Energy shaping is a control strategy wherein one converts a given mechanical system (called the open-loop system) to another mechanical system (called the closed-loop system) with desired properties. For example, if the open-loop system possesses $q_0$ as an unstable equilibrium point, one may wish to design a feedback which makes $q_0$ stable for the closed-loop system. While the strategy is attractive in many respects, it is difficult to implement since it leads to a system of quasi-linear partial differential equations that are generally over-determined.

In this talk we will (1) give an affine differential geometric formulation of the energy shaping problem and (2) give a complete description of part of the problem using techniques from the formal theory of partial differential equations.

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