Research Summary

My research has focused on developing the mathematical foundations of discrete geometry and mechanics to enable the systematic construction of geometric structure-preserving numerical schemes based on the approach of geometric mechanics, with a view towards obtaining more robust and accurate numerical implementations of feedback and optimal control laws arising from geometric control theory. This general approach is termed computational geometric mechanics, which is a subfield of geometric integration. It relies on a systematic and self-consistent discretization of geometry, mechanics, and control. The approach is based on discretizing Hamilton’s principle, which yields variational integrators that are automatically symplectic and momentum preserving, and exhibit good energy behavior for exponentially long times. Such discrete conservation laws typically impart long time numerical stability to computations, since they conserve exactly a discrete quantity that is always close to the continuous quantity of interest. Discrete Geometry. Classical field theories like electromagnetism and general relativity have a rich gauge symmetry, and it is important to distinguish between the physically relevant dynamics and the nonphysical gauge modes. A critical component to developing discretizations of field and gauge theories is the development of discrete exterior calculus [19; 20] and discrete connections on principal bundles [71]. This yields compatible discretizations of differential operators on unstructured meshes, and discrete analogues of Hodge– de Rham theory, the Poincaré lemma, Levi-Civita connections and their associated curvatures, as well as an explicit semi-global characterization of discrete reduced spaces arising in discrete reduction by symmetry. Discrete Mechanics. While Lagrangian systems have symplectic structures and momentum maps, they may also exhibit other important features like symmetries, nonlinear configuration spaces, or shocks and multiscale phenomena. It is desirable to preserve as many structural properties as possible, and towards this end a discrete theory of abelian Routh reduction was developed in [32] for systems with symmetry, as well as a generalization of variational integrators to flows on Lie groups and homogeneous spaces [22; 35; 40; 41; 51]. Two general approaches for constructing variational integrators have been developed, one based on the Galerkin approach, and the other on a boundary-value problem formulation. These methods are described in [64; 68], and examples of generalized Galerkin variational integrators that combine discrete variational mechanics with techniques from approximation theory and numerical quadrature, and yield adaptive symplectic-momentum methods for problems with multiple scales and shocks are described in [63]. Related techniques that use collocation on the prolongation of the Hamiltonian vector field are described in [67]. It is also possible to handle degenerate Hamiltonian systems without the need to transform into the Lagrangian framework [70]. A theory of discrete Dirac mechanics and discrete Dirac structures was developed in [65; 66], which provide a unified treatment of Lagrangian and Hamiltonian mechanics, and symplectic and Poisson structures, respectively. These results are related by the property that Hamilton–Pontryagin integrators preserve the discrete Dirac structure, and they provide the appropriate geometric framework for studying complex hierarchical interconnected systems. The foundations for extending Dirac structures and Dirac integrators to the setting of Hamiltonian PDEs is laid forth in [69; 95–97]. Another related aspect of the work focused on generating functionals in symplectic and multisymplectic maps, in the form of extensions of Hamilton–Jacobi theory to the setting of discrete [86; 87], and degenerate [72] systems. Computational Geometric Control Theory. A theory of geometric control was developed that builds upon the discrete-time dynamics associated with variational integrators, and which does not introduce any additional approximations. This results in numerical control algorithms which are geometric structurepreserving and exhibit good long-time behavior. This approach allowed for the development of a discrete method of controlled Lagrangians [7–9; 11] for stabilizing relative equilibria. Motivated by applications in robotics, astrodynamics, and autonomous vehicles, discrete optimal control techniques on Lie groups were developed in [10; 31; 36; 37; 45; 50] and various extensions were studied, such as geometric phase based control [44], reconfiguration of formations of spacecraft using combinatorial optimization [42], time optimal control [46], articulated multi-body systems in space [48] and in perfect fluids [52; 54], tethered spacecraft [53; 56; 57], and autonomous UAVs [55; 59; 61]. Additional applications include global state estimation [38; 43; 49] and uncertainty propagation on Lie groups [39; 47]. The geometric approach to the analysis of mechanical systems can also be used to provide important dynamical systems insights into problems evolving on nonlinear manifolds, as described in [15; 58].