A look at fast varying and state dependent delays from a systems theory point of view

The aim of this note is to discuss conditions on delay derivatives, frequently encountered in the literature, from a systems theory point of view. First an overview of potential problems when the delay rate exceeds one is given, including the violation of causality and the violation of the principe of existence and uniqueness of solutions. Second, the required assumptions on the delay variation to overcome these problems are stated, allowing to define a state space in a rigorous way. Finally, it is shown that a stability analysis of systems with a fast varying delay, where no assumptions on the delay rate are made, can be performed in a meaningful and practically relevant way in some cases, but should be interpreted in terms of relaxations of solutions. A look at fast varying and state dependent delays from a systems theory point of view Wim Michiels∗ and Erik I. Verriest† Abstract The aim of this note is to discuss conditions on delay derivatives, frequently encountered in the literature, from a systems theory point of view. First an overview of potential problems when the delay rate exceeds one is given, including the violation of causality and the violation of the principe of existence and uniqueness of solutions. Second, the required assumptions on the delay variation to overcome these problems are stated, allowing to define a state space in a rigorous way. Finally, it is shown that a stability analysis of systems with a fast varying delay, where no assumptions on the delay rate are made, can be performed in a meaningful and practically relevant way in some cases, but should be interpreted in terms of relaxations of solutions.The aim of this note is to discuss conditions on delay derivatives, frequently encountered in the literature, from a systems theory point of view. First an overview of potential problems when the delay rate exceeds one is given, including the violation of causality and the violation of the principe of existence and uniqueness of solutions. Second, the required assumptions on the delay variation to overcome these problems are stated, allowing to define a state space in a rigorous way. Finally, it is shown that a stability analysis of systems with a fast varying delay, where no assumptions on the delay rate are made, can be performed in a meaningful and practically relevant way in some cases, but should be interpreted in terms of relaxations of solutions.

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