplines--analysis versus differential geometry; and that potentially productive interactions may take place between P.D.E. control theory and differential geometry. These will be well served by expanding the traditionally analysis-based P.D.E.s approaches into fields such as Riemannian (and, in the time variable case, Lorentian) geometry.
Of course, the introduction and use of differential geometric methods in more general P.D.E.s theory has long been established. However, the use and role of differential geometric methods toward the solution of any of the modeling, control, and optimization problems mentioned in points a), b), c) above is largely unexplored. It opens up a highly promising area of research.
By contrast, the introduction of differential geometric methods in the study of control problems (such as exact controllability, feedback stabilization, optimization, filtering, etc.) for dynamical systems modeled by ordinary differential equations (O.D.E.s) dates as far back as the early 1970s.
In keeping with the stated character and goals, the proposed conference is intended to be highly focused and exploratory. The conference will feature high caliber speakers from both fields, geometry and P.D.E.s, and seeks likewise to attract a mixed audience of geometers and P.D.E. control theorists. A particular effort will be made to include a representative group of young mathematicians from both fields. The conference's distinctive theme will be "control of P.D.E.s opened to geometry, and geometry infused into P.D.E.s control".
Mail Address:
Summer Research Conferences Coordinator American Mathematical Society P.O. Box 6887 Providence, RI 02940
Date received: February 17, 1999
© 2008 Atlas Conferences Inc.