Nonlinearizable single-input control systems do not admit stationary symmetries

Geometric reduction in optimal control theory with symmetries

“…Moreover, Ψ being a symmetry of X means that the induced diffeomorphism ϕ is a symmetry of X in the sense of [18] and [28].…”

Normal Forms of Nonlinear Control Systems

“…As an application of canonical form, Respondek-Tall [29] and [30] studied the symmetry of nonlinear systems. For linearly controllable and analytic systems that are not feedback linearizable, the group of stationary symmetries contains at most two elements and the group of non stationary symmetries consist of at most two 1-parameter families.…”

A non-linear symmetry-preserving observer for velocity-aided inertial navigation

“…It has a strong differential geometric structure, which yields a local convergence property around any trajectory of the aircraft. Symmetries have been used in control theory for feedback design and optimal control, see for instance [3], [4], [8], [9], [5], [10] but seemingly much less for observer design [2], [1]. This paper also proposes a theory for invariant observers extending the ideas and results of [1]; it can be seen as the counterpart for observer design of [5] for invariant tracking controller design.…”