When is a linear system optimal

Expand (1) in a term which is bilinear in w ∆, and one which is quadratic in ∆. We obtain (1) ∞ ! ∞ ∆ " Φ $ d dt d dt w & dt # ∞ ! ∞ QΦ ∆ dt The trajectory w C∞ is said to be stationary with respect to Φ, relative to variations ∆, if the linear term in ∆ in (1) vanishes, i.e. if ∞ ! ∞ ∆ Φ $ d dt d dt wdt 0 ( ESAT, K.U. Leuven, B-3001 Leuven, Belgium, email: Jan.Willems@esat.kuleuven.ac.be. This research is supported by the Belgian Federal Government under the DWTC program Interuniversity Attraction Poles, Phase V, 2002–2006, Dynamical Systems and Control: Computation, Identification and Modelling, by the KUL Concerted Research Action (GOA) MEFISTO–666, and by several grants en projects from IWT-Flanders and the Flemish Fund for Scientific Research.