Pole Assignment for a Vibrating System with Aerodynamic Effect

This paper deals with a pole assignment problem by single-input state feedback control arising from a one-dimensional vibrating system with aerodynamic effect. On the practical side, we derive explicit formulae for the required controlling force terms, which can reassign part of the spectrum to the desired values while leaving the remaining spectrum unchanged. On the mathematical side, unlike the classical Sturm–Liouville problem, our eigenvalue problem is associated with a cubic pencil with unbounded operators as coefficients and has many interesting new features, one of which is that a new controllability condition appears. This condition together with the known controllability condition in the quadratic case are necessary and sufficient. This sheds light on the adjustment of the model parameters. We also analyze the spectrum of the associated noncompact operator and in particular show that the discrete spectrums of controlled and uncontrolled systems lie outside a closed interval on the negative real axis.