466 research outputs found
On the Relation of Delay Equations to First-Order Hyperbolic Partial Differential Equations
This paper establishes the equivalence between systems described by a single
first-order hyperbolic partial differential equation and systems described by
integral delay equations. System-theoretic results are provided for both
classes of systems (among them converse Lyapunov results). The proposed
framework can allow the study of discontinuous solutions for nonlinear systems
described by a single first-order hyperbolic partial differential equation
under the effect of measurable inputs acting on the boundary and/or on the
differential equation. An illustrative example shows that the conversion of a
system described by a single first-order hyperbolic partial differential
equation to an integral delay system can simplify considerably the solution of
the corresponding robust feedback stabilization problem.Comment: 32 pages, submitted for possible publication to ESAIM COC
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
Strict Lyapunov functions for semilinear parabolic partial differential equations
International audienceFor families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type
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