Relative critical points

Relative equilibria of Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic, Poisson, variational - generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings.

Treating the (dual) algebra elements as parameters yields functions invariant only with respect to the isotropy subgroup of the given parameter; if the algebra elements are regarded as variables transformed by the (co)adjoint action, the relevant functions are invariant with respect to the full symmetry group. A generating set of invariant functions can be used to reverse the usual perspective: rather than seeking the critical points of a specific function, one can determine famililies of functions that are critical on specified orbits. This approach can be used in the design of conservative models when the underlying dynamics must be inferred from limited quantitative and/or qualitative information.

Optimal control with moderation incentives

Optimal solutions of generalized time minimization problems, with purely state-dependent cost functions, take control values on the boundary of the admissible control region. Augmenting the cost function with a control-dependent term rewarding sub-maximal control utilization moderates the response. A moderation incentive is a cost term of this type that is identically zero on the boundary of the admissible control region.

Two families of moderation incentives on spheres are considered here: the first, constructed by shifting a quadratic control cost, allows piecewise smooth solutions with controls moving on and off the boundary of the admissible region; the second yields solutions with controls remaining in the interior of the admissible region. Two simple multi-parameter control problems, a controlled velocity interception problem and a controlled acceleration evasion problem, illustrate the approach.