ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs

Decay Estimates for 1-D Parabolic PDEs with Boundary Disturbances

In this work decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the 2 L and 1 H norms of the solution and discontinuous disturbances are allowed. Although an eigenfunction expansion for the solution is exploited for the proof of the decay estimates, the estimates do not re...

ISS in Different Norms for 1-D Parabolic PDES With Boundary Disturbances

For 1-D parabolic PDEs with disturbances at both boundaries and distributed disturbances we provide ISS estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an eigenfunction expansion of the solution, and (ii) a finite-difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximu...

Sup - norm Estimates in Glimm ’ s Method *

We give a proof that Total Variation {a,,(.)} < 1 can be replaced by Sup{u,(.)} e 1 in Glimm's method whenever a coordinate system of Riemann invariants is present. The argument is somewhat simpler but in the same spirit as that given by Glimm in his celebrated paper of 1965. 0 1990 Academic Press, Inc. We consider the Cauchy problem u, + F(u), = 0, (1) 4-G 0) = u,(x), (2) where (1) denotes a s...

A comparison of adaptive software for 1-D parabolic PDEs

In this paper we describe BACOL, a high-order, spatially and temporally adaptive software package for solving systems of one-dimensional parabolic partial differential equations and then compare it with several related software packages. BACOL employs collocation at Gaussian points with a B-spline basis for the spatial discretization. A modification of DASSL is used for the time integration of ...

Infinite dimensional backstepping for nonlinear parabolic PDEs