Backstepping Control of Multidimensional Coupled First-Order Hyperbolic PDEs with Collinear Velocities

Technical Summary: Backstepping Control of Multidimensional Coupled First-Order Hyperbolic PDEs with Collinear Velocities

Problem Statement
This paper addresses the boundary stabilization problem for coupled multidimensional first-order hyperbolic partial differential equation (PDE) systems. While significant advances have been made in the boundary control of one-dimensional (1D) hyperbolic systems using backstepping and Fredholm techniques, extending these methods to arbitrary spatial dimensions remains challenging. The specific class of systems considered involves $n+m$ coupled states ($u$ and $v$) propagating in opposite directions within a bounded, open, and connected domain $\Omega \subset \mathbb{R}^N$. The core difficulty lies in the multidimensional nature of the transport operators and the coupling between states, which complicates the direct application of 1D control designs.

Methodology
The proposed approach relies on a geometric transformation that reduces the multidimensional system to a continuum of independent one-dimensional systems. The methodology proceeds through the following steps:

1. Collinearity Assumption: The study restricts the transport velocity fields to be collinear. Specifically, all velocity fields are assumed to be scalar multiples of a single base vector field $a(x)$. This implies that at any point $x$, transport occurs either in the direction of $a(x)$ or $-a(x)$.
2. Characteristic-Based Change of Variables: Building on a framework previously developed for scalar multidimensional equations, the authors introduce a coordinate transformation based on characteristic curves defined entirely within the spatial domain (rather than space-time).
   • Characteristic curves are defined as solutions to $\frac{d\phi}{ds} = a(\phi)$ with $\phi(0) = x$.
   • The domain is foliated by these curves. A new coordinate system $(\sigma, \rho)$ is introduced, where $\rho$ represents the entry point of a characteristic on the boundary $\Gamma_-$, and $\sigma$ represents the distance traveled along the characteristic.
3. Reduction to a Continuum: Under this transformation, the original multidimensional PDE system is reformulated as a continuum of coupled 1D first-order hyperbolic systems. Crucially, while the states within each 1D system remain coupled (as in standard 1D backstepping problems), the systems corresponding to different characteristics (different $\rho$) are decoupled from one another.
4. Backstepping Controller Design:
   • The 1D systems in the continuum are normalized to a standard domain $[0, 1]$.
   • Standard backstepping controllers, previously established for 1D coupled hyperbolic systems (referencing works [4], [13], [24]), are applied to each element of the continuum independently.
   • The control law is derived as an integral over the characteristic curve leading to the boundary point where control is applied.
5. Stability Analysis: The finite-time stability of the original multidimensional system is deduced from the finite-time stability of each 1D system in the continuum. This relies on the assumption that the transit times along the characteristics are uniformly bounded.

Key Contributions

• Extension to Coupled Multidimensional Systems: The paper extends the backstepping methodology from scalar multidimensional equations (as treated in [8]) to coupled multidimensional systems.
• Decoupling via Transformation: The primary theoretical contribution is demonstrating that under the collinearity assumption, a coupled multidimensional system can be transformed into a continuum of coupled 1D systems where the coupling is confined strictly within individual 1D systems. This eliminates the need for complex ensemble control techniques that handle coupling across the entire continuum.
• Explicit Control Law: The authors derive an explicit boundary control law (Equation 32) expressed in the original spatial variables. This law involves an integral over the upstream characteristic arc $C(z)$, utilizing kernel functions that satisfy specific hyperbolic PDEs (kernel equations) defined on the transformed domain.

Results
The paper establishes the following main result (Theorem 2):

• Under Assumptions 1–4 (smooth domain, Lipschitz velocity, non-trapping characteristics with uniformly bounded transit times, and collinear velocities), the proposed boundary control law achieves finite-time stabilization of the origin for the multidimensional system.
• The settling time $t_F$ for the entire system is bounded by $T_{max} \left( \frac{1}{\lambda_1} + \frac{1}{\mu_1} \right)$, where $T_{max}$ is the maximum transit time across the domain and $\lambda_1, \mu_1$ are the smallest transport velocities.
• The solution is proven to exist and be unique in the sense of "weak characteristic solutions," which aligns with classical solutions under appropriate regularity conditions.

Significance and Scope
The paper positions this work as a "first step" toward the backstepping control of general coupled first-order hyperbolic systems in multiple space dimensions.

• Limitations: The authors explicitly acknowledge that the collinearity assumption is restrictive. In higher dimensions, general velocity fields are not necessarily collinear, and the current method does not yet address systems where transport occurs along multiple non-aligned axes simultaneously.
• Future Directions: The authors suggest that while their current transformation decouples the continuum elements, the general non-collinear case would likely result in couplings between different elements of the continuum. They propose that future research could investigate whether existing techniques for controlling ensembles of coupled hyperbolic equations can be adapted to handle these residual couplings.
• Contribution to the Field: The work bridges the gap between the control of multidimensional PDEs and the control of continua of 1D PDEs, offering a constructive method for stabilization where the multidimensional complexity is managed through geometric transformation rather than direct high-dimensional control synthesis.