Stabilization of a chain of 3 hyperbolic PDEs with 2 inputs in arbitrary position

This paper addresses the stabilization of a chain of three coupled hyperbolic partial differential equations actuated by two control inputs applied at arbitrary nodes of the network. With the exception of configurations where one input is located at an endpoint, cases already well studied in the literature, all admissible two-inputs configurations are treated in this paper within a unified framework. The proposed approach relies on a backstepping transformation combined with a reformulation of the closed-loop dynamics as an Integral Difference Equation (IDE). This IDE representation reveals a common structural pattern across configurations and clarifies the role played by delayed dynamics in the stability analysis. Within this formulation, the stabilization problem can be handled using existing IDE control techniques. For most configurations, the stabilization of the PDE system requires an approximate spectral controllability assumption. Remarkably, one specific configuration can be stabilized without imposing any additional spectral condition. In contrast, we also provide an explicit example of a configuration for which the required spectral controllability property fails to hold.

💡 Research Summary

This paper investigates the exponential stabilization of a chain of three coupled hyperbolic partial differential equations (PDEs) using two boundary control inputs. The actuators can be placed at arbitrary boundary nodes of the network. The authors exclude configurations where one input is located at an endpoint of the chain, as these cases have been previously solved. For all remaining admissible two-input configurations, a unified design and analysis framework is proposed.

The system consists of three segments, each modeled by a pair of counter-propagating waves (u_i, v_i) with in-domain couplings (σ+_i, σ-_i) and boundary couplings (q_ij, ρ_ij) connecting adjacent segments. The core methodology involves two main steps. First, a backstepping transformation maps the original PDE system into an intermediate target system of transport equations with modified boundary conditions. Second, this target system is reformulated as an abstract scalar Integral Difference Equation (IDE). This IDE representation, featuring pointwise and distributed delays, reveals a common structural pattern across different actuator configurations and highlights the role of delayed dynamics in stability.

The analysis treats each configuration separately:

1. Configuration (U1, U4): This case naturally decouples into two independent subsystems: the first segment controlled by U1, and a chain of the second and third segments controlled at one end by U4. Stabilization is achieved by directly applying existing results for a single hyperbolic PDE and a two-PDE chain.
2. Configurations (U1, U3), (U4, U3), and (U4, U2): For these, the problem is equivalent to stabilizing an IDE of the form (3). Stabilization generally requires an “approximate spectral controllability” assumption (Assumption 4), which stipulates that three associated holomorphic functions (F0, F1, F2 derived from the IDE’s Laplace transform) have no common zeros.
   • For (U1, U3) and (U4, U3), one control input (V2) is chosen as a distributed-delay feedback of the state (equation (6)), reducing the system to a single-input IDE. This simplified IDE falls into the framework of existing control techniques for IDEs.
   • For (U4, U2), one control input (V2) is chosen as a distributed-delay feedback of the other input (V1) (equation (11)), again yielding a stabilizable single-input IDE.

A key contribution is the demonstration that the required spectral assumption is not universally valid. The paper constructs an explicit, non-degenerate example within the (U1, U3) configuration (with specific zero in-domain couplings) for which Assumption 4 fails. This shows a fundamental limitation where the proposed methodology is not applicable.

In summary, the paper provides a comprehensive treatment of stabilization for a three-PDE chain with two arbitrarily placed actuators. It unifies the analysis through an IDE framework, offers constructive control laws for most configurations under a spectral condition, and importantly identifies a configuration where stabilization via this approach is not possible due to a lack of spectral controllability.