Abstract State-Space Models for a Class of Linear Hyperbolic Systems of Balance Laws

The paper discusses and compares different abstract state-space representations for a class of linear hyperbolic systems defined on a one-dimensional spatial domain. It starts with their PDE representation in both weakly and strongly coupled forms. Next, the homogeneous state equation including the unbounded formal state operator is presented. Based on the semigroup approach, some results of well-posedness and internal stability are given. The boundary and observation operators are introduced, assuming a typical configuration of boundary inputs 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. Next, the classical additive statespace representation involving (A, B, C)-triple of state, input and output operators is considered. After short discussion on the appropriate Hilbert spaces, state-space equation in the so-called factor form is also presented. Finally, the resolvent of the system state operator A is discussed.