Local observability of invariant dynamics on compact Lie groups with square intergrable output map functions (Q1130412)

Let \(\Sigma\) be a control system on a compact Lie group given by a set of left invariant vector fields and \(S_\Sigma\) the subsemigroup generated by these vector fields. If \(V\) is a finite dimensional inner product space and \(h:G\to V\) the output map coming with \(\Sigma\), then \(\Sigma\) is called observable if \(h(S_\Sigma g_1) =h(S_\Sigma g_2)\) implies \(g_1=g_2\) and locally obervable if \(h(S_\Sigma g_1) =h(S_\Sigma g_2)\) implies \(g_1=g_2\) for \(g_2\) in a neighborhood of \(g_1\). The authors consider output maps in \(L^2(G,V)\) which makes the methods of earlier papers on (local) observability of such systems using smooth output maps unavailable. As a substitute they use the right regular representation of \(G\) on \(L^2(G,V)\). One considers the linear control system \(R(\Sigma)\) given by the action of \(S_\Sigma\) on \(L^2(G,V)\) with linear output function given by the inner product with \(h\). The main results of the paper are 1) \(R(\Sigma)\) is observable if and only if its isotypic components are observable, 2) the observability of \(R(\Sigma)\) implies the local observability of \(\Sigma\).