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# Discretized feedback control for systems of linearized hyperbolic balance laws

This work is supported by DFG HE5386/13-15.
• 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.

Mathematics Subject Classification: Primary: 37L45, 93B52, 35B35; Secondary: 35B30, 93B18, 93C73, 93D15, 93D05.

 Citation: • • Figure 1.  $\log_2$ of $\hat{L}^1_x$-error for Riemann invariants

Figure 2.  Lyapunov functions for ${G = \mathbb{1}}$ and ${\hat{\mu} = 0}$

Figure 3.  Pipeline with two compressors and stability domain for the $L^2$-norm

Figure 4.  Lyapunov functions for the steady states ${\bar{h}(x): = 3}$ and ${\bar{h}(x): = 3+10^{-3} \cos(2 \pi x)}$

Figure 5.  Non-decaying Lyapunov functions for the steady state ${\bar{h}(x): = 3+0.1 \cos(2 \pi x)}$

Table 1.  $\hat{L}^1_x$-error and EOS for Riemann invariants

 $\hat{L}^1_x$-error EOC △x d=0 d=1 d=2 d=3 d=4 d=0 d=1 d=2 d=3 d=4 $2^{-4}$ 0.52 3.61 24.11 146.27 953.65 $2^{-5}$ 0.27 1.87 12.46 75.48 492.38 0.95 0.95 0.95 0.95 0.95 $2^{-6}$ 0.13 0.94 6.27 37.97 247.28 0.99 0.99 0.99 0.99 0.99 $2^{-7}$ 0.07 0.46 3.11 18.80 121.91 1.01 1.01 1.01 1.01 1.02 $2^{-8}$ 0.03 0.23 1.51 9.12 58.63 1.04 1.04 1.04 1.04 1.06 $2^{-9}$ 0.02 0.11 0.70 4.26 26.98 1.10 1.10 1.10 1.10 1.12

Table 2.  Error of Lyapunov function for Euler]{$L^1_t$-, $L^2_t$-, $L^\infty_t$-error and EOC for Lyapunov functions, units in $0.01$

 ${L_t^1\text{-error}}$ EOC $\Delta x$ d=0 d=1 d=2 d=3 d=4 d=0 d=1 d=2 d=3 d=4 $2^{-4}$ 6.90 7.08 7.08 7.08 7.00 $2^{-5}$ 3.61 3.70 3.70 3.70 3.63 0.93 0.93 0.93 0.93 0.95 $2^{-6}$ 1.84 1.88 1.88 1.88 1.81 0.98 0.98 0.98 0.98 1.00 $2^{-7}$ 0.91 0.94 0.94 0.94 0.87 1.01 1.01 1.01 1.01 1.06 $2^{-8}$ 0.45 0.46 0.46 0.46 0.39 1.04 1.04 1.04 1.04 1.17 $2^{-9}$ 0.21 0.21 0.21 0.21 0.14 1.10 1.10 1.10 1.10 1.42 ${L_t^2\text{-error}}$ EOC d=0 d=1 d=2 d=3 d=4 d=0 d=1 d=2 d=3 d=4 $2^{-4}$ 7.90 8.04 8.04 8.04 7.96 $2^{-5}$ 4.15 4.23 4.23 4.23 4.15 0.93 0.93 0.93 0.93 0.94 $2^{-6}$ 2.12 2.16 2.16 2.16 2.08 0.97 0.97 0.97 0.97 1.00 $2^{-7}$ 1.06 1.08 1.08 1.07 1.00 1.00 1.00 1.00 1.00 1.06 $2^{-8}$ 0.51 0.52 0.52 0.52 0.44 1.04 1.04 1.04 1.04 1.16 $2^{-9}$ 0.24 0.25 0.25 0.25 0.17 1.09 1.09 1.09 1.09 1.42 ${L_t^\infty\text{-error}}$ EOC $\Delta x$ d=0 d=1 d=2 d=3 d=4 d=0 d=1 d=2 d=3 d=4 $2^{-4}$ 13.90 13.92 13.92 13.91 13.78 $2^{-5}$ 7.41 7.43 7.43 7.42 7.29 0.91 0.91 0.91 0.91 0.92 $2^{-6}$ 3.79 3.79 3.79 3.79 3.66 0.97 0.97 0.97 0.97 0.99 $2^{-7}$ 1.87 1.87 1.87 1.87 1.74 1.02 1.02 1.02 1.02 1.07 $2^{-8}$ 0.89 0.90 0.90 0.90 0.77 1.06 1.06 1.06 1.06 1.19 $2^{-9}$ 0.42 0.42 0.42 0.42 0.29 1.09 1.09 1.09 1.09 1.40

Table 3.  Estimated decay rate µe (top), guaranteed rate µg (middle) and observed rate µo (bottom) for $\mathsf{\hat{\mu }}$ : = 0.25 with constant and perturbed steady states

 Estimated rate constant p=4 p=3 p=2 p=1 d=0 0.2499 0.2498 0.2485 0.2254 -0.1563 d=1 0.2499 0.2495 0.2420 0.0374 -2.2991 d=2 0.2499 0.2458 0.1288 -1.3167 -15.8406 Guaranteed rate constant p=4 p=3 p=2 p=1 d=0 0.2499 0.2498 0.2487 0.2279 -0.1154 d=1 0.2499 0.2495 0.2428 0.0591 -2.0424 d=2 0.2499 0.2463 0.1412 -1.1566 -14.2202 Observed rate constant p=4 p=3 p=2 p=1 d=0 0.2572 0.2572 0.2572 0.2537 0.1343 d=1 0.3461 0.3459 0.3416 0.2126 -1.3244 d=2 0.2632 0.2615 0.2089 -0.6659 -10.7687
• Figures(5)

Tables(3)

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