stylistic:style_transfer
Thermodynamic passivity

where the notation sup 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 is finite for all initial states x. Otherwise, the system is considered active. Roughly speaking, the inner product ⟨ v (t), i (t) ⟩ {\displaystyle \langle v(t),i(t)\rangle } is the instantaneous power (e.g., the product of voltage and current), and E is the upper bound on the integral of the instantaneous power (i.e., energy). This upper bound (taken over all T ≥ 0) is the available energy in the system for the particular initial condition x. If, for all possible initial states of the system, the energy available is finite, then the system is called passive. If the available energy is finite, it is known to be non-negative, since any trajectory with voltage v (t) = 0 {\displaystyle v(t)=0} gives an integral equal to zero, and the available energy is the supremum over all possible trajectories. Moreover, by definition, for any trajectory { v (·), i (·)}, the following inequality holds: