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Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation $\star$
Jean-Michel Coron, Long Hu, Guillaume Olive
TL;DR
The paper seeks the best finite time for stabilizing a coupled linear hyperbolic system using one-sided boundary feedback. It uses successive Volterra and invertible Fredholm transformations, and reports closed-loop vanishing at the optimal time under its stated assumptions.
Problem
The paper asks for a full-state boundary feedback that makes the coupled linear hyperbolic system vanish in the best time.
Source: §1 · p. 2Method
A Volterra transformation followed by an invertible Fredholm transformation maps the system to a target vanishing at Topt.
Source: §1 · pp. 2–3Results
Theorem 1 gives a bounded feedback F for which every L2 initial state yields a solution vanishing after Topt.
Source: §1 · p. 3Takeaways & Limitations
Because the transformation is invertible, the target and original systems share stability properties.
Source: §1 · p. 3The result assumes a C1 diagonal Λ with distinct, nonzero characteristic speeds, and continuous Σ with constant boundary coupling Q.
Source: §1 · p. 1
Abstract
from arXiv · showhide
This paper is devoted to a simple and new proof on the optimal finite control time for general linear coupled hyperbolic system by using boundary feedback on one side. The feedback control law is designed by first using a Volterra transformation of the second kind and then using an invertible Fredholm transformation. Both existence and invertibility of the transformations are easily obtained.
1 Introduction
The paper seeks a full-state boundary feedback that makes a coupled linear hyperbolic system vanish in the best finite time. It proposes a two-stage backstepping construction and establishes vanishing at the optimal time for the stated system.
1 Introduction
- The stated system assumes a diagonal C1 matrix Λ with distinct, nonzero characteristic speeds, and continuous internal coupling Σ with constant boundary coupling Q.Source: §1 · p. 1
- The paper seeks a full-state feedback F that makes the closed-loop system vanish and finds the best time for that outcome.Source: §1 · p. 2
- For inhomogeneous systems with sufficiently strong coupling, dissipative static boundary feedback can fail: some open-loop controllable systems cannot be stabilized by that feedback.Source: §1 · p. 2
- An earlier controller had a control time larger than the expected theoretical optimum, while a minimum-time result was known for constant-coefficient systems.The paper presents its approach as an easier way to establish this kind of controller and generalizes the constant-matrix result.Source: §1 · pp. 2–3
- A Volterra transformation followed by an invertible Fredholm transformation yields a target system vanishing at Topt, with only a simple trace coupling.The Fredholm kernel's cascade structure supports invertibility, so the original and target systems share stability properties.Source: §1 · pp. 2–3
- Theorem 1 asserts existence of a bounded feedback F from L2(0, 1)^n to R^m for which every L2 initial state yields a solution in C0([0, +∞); L2(0, 1)^n).Topt is given by (7).Source: §1 · p. 3
2 New target system
The section constructs a target system whose solutions vanish at the optimal time, then transfers that behavior back to the original system through invertible transformations. Choosing the feedback is essential: the earlier choice H = 0 does not attain Topt.
2 New target system
- A Volterra backstepping transformation relates the original system to the target system of [14], with a feedback chosen for each H.The bounded map V is invertible, and the resulting feedback makes the transformed solution correspond to a solution of the original system.Source: §2 · p. 3
- Choosing H through a second backstepping step reduces the earlier target system's vanishing time from tF to Topt.The earlier choice H = 0 did not give the expected optimal time.Source: §2 · p. 3
- The target system vanishes at Topt, with characteristics and the cascade structure yielding successive extinction of z− and z+.The proof first establishes z− extinction and then z+ extinction by Topt.Source: §2 · p. 4
- Proposition 2 requires the boundary condition z−(t, 1) = 0, so its construction does not allow the same freedom in choosing H.The condition is needed in the proof of the Fredholm transformation result.Source: §2 · p. 4
- The composed transformations transfer target-system extinction at Topt to the original closed-loop system.The target solution vanishes for t ≥ Topt, and the proof then obtains the same vanishing time for u.Source: §2 · p. 4
3 Existence of an invertible Fredholm transformation
The section constructs a Fredholm transformation by solving a kernel system with a cascade structure. Its kernel admits a unique weak solution, and the resulting transformation and feedback satisfy the required existence and invertibility properties.
3 Existence of an invertible Fredholm transformation
- The Fredholm map uses an integral kernel with a cascade structure, and the resulting map is invertible.The kernel entries are specified below the diagonal, while the other components are unchanged.Source: §3 · p. 4
- The imposed extra condition on the kernel simplifies its system, while the boundary condition at x = 0 for γ is automatically satisfied.The simplification follows from the structures of G and K.Source: §3 · p. 5
- The reduced K1 system is uncoupled with positive speeds and has a unique weak solution, given explicitly by characteristics.Its speeds are λi(x)/λj(y) > 0.Source: §3 · pp. 5–6
- The constructed feedback H is bounded, completing the proof of Proposition 2.The kernel construction uses the monotonicity and invertibility of φi.Source: §3 · p. 6
- The paper leaves open which target systems can be achieved by general linear transformations.It contrasts this open question with a finite-dimensional controllable-system result for a family of target systems.Source: §3 · p. 6
A Invertibility of the Fredholm transformation
The appendix proves that the Fredholm transformation is invertible for kernels with the stated cascade structure, and that its inverse has the same form.
A Invertibility of the Fredholm transformation
- For any square-integrable kernel with the stated cascade structure, the Fredholm transformation is invertible and its inverse has the same form.The inverse kernel has the same structure as the original kernel.Source: A Invertibility of the Fredholm… · p. 7
- The inverse-kernel matrix Θ is lower triangular with zero diagonal entries, matching the structure specified for K1.Source: A Invertibility of the Fredholm… · p. 7
- For indices m+1 through n, the proof gives z_i = γ_i when γ = Fz.The induction argument then establishes the lemma using coefficients that depend only on the indicated k_pj terms.Source: A Invertibility of the Fredholm… · p. 7