document: Storage functionals and Lyapunov functions for passive dynamical systems For nonlinear time-invariant input-output dynamical systems the passivity conditions are obtained under some restrictions . The conditions imply storage functions satisfying a dissipation inequality . A class of storage functions allowing unique reconstruction of a passive dynamical system is defined . These results are illustrated by an example of a linear system with fading memory . An important , for practical application , class of the linear relaxation systems without direct input-output interaction is considered . A necessary condition for dynamical systems to be of the relaxation type is obtained for this class . The condition is connected with the existence of a unique quadratic Lyapunov function satisfying the complete monotonicity condition . This unique Lyapunov function corresponds to a `` standard '' thermodynamic potential in a compact family of potentials in the nonequilibrium thermodynamics . The results obtained can be useful in automatic control , mechanics of viscoelastic materials , and various applications in physics and the system theory
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extractive_keyphrases: storage functionals passive dynamical systems passivity conditions dissipation inequality linear system fading memory necessary condition unique quadratic lyapunov function complete monotonicity condition thermodynamic potential nonequilibrium thermodynamics automatic control mechanics viscoelastic materials
abstractive_keyphrases: nonlinear time-invariant input-output dynamical systems