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ISS With Respect to Boundary Disturbances for 1-D Parabolic PDES
Iasson Karafyllis, Miroslav Krstic
TL;DR
Boundary disturbances in PDEs pose a challenge for establishing ISS: common transformations introduce unbounded input operators and often yield stability only with respect to disturbance derivatives. This paper establishes boundary-disturbance ISS for 1-D parabolic PDEs using a disturbance-dependent, time-varying state space, and applies the results to transport PDE gains and actuator errors under boundary feedback.
Problem
Most existing PDE ISS results address distributed disturbances, while boundary-disturbance analyses using standard transformations involve unbounded input operators and typically yield ISS only with respect to the disturbance and some of its time derivatives.
Source: §1 · p. 1Method
The analysis avoids transforming the boundary disturbance into a domain input and studies the system in disturbance-dependent, time-varying state spaces.
Source: §1 · p. 2, §5 · p. 16Results
The paper establishes ISS with respect to boundary disturbances for 1-D parabolic PDEs under its stated assumptions.
Source: §2 · p. 4, §5 · p. 16Takeaways & Limitations
The results are used to compare inlet-disturbance gains for transport PDEs and establish ISS for control actuator errors in parabolic systems under boundary feedback.
Source: §1 · p. 2, §5 · p. 16The stated result requires a sufficiently regular disturbance for classical solutions; the authors say weak solutions would require less demanding disturbance regularity.
Source: §2 · p. 5
Abstract
from arXiv · showhide
Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-to-state 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.
1. Introduction
The introduction frames boundary-disturbance ISS as a challenge because standard transformations yield an unbounded input operator and typically establish ISS only with respect to the disturbance and some time derivatives. The paper instead analyzes 1-D parabolic PDEs in disturbance-dependent, time-varying state spaces, with applications to transport gains and boundary-feedback actuator errors.
1. Introduction
- Standard transformations express boundary disturbances through unbounded operators and typically yield ISS only with respect to the disturbance and some of its time derivatives.Most existing PDE ISS results address distributed disturbances; the cited exception is.Source: §1 · p. 1
- The proposed methodology avoids transforming the boundary disturbance into a domain disturbance by letting the state space vary with the disturbance over time.The state evolves in a parameterized convex set.Source: §1 · p. 2
- For 1-D parabolic PDEs, the proof estimates generalized Fourier coefficients and uses Parseval’s identity for a weighted L2 norm, rather than a Lyapunov functional.The authors say finding an ISS-Lyapunov functional is difficult, not impossible, because the state space changes at each instant.Source: §1 · p. 2
- The results are applied to compare inlet-disturbance gains for transport PDEs and study ISS with respect to actuator errors under backstepping boundary feedback.The introduction also notes that diffusion and the transport device’s exit boundary condition are illustrated in the gain comparison.Source: §1 · p. 2
2. Problem Description and Main Result
For the 1-D parabolic PDE with boundary disturbance, the main result establishes a unique solution evolving in disturbance-dependent state sets and an ISS estimate with respect to the boundary input.
2. Problem Description and Main Result
- For every admissible disturbance and initial state, the evolution equation has a unique solution that remains in the time-varying set X_d(t).The theorem also gives an estimate for the solution for all t ≥ 0 and ε > 0.Source: §2 · p. 4
- The solution satisfies an ISS estimate with respect to the boundary input, with a gain specified by the theorem.The estimate is stated for all t ≥ 0 and ε > 0.Source: §2 · p. 4
- The gain estimate is nonconservative and can be computed without exact knowledge of the Sturm–Liouville operator’s eigenvalues or eigenfunctions.The paper states that Assumption (H) is the only required information about them.Source: §2 · p. 5
- The state is restricted to a disturbance-dependent convex set, and sufficiently regular disturbances are required for classical solutions.The paper notes that weaker disturbance regularity would suffice if weak solutions were studied instead.Source: §2 · p. 5
3. Proof of Main Result
The proof establishes existence and uniqueness of classical solutions, then estimates generalized Fourier coefficients and combines them with Parseval’s identity to bound the L2 norm.
