389 research outputs found
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 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.Comment: 35 pages, submitted for possible publication to ESAIM-COC
Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients
Recently, the problem of boundary stabilization for unstable linear
constant-coefficient coupled reaction-diffusion systems was solved by means of
the backstepping method. The extension of this result to systems with advection
terms and spatially-varying coefficients is challenging due to complex boundary
conditions that appear in the equations verified by the control kernels. In
this paper we address this issue by showing that these equations are
essentially equivalent to those verified by the control kernels for first-order
hyperbolic coupled systems, which were recently found to be well-posed. The
result therefore applies in this case, allowing us to prove H^1 stability for
the closed-loop system. It also shows an interesting connection between
backstepping kernels for coupled parabolic and hyperbolic problems.Comment: Submitted to IEEE Transactions on Automatic Contro
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
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