Exponential Turnpike in Optimal Boundary Control of Multi-Dimensional First-Order Hyperbolic Systems

RESUMO

We establish an exponential turnpike property for optimal boundary control of multi-dimensional first-order linear hyperbolic systems in which both the control and observation operators are unbounded and the cost functions may be nonquadratic. The main assumptions are internal and boundary dissipation conditions. These conditions ensure that the associated input-to-state and input-to-output maps are bounded, with operator norms independent of the time horizon T. This uniform boundedness is essential to the proof, which combines weighted energy estimates for the forward and adjoint equations, input-output duality, and strong convexity. Finally, we apply the result to the linearized two-dimensional Saint-Venant equations with sluice-gate control and illustrate the turnpike behavior numerically.