Characterizations of integral input-to-state stability for bilinear   systems in infinite dimensions
  For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions. 
Andrii Mironchenko and Hiroshi Ito
math.DS math.AP math.OC