Approximate Feedback Linearization for a Nonlinear Hyperbolic PDE Class -- Part I: Volterra Truncation

Abstract: Backstepping for nonlinear PDEs yields exact feedback linearizing laws in the form of infinite Volterra series -- elegant in theory, but with challenges for implementation. This paper shows that even very low-order truncations of such controllers, no longer exactly linearizing, retain the stabilizing power. The key insight is that higher-order terms become negligible near the origin, so stability is recovered for any fixed truncation order by restricting the initial condition size. We establish spatial sup-norm results: finite-time practical stability and asymptotic stability characterized by a class-$\mathcal{KL}$ estimate. The region-of-attraction estimate grows with the truncation order and shrinks with the growth rate of the nonlinearity. The analysis overcomes the lack of pointwise kernel bounds and resolves well-posedness of the nonlinear closed loop, showing that surprisingly simple approximations already capture the essence of nonlinear PDE feedback linearization.

• The paper analyzes finite-order truncations of infinite Volterra series to achieve local asymptotic stabilization in nonlinear hyperbolic PDEs.
• It rigorously establishes stability properties in both the sup-norm and L2 norm, demonstrating geometric decay of the truncation residual with increasing order.
• The work provides practical design guidelines for implementable feedback controllers and suggests potential extensions using neural operator surrogates.

Approximate Feedback Linearization for Nonlinear Hyperbolic PDEs: Volterra Truncation

Introduction and Problem Setting

This work addresses the practical implementation of nonlinear feedback linearization for a class of nonlinear hyperbolic partial differential equations (PDEs) with spatial Volterra-type nonlinearities. The classical backstepping approach in this context leads to feedback controllers given by infinite Volterra series, which, while offering exact linearization and local exponential stabilization (Krstic, 30 Apr 2026), are infeasible for implementation due to the necessity of evaluating infinitely many nested integrals and solving infinite-dimensional kernel PDEs in real time.

The main contribution of this paper is the analysis of finite-order truncations of such infinite Volterra feedback controllers. Specifically, for the nonlinear hyperbolic PDE

$$u_t(x, t) = u_x(x, t) + F[u](x, t), \qquad x \in [0, 1),\ t \geq 0,$$

with $u(1, t) = U(t)$, and $F$ a spatial Volterra series in the state, the feedback law is obtained by truncating the infinite Volterra series at order $N$, yielding an implementable operator $K_N[u]$ used for boundary control: $U(t) = K_N[u](1, t)$. The paper rigorously establishes the stabilization properties and domain of attraction of such truncated feedback laws.

Technical Contributions

Truncation and Stability in the Sup-Norm and $L^2$ Norm

A notable technical point is the demonstration that finite-order truncations can still accomplish local asymptotic stabilization, provided the magnitude of initial conditions is suitably restricted. The precise region of attraction depends on the truncation order $N$ and on the analyticity radius of the plant nonlinearity. The analysis systematically addresses well-posedness, forward invariance, practical and asymptotic stability, with a focus on the spatial sup-norm (maximum norm), which is typically more stringent than the $L^2$ norm prevalent in earlier backstepping literature.

The instability introduced by truncation is localized at the boundary via a residual (the discarded Volterra tail), which acts as a perturbation in the closed-loop dynamics. The closed-loop system, reduced by the backstepping transformation, is a transport equation with nonhomogeneous boundary conditions governed by this tail. The magnitude of this residual is shown to decrease geometrically with the truncation order $N$.

The main theorem asserts the following stability properties under appropriate choices of $u(1, t) = U(t)$0 and initial condition norm:

• Forward invariance: The solution maintains its spatial sup-norm within a pre-defined ball.
• Practical stability: The state remains bounded by an explicit function of the initial data and the truncation residual.
• Finite-time attractivity: After a finite time (specifically, after the transport crossing time), the state enters and remains in a ball whose radius is a function of the truncation residual.
• Asymptotic stability (class-$u(1, t) = U(t)$1 estimate): The solution decays in sup-norm to zero as $u(1, t) = U(t)$2 according to a class-$u(1, t) = U(t)$3 function, despite the nonzero boundary residual.

