We propose an extension of the Popov’s hyperstability theory which applies to a class of single-control distributed systems in which the linear part depends explicitely upon the distributed parameter, z. The nonlinearness of these systems is expressed by means of the control. The main features of our results are the following: (i) The hyperstability conditions that we obtain involve specific z-dependent functions which we can consider as being extensions of the transfer function concept; (ii) they also involve integrals with respect to the distributed parameter, which express an averaging effect of this latter. Then systems in which the admissible controls are defined via time-varying conditions are investigated. For such systems, we define the concept of “average hyperstability” in time, and average hyperstability conditions are given. Similar problems are solved for multi-control distributed systems. As an application we show how these results yield a broad class of absolute stability conditions for distributed systems: they are space averaging conditions and they may apply when other criteria are in-applicable. Three examples are given: the last one illustrates how a space-describing function approach can be used to determine the distributed transfer function of the system.

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