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On the investigation of dissipativity by a discrete observation of the storage function

In proving uniform asymptotic stability of the equilibrium point of a dynamical system, the theorem of Liapunov requires the existence of a Liapunov function whose derivative, along the flow of the system, is negative definite. This condition may be relaxed. In order to prove uniform asymptotic stability, it is sufficient to find a time T>0 and a Liapunov function which is, along the flow of the system, decreasing when considered every time T. As to the nature of dissipativity, a similar result is obtained. The definition of dissipativity requires the existence of a storage function whose rate of increase is instantaneously not larger than the supply rate. But in the present paper, we establish that in order to prove dissipativity of a system, it is sufficient to find a time T>0 and a storage function whose increase is, every time T, not larger than the integral of the supply rate.