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1 2001

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Storage functionals and Lyapunov functions for passive dynamical systems

Borukhov V., Shnip A., Zelenyak D.

Systems Science

2001

Vol. 27, nr 2

27-35

For nonlinear time-invariant input-output dynamical systems the passivity conditions are obtained under some restrictions. The conditions imply storage functions satisfying 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 new 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 non-equilibrium thermodynamics. The results obtained can be useful in automatic control, mechanics of viscoelastic materials, and various applications in physics and the system theory.