Observable linear pairs (Q1384162)

It is known that the observability of a linear control system \(\dot{x}=Ax+Bu\), \(h(x)=Cx\) (\(x\) is the state, \(u\) is the input and \(h(x)\) is the output) depends on the matrices \(A\) and \(C\) only. In the paper under review, the authors extend the observability criteria to linear control systems on a Lie group \(G\). The role of a pair \((A,C)\) is played by the so-called linear pair \((X,h)\), where \(X\) is an infinitesimal automorphism of \(G\) and the output function \(h\) is a Lie group homomorphism defined on \(G\). By definition, a linear pair \((X,h)\) is observable at \(g \in G\) if for all \(\overline{g} \in G\), \(g \neq \overline{g}\), there is \(t \geq 0\) such that \(h(X_t(g)) \neq h(X_t(\overline{g}))\) (\(X_t\) is the flow on \(G\) defined by \(X\)). It is locally observable at \(g\) if the above holds in a neighbourhood of \(g\). If the pair \((X,h)\) is observable (locally observable) at all \(g \in G\), then it is called observable (locally observable). After obtaining the criteria for local observability of the pair \((X,h)\), the authors show that the condition for observability of \((X,h)\) can be expressed in terms of the Lie algebra \({\mathcal I}\) of the equivalence class of the identity element of \(G\) (equivalence being defined by \(g \sim \overline{g} \Leftrightarrow h(X_t(g)) = h(X_t(\overline{g}))\) for every \(t \geq 0\)) and the set of fixed points of the action on \(G\) generated by \(X\). An algorithm for the computation of the Lie algebra \({\mathcal I}\) is given, and finally, the observability criteria are illustrated on two examples.