Feedback Linearization of Hyperbolic PDEs with Volterra Nonlinearities

Abstract: Alberto Isidori's framework of geometric nonlinear control, and particularly of feedback linearization, is the inspiration behind PDE backstepping: apply a transfromation of the state to cast the plant into a canonical form, bring all the non-canonical effects within the "span" of (boundary) control, and close the design with a feedback that makes the closed loop evolve in accordance with well-studied stable dynamics. The specificity of this approach is that, for PDEs, there is not one canonical form (like Brunovsky for ODEs) but the canonical forms are PDE-class-specific. When conducting this process for nonlinear PDEs, where the "transformation of the state" is performed using a nonlinear Volterra series indexed by the spatial variable, enormous technical challenges arise. One has to deal with kernels governed by PDEs on simplex domains growing in dimension to infinity, capture the growth rates of these kernels of the "direct transformation," and conduct the same for the "inverse transformation" without directly studying its Volterra kernels. So far, this agenda has been executed only once, two decades ago: for parabolic PDEs by Vazquez and Krstic [Automatica, 2008]. Generalization attempts have not followed because of the immense complexity involved in feedback-linearizing nonlinear PDEs. In this paper, dedicated to Professor Isidori, we convert the PDE feedback-linearizing methodology of 2008 from the parabolic to a hyperbolic class and, for a transport-adapted subclass of Chen-Fliess series, construct controllers without kernel PDEs.

• The paper introduces a feedback-linearizing transformation for hyperbolic PDEs with spatial Volterra nonlinearities and demonstrates local exponential stability.
• The methodology employs recursive kernel equations and contraction mapping to construct both the transformation and its Lipschitz continuous inverse.
• Numerical simulations and rigorous L2 analysis validate the approach for transport-adapted Chen–Fliess systems and practical control applications.

Feedback Linearization of Hyperbolic PDEs with Volterra Nonlinearities

Introduction and Motivation

This work extends the methodological framework of feedback linearization from parabolic to first-order hyperbolic partial differential equations (PDEs) with spatial Volterra nonlinearities. This approach is rooted in the geometric nonlinear control paradigm, where feedback linearizability is treated as a structural property of the system, not merely as a design procedure. The paper’s central objective is to develop a feedback-linearizing transformation for hyperbolic PDEs whose nonlinearities are given by a Volterra series expansion, beginning at quadratic order and expressed in terms of spatial variables. The technical obstacles addressed include the construction and boundedness of the transformation kernels and ensuring both the well-posedness and stability of the closed-loop system.

Problem Formulation

The principal system under consideration is the transport PDE:

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

subject to the boundary input $u(1,t) = U(t)$ and initial condition $u(x,0) = u_0(x)$. The nonlinearity $F[u](x,t)$ is a spatial Volterra series,

$$F[u](x,t) = \sum_{n=2}^\infty \int_{T_n(x)} f_n(x,\xi_1,\dots,\xi_n) \prod_{i=1}^n u(\xi_i,t)\, d\xi_n \cdots d\xi_1,$$

with the kernels $f_n$ satisfying essential boundedness on simplex domains. The linear Volterra term is preliminarily removed via linear backstepping, leaving focus on purely nonlinear effects. The challenge is to construct a nonlinear Volterra transformation and associated boundary feedback such that the closed-loop PDE is mapped to a canonical transport equation with homogeneous boundary conditions.

Methodology

The transformation is constructed as follows:

$w(x,t) = u(x,t) - K[u](x,t),$

where $K[u](x,t)$ is itself a Volterra series with kernels $k_n$ to be determined recursively. The boundary feedback is selected as $U(t) = K[u](1,t)$ to enforce $u(1,t) = U(t)$0. Substitution into the PDE and enforcement of the linear target dynamics lead to a hierarchy of kernel equations for $u(1,t) = U(t)$1, each posed on a simplex and governed by linear first-order (transport) PDEs. These equations are solvable explicitly along characteristics due to their transport structure, and are coupled only to lower-order kernels. This structure enables recursive construction without solving high-dimensional kernel PDEs.

Kernel Boundedness and Convergence

A critical contribution is the thorough analysis of kernel growth. Using Lyapunov functionals on simplex domains and recursive $u(1,t) = U(t)$2 estimates, the boundedness and convergence of the Volterra transformation are ensured under mild growth conditions on the original nonlinear kernels. The convergence radius is quantified explicitly, and global convergence is shown for systems with entire or polynomial-type nonlinearities.

