562 research outputs found
A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances
In this paper, we introduce a weak maximum principle-based approach to
input-to-state stability (ISS) analysis for certain nonlinear partial
differential equations (PDEs) with boundary disturbances. Based on the weak
maximum principle, a classical result on the maximum estimate of solutions to
linear parabolic PDEs has been extended, which enables the ISS analysis for
certain {}{nonlinear} parabolic PDEs with boundary disturbances. To illustrate
the application of this method, we establish ISS estimates for a linear
reaction-diffusion PDE and a generalized Ginzburg-Landau equation with
{}{mixed} boundary disturbances. Compared to some existing methods, the scheme
proposed in this paper involves less intensive computations and can be applied
to the ISS analysis for a {wide} class of nonlinear PDEs with boundary
disturbances.Comment: 14 page
ISS Estimates in the Spatial Sup-Norm for Nonlinear 1-D Parabolic PDEs
This paper provides novel Input-to-State Stability (ISS)-style maximum
principle estimates for classical solutions of highly nonlinear 1-D parabolic
Partial Differential Equations (PDEs). The derivation of the ISS-style maximum
principle estimates is performed by using an ISS Lyapunov Functional for the
sup norm. The estimates provide fading memory ISS estimates in the sup norm of
the state with respect to distributed and boundary inputs. The obtained results
can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary
conditions. Three illustrative examples show the efficiency of the proposed
methodology for the derivation of ISS estimates in the sup norm of the state.Comment: 20 pages, submitted to ESAIM COCV for possible publicatio
Well-posedness and robust stability of a nonlinear ODE-PDE system
This work studies stability and robustness of a nonlinear system given as an
interconnection of an ODE and a parabolic PDE subjected to external
disturbances entering through the boundary conditions of the parabolic
equation. To this end we develop an approach for a construction of a suitable
coercive Lyapunov function as one of the main results. Based on this Lyapunov
function we establish the well-posedness of the considered system and establish
conditions that guarantee the ISS property. ISS estimates are derived
explicitly for the particular case of globally Lipschitz nonlinearities.Comment: 41 page
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