3. Proof of Main Result
- The proof first establishes existence and uniqueness of a classical solution for the PDE problem.The authors state that the cited results would yield a less smooth solution.Source: §3 · p. 5
- For each sufficiently regular boundary disturbance and admissible initial state, Corollary 3.2 gives a unique classical solution that remains in X_d(t).The trajectory belongs to the disturbance-dependent state set at every time.Source: §3 · p. 6
- The proof estimates generalized Fourier coefficients for the original equations and combines those estimates with Parseval’s identity to bound the L2 norm.The boundary-to-domain transformation is used only to establish existence and uniqueness, not to derive the estimates.Source: §3 · p. 8
4. Applications
The applications examine boundary-disturbance analysis in two settings: gain estimates for a 1-D transport PDE with inlet disturbances and an ISS estimate for control actuator errors in a boundary-feedback parabolic system.
4.I. Gains for Transport PDEs With Respect to Inlet Disturbances
- Under condition (4.12), the solution of PDE (4.1), with boundary (4.2) or (4.3) and initial condition y0, is unique, exists for all t ≥ 0, and satisfies the stated estimate for all t ≥ 0 and ε > 0.Source: §4 · p. 11
- With no reaction, transport-PDE gains decrease as a increases and are smallest for Dirichlet conditions; they are below advection gains only when D/v is small and a is large.Figure 1 compares the gains for a = 0, 1, and +∞ with the advection-equation gain.Source: §4 · p. 12, Figure 1 · p. 13
- Although the advection-equation solution is independent of D, its gain depends on D through the weighted L2 norm used in the analysis.The gain parameterization includes D even though the solution formula does not.Source: §4 · p. 13
ISS With Respect to Control Actuator Errors for Boundary State Feedback
- A Volterra transformation converts actuator errors into boundary disturbances for a heat equation whose free parameter c sets the convergence rate.The boundary-feedback kernel is obtained from the Volterra transformation, and its inverse recovers the original solution.Source: §4 · p. 14
- For the transformed-state estimate, gain G is specified separately for c > 0 in (4.35) and c = 0 in (4.36).Source: §4 · p. 15
- The closed-loop parabolic system satisfies the ISS property with respect to control actuator errors.The result applies to every stated actuator error d and compatible initial condition; the solution is unique and defined for all t ≥ 0.Source: §4 · pp. 15–16
5. Concluding Remarks
The paper proposes an ISS methodology for 1-D parabolic PDEs with boundary disturbances, analyzing stability in time-varying state-space subsets. It applies the results to transport-PDE gains and actuator errors under boundary feedback, while identifying the L∞ norm as future work.
5. Concluding Remarks
- The proposed ISS methodology for 1-D parabolic PDEs avoids transforming boundary disturbances into domain inputs and analyzes stability in time-varying state-space subsets.The paper uses the L2 norm in its analysis.Source: §5 · p. 16
- The results are applied to compare transport-PDE gains for inlet disturbances and establish ISS with respect to actuator errors under parabolic boundary feedback.Source: §5 · p. 16
- Establishing ISS with the L∞ norm remains future work because the analogous Parseval identity is unavailable.The paper’s analysis uses the L2 norm.Source: §5 · p. 16
Appendix
This appendix proves auxiliary results for Lemma 2.1 and Theorem 3.1, including a boundary-value construction and regularity properties of the constructed solution. It also establishes uniqueness for solutions with the stated initial condition and that the solution remains in the space X for all nonnegative times.
Proof of Lemma 2.1
- A function constructed from the unique solution of an auxiliary boundary-value problem satisfies (2.8)–(2.9), supporting the proof of Lemma 2.1.The proof then uses the Sturm–Liouville eigenfunctions as an orthonormal basis to obtain Parseval identities.Source: Appendix · p. 17
Proof of Theorem 3.1
- For initial states in X_0, the eigenfunction expansion converges uniformly and absolutely, and the proof defines the state using a convergent series.The cited results are Theorem 9.3 in, Theorem 7.5.4 in, and Theorem 3.11.3.4 in.Source: Appendix · pp. 18–19
- The series-defined state has a continuous time derivative for t > 0, established by uniform and absolute convergence of its termwise differentiated series.The proof gives the derivative representation in (A.9).Source: Appendix · p. 20
Appendix
- The spatial derivative of x exists and is continuous for positive times and interior spatial points.Its existence is shown through uniform and absolute convergence of the differentiated series on each compact time interval.Source: Appendix · p. 21
- Any two solutions in the stated class with the same initial profile are identical for all t >= 0.The proof applies Parseval’s identity to their difference and shows that all its expansion coefficients vanish.Source: Appendix · pp. 22–23