Introduction

Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-tostate stability (ISS) properties and finite gains cannot be established with respect to disturbances at the boundary. This intuition has been reinforced by many unsuccessful attempts, as well as by the success in establishing ISS only with respect to the derivative of the disturbance. Contrary to this intuition, we establish such a result for parabolic PDEs. Our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time-varying subsets of the state space. The obtained results are used for the comparison of the gain coefficients of transport PDEs with respect to inlet disturbances and for the establishment of the ISS property with respect to control actuator errors for parabolic systems under boundary feedback control.

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“…FACT (see Chapter 11 in [3] and pages 498-505 in [13]): All eigenvalues of the Sturm-Liouville operator…”

Section: Problem Description and Main Resultsmentioning

confidence: 99%

ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs

Karafyllis

Krstić

2016

IEEE Trans. Automat. Contr.

115

120

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“…are real constants satisfying the following assumptions: FACT (see Chapter 11 in [3] and pages 498-505 in [22]): All eigenvalues of the Sturm-Liouville operator ]) 1 , 0 ([ : In this paper (as in [14]), we also make the following assumption for the Sturm…”

Section: Introductionmentioning

confidence: 99%

ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances

Karafyllis¹,

Krstić²

2017

SIAM J. Control Optim.

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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 maximum principle for the heat equation. Finally, the obtained results are applied to quasi-static thermoelasticity models that involve nonlocal boundary conditions. Small-gain conditions that guarantee the global exponential stability of the zero solution for such models are derived.

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“…. It is well-known (Chapter 11 in [3] and pages 498-505 in [34]) that all eigenvalues of the SL operator In the present work, we use the following assumption for the SL operator ) 1 , 0 ( :…”

Section: Resultsmentioning

confidence: 99%

Decay estimates for 1-D parabolic PDES with boundary disturbances

Karafyllis

Krstić

2018

ESAIM: COCV

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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 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. 0

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Introduction

Cited by 18 publications

References 0 publications

ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs

ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs

ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances

Decay estimates for 1-D parabolic PDES with boundary disturbances

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