Decay Estimates for 1-D Parabolic PDEs with Boundary Disturbances
Provides theoretical decay estimates for parabolic PDEs with disturbances, useful for control and stability analysis in applied mathematics and engineering.
The paper derives decay estimates for solutions of 1-D linear parabolic PDEs with boundary and distributed disturbances, in L2 and H1 norms, without requiring eigenvalue knowledge. The results are applied to stability analysis of PDEs with nonlocal terms.
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 L2 and H1 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 require knowledge of the eigenvalues and the eigenfunctions of the corresponding Sturm-Liouville operator. Examples show that the obtained results can be applied for the stability analysis of parabolic PDEs with nonlocal terms.