Boundary predictive control with Riemann invariants approach for 2 × 2 hyperbolic systems

Many physical systems are commonly expressed as collections of conservation laws which are described by 2 × 2 hyperbolic partial differential equations (PDEs). To address the control problem of such systems, in this paper, a boundary predictive control algorithm with Riemann invariants approach is presented. An adjoint-based approach is applied to the static optimization problem of model predictive control in order to deduce the corresponding adjoint equations. Notably, the adjoint equations are inherently of the same type as the original equations, both being 2 × 2 hyperbolic PDEs. Furthermore, the eigenvalues of the coefficient matrices of the adjoint equations and the original equations are proven to be identical. Consequently, along these identical characteristic curves, we reformulated the gradient expression under the same Riemann invariant coordinate. This transformation converts an infinite-dimensional problem, initially addressed across the entire Banach space, into a one-dimensional problem tackled along the characteristic curves. Notably, computing the Riemann-based gradients only requires solving Riemann invariants along characteristic curves, without the need to solve the adjoint equations. The dynamics of the Riemann invariants are described by ordinary differential equations along the characteristic curves, greatly reducing redundant computations. The presented algorithm possesses general applicability to 2x2 hyperbolic systems, a claim demonstrated by validation through two illustrative examples, an open-channel system and a traffic flow scenario.