Stability of mechanical systems with kinetic minus potential energy Lagrangians is a classical subject. In the best cases, stability (and more) can be directly inferred from the properties of the potential function. For this reason, the idea of using feedback to shape the closed-loop energy of a mechanical system is an attractive thing to do. If one wishes to only shape the potential energy, then it is possible to understand the set of closed-loop potential functions using the classical Frobenius Theorem; this has been known since the work of van der Schaft in 1986. However, the set of systems that can be stabilised using energy shaping is enlarged if one also allows the shaping of the closed-loop kinetic energy. Unfortunately (or fortunately if you like geometry), this complicates the problem enormously by its leading to overdetermined quasi-linear partial differential equations for the closed-loop kinetic energy.
In this talk we will discuss some of the many open problems in this idea of energy shaping. We will also indicate what few results exist concerning the integrability of the energy shaping partial differential equations.
No online version avaliable.