State-space representations for 2×2 hyperbolic systems with boundary inputs

The paper discusses different abstract state-space representations for a class of linear distributed parameter systems of hyperbolic type defined on a one-dimensional spatial domain. It starts with the homogeneous state equation including the unbounded formal state operator. Based on the semigroup approach, some theoretical results of well-posedness and internal stability for the considered systems are given here. Next, the boundary and observation operators are introduced, taking a typical boundary inputs configuration as well as pointwise observations of the state variables. Consequently, the homogeneous state equation is extended to the so-called boundary control state/signal form. Finally, the most classical state-space representation involving typical (A, B, C)-triple of state, input and output operators is considered together with the definition of the Pritchard-Salamon class of infinite-dimensional systems.