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This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise: equivalence by endogenous dynamic feedback) is conjugation by diffeomorphisms. These diffeomorphisms are very much related to Lie-Bäcklund transformations. It is proved in this framework that dynamic equivalence of single-input systems is the same as static equivalence.
This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and two controls. Some of them are known to be flat, and this implies
admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a
lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about “-flatness”...
This paper presents a method to design explicit control Lyapunov functions for affine
and homogeneous systems that satisfy the so-called “Jurdjevic-Quinn conditions”.
For these systems a positive definite function
is known that can only be made
non increasing by feedback. We describe how a control Lyapunov function can be
obtained a deformation of this “weak” Lyapunov function. Some examples are
presented, and the linear quadratic situation is treated as an illustration.
The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on (3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S associated to the dynamics...
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