A Krasovskii-LaSalle theorem for behavior: Output persistent excitation and detectability

This paper studies stability properties for those systems modeled as behaviors that roughly speaking, describe systems using the view-point of signals. Popular examples include of continuous-time systems, discrete-time systems, switched systems, hybrid systems and time-delay systems. By introducing the output persistently exciting (for short, OPE) condition, a general result regarding the OPE conditions of two behaviors is proposed. An output zeroing system and a detectability condition are then used to help the verification of the OPE condition. From this, a type of Krasovskii-LaSalle theorem can be proposed for behavior. The achieved result could be applied to general dynamic systems. Particularly, it is applied to time-delay time-varying systems as a demonstration. A simple example is also provided to show the effectiveness of the proposed result.