Extended Research Statement

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 [67]. 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 are endowed with 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 of these structural properties as possible, and towards this end we developed a discrete theory of abelian Routh reduction [31] for systems with symmetry, as well as a generalization of variational integrators to flows on Lie groups and homogeneous spaces [22; 34; 39; 40; 50]. More generally, generalized Galerkin variational integrators [60] 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. It is also possible to handle degenerate Hamiltonian systems without the need to transform into the Lagrangian framework [66]. Computational Geometric Control Theory. By developing a theory of geometric control that is predicated on the discrete-time dynamics associated with variational integrators, and does not introduce any additional approximations, one obtains numerical control algorithms which are geometric structure-preserving and exhibit good long-time behavior. This approach allows us to develop a discrete method of controlled Lagrangians [7–9; 11] for stabilizing relative equilibria. Motivated by applications in robotics, astrodynamics, and autonomous vehicles, we have also developed discrete optimal control on Lie groups [10; 30; 35; 36; 44; 49] and in that setting, studied geometric phase based control [43], reconfiguration of formations of spacecraft using combinatorial optimization [41], time optimal control [45], articulated multi-body systems in space [47] and in perfect fluids [51; 53], tethered spacecraft [52; 55; 56], and autonomous UAVs [54; 58; 59]. We also studied global state estimation [37; 42; 48] and uncertainty propagation on Lie groups [38; 46]. Immediate Research Plans. We have developed discrete Dirac mechanics and discrete Dirac structures [62; 63], 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. Hierarchical, modular and interconnected models of complicated engineering systems are ubiquitous, and the immediate plan is to generalize discrete Dirac mechanics to the case of interconnected Lagrange–Dirac mechanical systems, which will lead to a unified, and intrinsically modular treatment of interconnected systems in continuous time, discrete time, and in computations. This will provide a Lagrangian alternative to the port-Hamiltonian framework, which will more naturally handling covariant systems, allow more general interconnection constraints, and have both a variational and geometric description of the dynamics. Discrete Dirac structures are intimately related to the geometry of symplectic maps and generating functions, and this provides a natural framework for developing a discrete Hamilton–Jacobi theory [81; 82] and its nonholonomic generalization, as well as the discrete Hamilton–Jacobi–Bellman equation. A combination of discrete Hamilton–Jacobi theory and discrete Hamiltonian mechanics [66] leads to arbitrarily high order of accuracy generalizations of the Bellman equation.