Detectability via observability in a nonuniform framework: dual relationship with controllability and stabilizability

In this paper we propose a new observability property for nonautonomous linear control systems in finite dimension: the nonuniform complete observability, which is more general than the uniform complete observability. A dual relationship is established between this new notion of observability and the recently defined nonuniform complete controllability property, with the aim of obtaining the main result, which proves that nonuniform complete observability guarantees the nonuniform exponential detectability, a concept related to exponential stability in this framework.