| This dissertation contributes to the areas of discrete mechanics and geometric computational control theory, addressing stabilization of relative equilibria of discrete mechanical systems. Recent studies have shown that the energy shaping is an effective approach to stabilization of equilibria and relative equilibria of mechanical systems. Mechanical systems have a very reach structure, and taking this structure into account when numerical models are designed is important. Examples of quantities/structures to preserve are the momentum and, in some cases, the symplectic form and the phase volume. Generic algorithms typically do not preserve these structures. In our investigation, we develop discrete analogues of energy shaping techniques of Bloch, Leonard, and Marsden. The major contributions are the use of nonconservative forcing for energy shaping in the discrete setting and the development of discrete models that preserve the stability properties of the related continuous-time systems. |