Inverse Transformation and Lipschitz Analysis

Rather than relying on formal series inversion (as in prior work on parabolic PDEs using Boyd–Chua inversion), the inverse transformation is constructed via a contraction mapping argument in $u(1,t) = U(t)$3. The nonlinear mapping is shown to be locally Lipschitz, and explicit small-gain conditions provide concrete estimates for the region of invertibility and stability. Both the transformation and its inverse are established to be Fréchet $u(1,t) = U(t)$4-diffeomorphisms in a neighborhood of the origin in $u(1,t) = U(t)$5.

Main Results: Stability and Well-Posedness

The closed-loop system, under the feedback-linearizing transformation, is shown to be locally exponentially stable in the $u(1,t) = U(t)$6 norm. The transformed variable $u(1,t) = U(t)$7 evolves according to a simple transport equation with zero boundary, ensuring finite-time extinction followed by exponential decay. All analytical estimates are translated back to the original variables using the explicit Lipschitz properties of the transformation and its inverse.

Well-posedness of the closed-loop system is rigorously established in the mild $u(1,t) = U(t)$8 sense, circumventing the limitations imposed by the presence of unbounded operators and nonlinearities of Volterra type. The solution semiflow is shown to be bi-Lipschitz continuous, and uniqueness follows directly from contraction properties in Banach space.

Specialization: Transport-Adapted Chen–Fliess Series

A significant computational advance is obtained when the system’s nonlinearity has a transport-adapted Chen–Fliess (gap-basis) representation with analytic and geometrically decaying coefficients. In this case, the recursive construction of backstepping kernels is reduced to a cascade of scalar ODEs (or quadratures) for the coefficient functions, rather than requiring characteristic integration on high-dimensional simplices. The kernel PDEs are thus avoided entirely, offering a fully one-dimensional construction for the infinite-dimensional control law. The analytic convergence of the resulting series is established in a divided-power coefficient algebra framework.

Numerical Illustration

The theoretical developments are illustrated on a nonlinear PDAE example, where the nonlinearity arises as the square of a spatial integral of the solution. The feedback law is constructed via the presented methodology, and numerical simulations demonstrate the stabilizing effect of successive truncations of the feedback-linearizing Volterra series. Notably, including only the linearizing control kernels up to a given degree removes the corresponding nonlinear obstruction, consistent with the formal structure of Volterra feedback linearization.

Implications, Limitations, and Future Directions

This work establishes, for the first time, a rigorous feedback-linearization procedure for nonlinear hyperbolic PDEs with Volterra nonlinearities, generalizing the approach previously limited to parabolic systems. The explicit technical machinery, including recursive $u(1,t) = U(t)$9 kernel analysis and constructive inversion via contraction mapping, represents a substantial advance in the nonlinear control of infinite-dimensional systems.

The practical implications are pronounced for classes of hyperbolic PDEs that admit causal polynomial representations, such as those arising in transport phenomena, fluid flows with specific nonlinearities, and systems admitting Koopman or Chen–Fliess expansions. The reduction of infinite-dimensional design to sequences of coupled ODEs in these cases opens the way for effective computational implementation and real-time control.

The region of attraction, as in previous nonlinear PDE stabilization results, remains generally local due to the nonlinearity, and closed-form expansion of the region is determined through explicit small-gain conditions. The methods are inherently class-specific, consistent with the lack of a general theory for feedback linearization in PDEs.

Potential future developments include extending these techniques to more general boundary conditions, multidimensional domains, and broader classes of nonlinear PDEs, as well as investigating robustness under model uncertainties and actuator constraints. Furthermore, the algebraic analysis of the coefficient recursion and possible closure within broader functional-analytic structures could yield deeper insight into the universality and limitations of feedback linearization in infinite dimensions.

Conclusion

The presented results provide a rigorous, analytically explicit, and computationally tangible approach to feedback linearization of first-order hyperbolic PDEs with Volterra nonlinearities. By leveraging transport-adapted Volterra and Chen–Fliess representations, the theoretically grounded design bypasses high-dimensional kernel PDEs, admits explicit gain and convergence ratios, and guarantees local exponential closed-loop stability via bi-Lipschitz transformations on $u(x,0) = u_0(x)$0. This establishes a new archetype for nonlinear feedback control design in infinite-dimensional systems and sets a foundation for further developments in PDE control theory.