Discretized feedback control for systems of linearized hyperbolic balance laws

Abstract

Physical systems such as water and gas networks are usually operated in a state of equilibrium and feedback control is employed to damp small perturbations over time. We consider flow problems on networks, described by hyperbolic balance laws, and analyze the stability of the linearized systems. Sufficient conditions for exponential stability in the continuous and discretized setting are presented. The analysis is extended to arbitrary Sobolev norms. Computational experiments illustrate the theoretical findings.

Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition

“…We rather use discrete Lyapunov functions to prove exponential decay of our solution. In a L 2 framework, the articles [18,5] prove a stabilization result for discretized linear balance laws coupled in the domain and at the boundary using such techniques. However, it does not seem to be generalizable to a BV context for nonlinear scalar conservation laws.…”

“…) where a n j was defined in (18). Because of the Harten's condition (19) we have for 1 ≤ i ≤ d and 1 ≤ j ≤ N − 1:…”

Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity

“…For β → 0, we have lim β→0 a(β) = −θ > −1 and we have lim β→0 β e β −1 = 1. Hence, there exists a β * * > 0 such that for all β ≤ min{β * , β * * } and for a(β) as given by Equation (27), the inequalities (26), ( 27) and ( 19) hold true. Using the inequality (26) and the particular choice for a(β) and (β), we obtain for all β sufficiently small…”

“…The Lyapunov method is also used for ISS of (local) hyperbolic systems in [11,20]. For the numerical analysis of asymptotic behavior of numerical solutions discretized by a first-order finite volume scheme, a discrete Lyapunov function is used to prove exponential stability results for hyperbolic systems in [24][25][26][27][28] and for scalar conservation laws with nonlocal velocity in [10], and ISS results for (local) hyperbolic systems could be established recently in [29,30]. Please note that the previous given references refer to ISS for hyperbolic systems.…”

Chance‐constrained optimal inflow control in hyperbolic supply systems with uncertain demand

“…With model (), we have a very generic building block for the energy system that can also be seen in the general light of the study of flows on networks. For example, the stochastic optimal control approach might also be useful for control tasks related to water flow considerations (see References 34,35).…”