Strong input-to-state stability for infinite-dimensional linear systems

This paper deals with strong versions of input-to-state stability for infinite-dimensional linear systems with an unbounded control operator. We show that strong input-to-state stability with respect to inputs in an Orlicz space is a sufficient condition for a system to be strongly integral input-to-state stable with respect to bounded inputs. In contrast to the special case of systems with exponentially stable semigroup, the converse fails in general.

[1]  G. Burton.  Sobolev Spaces , 2013 .

[2]  K. Knopp.  Theory and Application of Infinite Series , 1990 .

[3]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[4]  F. Smithies,et al.  Convex Functions and Orlicz Spaces , 1962, The Mathematical Gazette.

[5]  Jonathan R. Partington,et al.  Infinite-Dimensional Input-to-State Stability and Orlicz Spaces , 2016, SIAM J. Control. Optim..

[6]  George Weiss,et al.  Admissibility of unbounded control operators , 1989 .

[7]  Roland Schnaubelt,et al.  Observability of polynomially stable systems , 2007, Syst. Control. Lett..

[8]  F. Mazenc,et al.  Strict Lyapunov functions for semilinear parabolic partial differential equations , 2011 .

[9]  Yu. S. Ledyaev,et al.  Asymptotic Stability and Smooth Lyapunov Functions , 1998 .

[10]  H. Logemann,et al.  The Circle Criterion and Input-to-State Stability , 2011, IEEE Control Systems.

[11]  Iasson Karafyllis,et al.  ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs , 2015, IEEE Transactions on Automatic Control.

[12]  Fabian R. Wirth,et al.  Characterizations of Input-to-State Stability for Infinite-Dimensional Systems , 2017, IEEE Transactions on Automatic Control.

[13]  F. Wirth,et al.  Restatements of input-to-state stability in infinite dimensions : what goes wrong ? , 2016 .

[14]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[15]  Iasson Karafyllis,et al.  ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances , 2016, SIAM J. Control. Optim..

[16]  M. Morse,et al.  Functionals F Bilinear Over the Product A × B of Two Pseudo-Normed Vector Spaces: II. Admissible Spaces A , 1950 .

[17]  Sergey Dashkovskiy,et al.  Input-to-state stability of infinite-dimensional control systems , 2012, Mathematics of Control, Signals, and Systems.

[18]  Eduardo Sontag.  Input to State Stability: Basic Concepts and Results , 2008 .

[19]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[20]  Hiroshi Ito,et al.  Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions , 2014, 1406.2458.