Input-to-State Stability of Non-uniform Linear Hyperbolic Systems of Balance Laws via Boundary Feedback Control
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Input-to-State Stability of Non-uniform Linear Hyperbolic Systems of Balance Laws via Boundary Feedback Control Gediyon Y. Weldegiyorgis1
· Mapundi K. Banda1
Accepted: 10 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)Lyapunov function in L 2 −norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function. Keywords Lyapunov function · Hyperbolic PDE · System of balance laws · Feedback control Mathematics Subject Classification 65Kxx · 49M25 · 65L06
1 Introduction We consider a k × k system described by the following linear hyperbolic system of balance laws with variable coefficients ∂t W (x, t) + (x)∂x W (x, t) + (x)W (x, t) = 0, (x, t) ∈ [0, l] × [0, +∞), (1) where W := W (x, t) diag{+ (x), −− (x)}, −− (x) = diag{λi (x) and (x) ∈ Rk×k is a
B
: [0, l] × [0, +∞) → Rk is a state vector, (x) = with + (x) = diag{λi (x) > 0 : i = 1, . . . , m} and < 0 : i = m + 1, . . . , k}, is a non-zero diagonal matrix non-zero matrix. Corresponding to the diagonal entries of
Mapundi K. Banda [email protected] Gediyon Y. Weldegiyorgis [email protected]
1
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, Gauteng, South Africa
123
Applied Mathematics & Optimization
(x), the state vector W is specified by W = [W + , W − ] , where W + ∈ Rm and W − ∈ Rk−m . The system (1) is subject to an initial condition set as W (x, 0) = W0 (x), x ∈ (0, l),
(2)
for some function W0 : (0, l) → Rk and linear feedback boundary conditions with disturbances defined by
+ W (l, t) W + (0, t) = K + Mb(t), t ∈ [0, +∞), W − (l, t) W − (0, t)
(3)
0 K− , with K − ∈ where K ∈ is a constant matrix of the form K = K+ 0 Rm×(k−m) and K + ∈ R(k−m)×m , M ∈ Rk×k is a non-zero constant diagonal matrix, and b ∈ Rk is a vector of disturbance functions. It is for such a system that the Input-to-State Stability (ISS) will be discussed in this paper. In science and engineering, many important physical phenomena, in particular flow of fluids such as flow of shallow water, gas, traffic and electricity, have mathematical models that describe the dynamic behaviour of the flow in terms of mathematical equations. These mathematical models are mainly represented by hyperbolic systems of balance laws, e.g. Saint-Venant equations, isentropic Euler equations, or Telegrapher’s equations. The solution of linear hyperbolic systems of balance laws under an initial condition, boundary conditions and initial-boundary compatibility conditions exist and are unique (see [5,26]). Stabilisation problems with boundary controls (also called boundary feedbacks or bo
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