20,445 research outputs found
On continuity of solutions for parabolic control systems and input-to-state stability
We study minimal conditions under which mild solutions of linear evolutionary
control systems are continuous for arbitrary bounded input functions. This
question naturally appears when working with boundary controlled, linear
partial differential equations. Here, we focus on parabolic equations which
allow for operator-theoretic methods such as the holomorphic functional
calculus. Moreover, we investigate stronger conditions than continuity leading
to input-to-state stability with respect to Orlicz spaces. This also implies
that the notions of input-to-state stability and integral-input-to-state
stability coincide if additionally the uncontrolled equation is dissipative and
the input space is finite-dimensional.Comment: 19 pages, final version of preprint, Prop. 6 and Thm 7 have been
generalised to arbitrary Banach spaces, the assumption of boundedness of the
semigroup in Thm 10 could be droppe
A De Giorgi Iteration-based Approach for the Establishment of ISS Properties for Burgers' Equation with Boundary and In-domain Disturbances
This note addresses input-to-state stability (ISS) properties with respect to
(w.r.t.) boundary and in-domain disturbances for Burgers' equation. The
developed approach is a combination of the method of De~Giorgi iteration and
the technique of Lyapunov functionals by adequately splitting the original
problem into two subsystems. The ISS properties in -norm for Burgers'
equation have been established using this method. Moreover, as an application
of De~Giorgi iteration, ISS in -norm w.r.t. in-domain disturbances
and actuation errors in boundary feedback control for a 1- {linear}
{unstable reaction-diffusion equation} have also been established. It is the
first time that the method of De~Giorgi iteration is introduced in the ISS
theory for infinite dimensional systems, and the developed method can be
generalized for tackling some problems on multidimensional spatial domains and
to a wider class of nonlinear {partial differential equations (PDEs)Comment: This paper has been accepted for publication by IEEE Trans. on
Automatic Control, and is available at
http://dx.doi.org/10.1109/TAC.2018.2880160. arXiv admin note: substantial
text overlap with arXiv:1710.0991
Input-to-State Stability with Respect to Boundary Disturbances for a Class of Semi-linear Parabolic Equations
This paper studies the input-to-state stability (ISS) properties based on the
method of Lyapunov functionals for a class of semi-linear parabolic partial
differential equations (PDEs) with respect to boundary disturbances. In order
to avoid the appearance of time derivatives of the disturbances in ISS
estimates, some technical inequalities are first developed, which allow
directly dealing with the boundary conditions and establishing the ISS based on
the method of Lyapunov functionals. The well-posedness analysis of the
considered problem is carried out and the conditions for ISS are derived. Two
examples are used to illustrate the application of the developed result.Comment: Manuscript submitted to Automatic
Input-to-state stability of unbounded bilinear control systems
We study input-to-state stability of bilinear control systems with possibly
unbounded control operators. Natural sufficient conditions for integral
input-to-state stability are given. The obtained results are applied to a
bilinearly controlled Fokker-Planck equation.Comment: 20 pages, completely new version based on the few preliminary ideas
in v1. Compared to v1, the results have been significantly generalized and
extende
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