with a state representation S, and initial state x, define available energy E_A as:

where the notation sup_x->T>=0 indicates that the supremum is taken over all T >= 0 and all admissible pairs {v(·), i(·)} with the fixed initial state x (e.g., all voltage-current trajectories for a given initial condition of the system). A system is considered passive if E_A is finite for all initial states x.