23 research outputs found
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
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
Event-triggered gain scheduling of reaction-diffusion PDEs
This paper deals with the problem of boundary stabilization of 1D
reaction-diffusion PDEs with a time- and space- varying reaction coefficient.
The boundary control design relies on the backstepping approach. The gains of
the boundary control are scheduled under two suitable event-triggered
mechanisms. More precisely, gains are computed/updated on events according to
two state-dependent event-triggering conditions: static-based and dynamic-based
conditions, under which, the Zeno behavior is avoided and well-posedness as
well as exponential stability of the closed-loop system are guaranteed.
Numerical simulations are presented to illustrate the results.Comment: 20 pages, 5 figures, submitted to SICO