DISCRETE OBSERVABILITY AND NUMERICAL QUADRATURE

We consider the problem of approximate observability of a one dimensional diffusion equation on a finite spatial domain with spatial point measurements. The problem of the optimal selection of the measurement points is considered under three conditions: 1) no preassigned measurement nodes; 2) one preassigned node and; 3) two preassigned nodes. The main observation of this paper is that the optimal choice is intimately related to three classical procedures in numerical analysis: 1) Gaussian quadrature; 2) Radau quadrature and; 3) Lobatto quadrature. We also show that the existence of Radau and Lobatto quadrature is closely related to classical root locus theory.