Geometric Methods in the Control of PDEs

Technical Aspects

(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. The latter include coercivity (Lax-Milgram) type estimates for the static case, as well as the dynamic estimates described in more details in point b) below.
In shell theory, the goal is to relate the complex motion of a shell to simpler, better understood motions as far as possible. Recent research of the past few years based on the Bochner techniques in differential geometry, points to the expectation that the mathematical model of a shell can be intrinsically decomposed and expressed as a wave equation motion strongly coupled with a plate equation motion. As these latter dynamics are now well understood, the implication of this decomposition is that the study of a shell can then be drastically simplified, by bringing to bear a combination of mathematical techniques that have already proved successful in analyzing the two basic dynamical components.

(b) For linear homogeneous equations, the inequalities in question involve a suitable L-2 boundary trace being bounded from above (trace regularity) and from below (continuous observability inequality) by the initial energy. Reverse inequalities are related to certain inverse-type problems: recovery of the initial data from boundary observations. Corresponding reverse inequalities refer to solutions of originally conservative problems, which are subject to boundary or interior dissipation. The importance of these inequalities is that they guarantee regularity and, respectively, surjectivity of the input-solution map (from the boundary data to the solution) . In the dissipative case they guarantee uniform decay rates of solutions (uniform stabilization). These fundamental properties form the foundation for the theory of optimal control of PDEs. As a by-product, one obtains a solution of a stability problem: that of enforcing uniform decay rates for all solutions of originally conservative (energy preserving) PDEs. 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 inequalities in the general case of variable coefficients. Riemann geometric methods appear to bring two advantages:
(i) they essentially reduce the analysis to the constant coefficient 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.

(c) The overall system may consist either of two PDEs 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 PDE 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. This topic will also be the subject of an NSF-CBMS Conference, to be held at the University of Nebraska, Lincoln, 5-9 August 1999.

This conference appears to be at the leading edge of what the organizers hope will be a productive interdisciplinary field.

Of course, the introduction and use of differential-geometric methods in more general PDE 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, was largely unexplored until very recently. It opens up a highly promising area of research, with unexpected links which have been already made (e.g. the use of covariant differential calculus for Levi-Civita connections associated with the differential, variable coefficient, principal part, elliptic operator, to obtain general Carleman-type estimates for PDE solutions), with many more still to be discovered.

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 (ODEs) dates as far back as the early '70s. At present, the label "geometric control theory" is reserved to this established branch, whose status is well documented (see e.g. the proceedings of a Symposium "Motion, control, geometry", held in 1994 at the National Academy of Sciences in Washington, published by the Board of Mathematical Sciences, National Research Council, 1997.) However, in the case of control theory for ODEs, the role played by differential geometry is quite different from the one that occurs in PDEs.