Characterization of the Controlled States in W₂⁽¹⁾ of Linear Hereditary Systems

The question of controlling a linear retarded functional differential equation from an initial function to a terminal function, both functions belonging to the Sobolev space W₂⁽¹⁾, is considered. Necessary and sufficient conditions for full controllability in this function space are derived. These conditions result in computable algebraic criteria. Null controllability in function space is also investigated and conditions that are necessary and sufficient are obtained. In the absence of full controllability, methods for characterizing the attainable set in W₂⁽¹⁾ are discussed in detail along with a number of illustrative examples.