A secondary result mirrors the sup-norm statements in the $u(1, t) = U(t)$4 norm, demonstrating that the same truncated feedback achieves analogous stabilization in the energy norm—though with more conservative constants.

Quantitative Analysis and Operator Bounds

The analysis develops new quantitative tools for bounding the infinite-dimensional, spatially nonlocal Volterra kernels. In the absence of direct pointwise kernel bounds, it utilizes $u(1, t) = U(t)$5 estimates for the kernel slices, combined with Cauchy--Schwarz and combinatorial arguments, to derive uniform $u(1, t) = U(t)$6 kernel bounds and to control the truncation residual in sup-norm. The boundedness and Lipschitz continuity of the truncated Volterra operator $u(1, t) = U(t)$7, as well as the local invertibility of the backstepping transformation in sup-norm, are established via fixed-point theorems on function spaces.

A geometric decay rate in $u(1, t) = U(t)$8 is established for the residual, forming the theoretical basis for truncated design: as $u(1, t) = U(t)$9, the practical behavior approaches that of the exact feedback, and the region of attraction expands accordingly. Notably, even at low $F$0 (orders 2 or 3), substantial basins of attraction are demonstrated, as reflected in the explicit quantitative estimates.

Well-Posedness and Local Analysis

The truncated feedback controller destroys the closed-loop equivalence with the original target transport system, necessitating new existence and uniqueness arguments for solutions of the closed-loop PDEs. Local well-posedness is established by constructing canonical mild solutions via a Banach fixed-point argument. Forward invariance and global-in-time existence on a local ball are secured by showing that the solution cannot escape the invariant region before blow-up.

Crucially, the results are inherently local: global stabilization is precluded by known counterexamples in boundary control of nonlinear PDEs with spatially analytic nonlinearities. Thus, the size of the stabilizable initial condition set is always strictly less than global.

Numerical and Practical Implications

The practical implication is that robust, implementable feedback linearization of nonlinear hyperbolic PDEs is possible even with low-order Volterra truncations, provided initial conditions are appropriately moderate. Each controller evaluation is computationally nontrivial, requiring knowledge of $F$1-dimensional spatial kernels and $F$2-fold integrations, which are plant-dependent. Any change in plant coefficients mandates recomputation of kernels. Extensions to drastically reduce online computation cost via neural operator surrogates are hinted at and developed in a companion paper (Part II).

The results provide explicit design procedures: for given plant parameters and a chosen truncation order, one can compute the sizes of admissible initial condition balls and region of attraction using the derived formulas.

Theoretical Implications and Future Prospects

On the theoretical level, this work rigorously addresses the classic gap between infinite-dimensional control designs and their implementable finite approximations. It confirms that local stabilization properties persist under truncation in the context of hyperbolic PDEs with Volterra nonlinearities, provided appropriate norm-based control of the residual is maintained.

An open problem remains for the $F$3 stabilization under exact infinite-series feedback, as truncation is essential to obtain residuals that vanish locally without losing well-posedness. The paper conjectures that similar results could be extended to Volterra control of parabolic PDEs, albeit with modifications to account for semigroup decay rather than transport crossing.

There are strong links to classical works on approximate feedback linearization in finite dimensions [Krener1984ApproximateLinearization, Kang1994ApproximateLinearization] and to Volterra theory from a broader systems perspective. Potential future developments include the integration of operator learning approaches—such as neural operators—for real-time feedback computation, and systematic extensions to more general classes of infinite-dimensional, nonlinear boundary control systems.

Conclusion

This paper provides a comprehensive quantitative framework for the local stabilization of nonlinear hyperbolic PDEs via approximate feedback linearization with truncated Volterra operators. It demonstrates, both in the sup-norm and $F$4 norm, that stability and finite-time attractivity are preserved under finite truncation, with explicit and computable guarantees on the region of attraction and the decay properties. The analysis bridges rigorous infinite-dimensional feedback theory with practical implementation requirements, forming the foundation for future developments in data-driven, model-informed control design for nonlinear distributed parameter systems.

Reference:

"Approximate Feedback Linearization for a Nonlinear Hyperbolic PDE Class -- Part I: Volterra Truncation" (2607.04361)