Links between Dynamic Physical Systems and Operator Theory Issues Concerning Energy Balances and Stability

This study relates the properties of operators with the well- known concepts of positive realness and passivity properties in dynamic systems and their associate transfer functions. Those concepts together with very close related ones are first examined from a physical point of view. Then, they are related to hyper-stability and properties of transfer functions while the hyper-stability theorem is revisited and interpreted. Finally, the above concepts are compared to the mathematical concepts of positivity and closely related ones in operator theory in Hilbert spaces.