Riesz Basis Property Of System Research Articles

Riesz basis property of system of root functions of second-order differential operator with involution

The properties of the root functions are studied for an arbitrary operator generated in L 2(−1, 1) by the operation with involution of the form Lu = −u″(x)+αu″(−x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (−1, 1), ν(x) is an absolutely continuous involution of the segment [−1, 1] and the coefficients q(x) and qν(x) are summable functions on (−1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L 2(−1, 1) for the system of the root functions of the operator.

Riesz basis property of a second-order hyperbolic system with collocated scalar input-output

The closed-loop form of a second-order hyperbolic system with scalar collocated sensor/actuator is considered. The Riesz basis property of the system is verified for diagonal semigroups based on an abstract result of one rank perturbation of discrete-type operators in Hilbert spaces. A simplified proof for the abstract result is presented.