Recently a class of Hamiltonian control systems was defined for open irreversible thermodynamic systems. These systems are Hamiltonian control systems defined on a symplectic manifold, however departing from standard Hamiltonian control systems, due to the property that the Hamiltonian function is homogeneous in the generalized momentum variables. In this paper we study the class of state feedbacks preserving the geometric structure of such Homogeneous Hamiltonian control systems and rendering the closed-loop system again a Homogeneous Hamiltonian control system. It is shown that only a constant control preserves the canonical Liouville form. Hence a non-trivial state feedback necessarily changes the geometric structure in closed-loop defined by a modified Pfaffian form. Finally we derive a matching equation on the nonlinear feedback and the closed-loop Pfaffian form.
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