<p>This paper introduces bilateral boundary control laws for linear <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(2+n+2)\times (2+n+2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> hyperbolic partial differential equation (PDE) systems characterized by <i>n</i> equations with zero transport velocities, two equations with identical positive velocities, and two equations with identical negative velocities. Zero transport velocities implies that the transport direction of the system state is parallel to the time axis, whereas equal transport velocities indicate that the propagation directions between the system states remain parallel. Bilateral boundary control signifies the presence of actuators at both ends of the domain. A full-state feedback control law is developed to ensure global exponential stability of the closed-loop system in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm. The presence of equal and zero transport velocities renders traditional backstepping design ineffective, leading to infinite controller gain. Consequently, we initially decouple states with identical velocities and apply the symmetric-Volterra transformation (an integral operator) solely to the subsystem with non-zero transport velocities, ensuring the target subsystem with zero transport velocities achieves input-to-state stability. The presence of actuators at both ends of the domain renders the existing Lyapunov functional inapplicable. Consequently, exponential stability of the target system is achieved through the construction of a modified Lyapunov functional. The emergence of zero velocity restricts the arbitrariness of the exponential convergence rate in hyperbolic systems. A key contribution of this study is the establishment of the convergence rate range for closed-loop systems. Additionally, the design of bilateral controllers facilitates the development of fault-tolerant control strategies for linear hyperbolic systems.</p>

Bilateral Boundary Control of One-Dimensional Linear Hyperbolic PDEs with Equal and Zero Transport Velocities

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发表日期：

2026年4月10日

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Wei Sun,
Jing Li,
Liangyu Xu

摘要

This paper introduces bilateral boundary control laws for linear $(2+n+2)\times (2+n+2)$ $(2 + n + 2) \times (2 + n + 2)$ hyperbolic partial differential equation (PDE) systems characterized by n equations with zero transport velocities, two equations with identical positive velocities, and two equations with identical negative velocities. Zero transport velocities implies that the transport direction of the system state is parallel to the time axis, whereas equal transport velocities indicate that the propagation directions between the system states remain parallel. Bilateral boundary control signifies the presence of actuators at both ends of the domain. A full-state feedback control law is developed to ensure global exponential stability of the closed-loop system in the $L^{2}$ $L^{2}$ norm. The presence of equal and zero transport velocities renders traditional backstepping design ineffective, leading to infinite controller gain. Consequently, we initially decouple states with identical velocities and apply the symmetric-Volterra transformation (an integral operator) solely to the subsystem with non-zero transport velocities, ensuring the target subsystem with zero transport velocities achieves input-to-state stability. The presence of actuators at both ends of the domain renders the existing Lyapunov functional inapplicable. Consequently, exponential stability of the target system is achieved through the construction of a modified Lyapunov functional. The emergence of zero velocity restricts the arbitrariness of the exponential convergence rate in hyperbolic systems. A key contribution of this study is the establishment of the convergence rate range for closed-loop systems. Additionally, the design of bilateral controllers facilitates the development of fault-tolerant control strategies for linear hyperbolic systems.