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A quantitative measure of observability and controllability

Observability and controllability are fundamental concepts of control systems. In this paper, we use dynamical optimization and its computational methods as the tool to define and to quantitatively measure the observability and reachability. In addition, the duality between the observability and reachability becomes straightforward under the definitions introduced in this paper. Several examples are given to illustrate the concepts, including an example in which a system being observable under linear control theory is proved to be weakly unobservable.