Maximally Degenerate Floquet Structure and Possible Nonexistence of Optimal Control in a Pendulum Swing-Up Problem

Abstract

This paper studies an infinite-horizon optimal control problem for a pendulum with quadratic control cost via its associated Hamiltonian system. The problem is strongly degenerate, as the linearization at the upright equilibrium has purely imaginary eigenvalues with multiplicity, making standard linear analysis inconclusive. Using Lie series and Chetaev's instability theorem, we show that the equilibrium is weakly unstable in a non-hyperbolic sense. We further identify a family of periodic orbits forming an invariant cylinder, whose monodromy matrix exhibits a maximally degenerate Floquet structure consisting of a single Jordan block at the unit multiplier. This structure implies slow, non-exponential dynamics and provides a mechanism for long-time transitions with arbitrarily small control effort. Although derived for a pendulum, this phenomenon arises from degeneracy in the optimal control formulation and may occur more broadly, suggesting the possible nonexistence of optimal control despite stabilizability.