On the relationship between stochastic and deterministic energy shaping techniques for stochastic port-Hamiltonian systems via invariant measure characterization

This paper derives two relevant results in the context of weak energy shaping for stochastic port-Hamiltonian systems (SPHS). Energy shaping is a technique systematically used to design feedback laws that shape the Hamiltonian of a controlled system to converge to a desired configuration under a general passivity condition. Such an approach has been recently extended to SPHS, but the resulting theory has limitations excluding relevant examples such as the additive noise case. However, it has been shown that the invariant measure of the SPHS allows the introduction of a weaker notion of convergence, hence obtaining a significant extension of the classical energy shaping theory for SPHS. In this paper, we continue investigating the weak energy shaping of SPHS, showing that the Fokker–Planck equation associated with an SPHS can be seen as the Fokker–Planck equation of an infinite-dimensional deterministic PHS. Moreover, we explicitly compute the invariant measure for a specific class of SPHS with additive noise.