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the first has to do with the possibility of determining the input-output system from its natural state set and the second deals with differentiability properties involving the natural state inherited from the input-output system, including differentiability of the natural state and natural state trajectories.
This book defines and develops the generalized adjoint of an input-output system. For a space of input-output systems, a generalized adjoint map from this space of systems to the space of generalized adjoints is defined.
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