3.4 Neumann Boundary Control
3.4.2 Exponential Constants for Neumann Boundary Control II
In this subsection we consider the Neumann controlled PDE (3.42) with an arbi-trarily large reaction parameter µ. Since the backstepping approach only works in one dimension we investigate
yt(x, t) =yxx(x, t) +µy(x, t) in (0,1)×(0,∞) (3.49a)
y(0, t) = 0 in (0,∞) (3.49b)
yx(1, t) = u(t) in (0,∞) (3.49c)
y(x,0) =y0(x) in (0,1). (3.49d)
As stage cost we use the one dimensional version of (3.43), i.e., l(y(n), u(n)) = 1
2ky(·, nT)k2L2(Ω)+λ
2|u(nT)|2. (3.50) Theorem 3.15
The Neumann controlled heat equation (3.49) with control (3.27), whereK is chosen such that K > µ−λ1, and stage cost (3.50) satisfies the exponential controllability condition (1.34). The corresponding constants are given byσ=e−2T(λ1−µ+K) ∈(0,1)
Proof. First, we note that the Volterra kernel for the Neumann control is exactly the same as in the Dirichlet case (only the feedback control and the eigenvalue change).
With the same calculation as in Theorem 3.10 we get l∗(y(n)) = 1
2ky(·, nT)k2L2(Ω) ≤ 1
2ξ(K)σnky(·,0)k2L2(Ω) =ξ(K)σnl∗(y0) (3.51) with ξ(K) := (1 + L1(K))2(1 + L2(K))2 and σ := e−2T(λ1−µ+K). With similar arguments as in Theorem 3.10 and Lemma 3.9 we obtain
|u(t)| ≤ K
3 Minimal Stabilizing Horizons
≤
√M
2 + ˜η(K)
! K
Z 1 0
y(x, t)2 dx 12
This yields
l(y(n), u(n)) = 1
2ky(·, nT)k2L2(Ω)+λ
2|u(nT)|2 ≤Cl˜ ∗(y(n)) (3.52) with ˜C =
1 +λK2√ M
2 + ˜η(K)2
. By combining (3.51) and (3.52) we obtain l(y(n), u(n))≤Cl˜ ∗(y(n))≤Cξ(K˜ )σnl∗(y0) =Cσnl∗(y0) (3.53) with C := ˜Cξ(K).
It should be mentioned that the smallest eigenvalue in the previous theorem is given by λ1 = π2/4 and, thus, is different to that in the Dirichlet case where we have λ1 =π2.
3.4.3 Numerical Results
In this section we investigate the Neumann controlled heat equation in the context of our theoretical results from the previous section. The case of Neumann control is remarkable because we obtained different exponential constants for different mag-nitudes of the reaction value µ. This is the opposite behaviour to the distributed and the Dirichlet boundary control. The focus of this section is on the investigation whether the different controls from Section 3.4.1 and 3.4.2 result in a different be-haviour of the minimal stabilizing horizon.
For this purpose we consider the Neumann controlled equation in one dimension (3.49). From the spectral properties we know that the uncontrolled equation is un-stable forµ≥π2/4. The smallest eigenvalue of the corresponding Dirichlet problem is given by CD =π2. Thus, for π2/4≤ µ < π2 we can stabilize (3.49) with control (3.44). Moreover, we can use the results from Theorem 3.14. For larger values of the reaction parameter (µ≥π2) we have to use the weaker results from Theorem 3.15.
We focus again on the analysis of the overshoot bound. It is visible that the overshoot constant in Theorem 3.14 (C = (1 +λK2M)) has a very similar structure to that in the distributed controlled case in Theorem 3.4 (C = (1 +λK2)). In contrast to this the overshoot constant for large values of µ(C = (1 +λK2(√2M + ˜η(K))2)ξ(K)) re-sembles that of the Dirichlet controlled case (C = (1 +λK2η(K))ξ(K)). Therefore, from our theory we expect a Dirichlet like behaviour for µ ≥ π2 and a distributed control like behaviour for π2/4 ≤ µ < π2. This is exactly what we observe in the numerical simulation.
First, we recapitulate the major differences between Dirichlet and distributed con-trol from Subsection 3.3.2. One observation was that it is always possible to stabilize the heat equation with distributed control with the smallest horizon by reducing the
3.4 Neumann Boundary Control value of the regularization parameter λ. Obviously, the arguments from Subsection 3.3.2 are also true for the Neumann controlled equation with µ < π2: We can bring C = (1 +λK2M) arbitrarily close to 1 by reducing the value of λ. Thus, it is pos-sible to stabilize the problem with a horizon N = 2 for sufficiently small λ. Since the control (3.44) does not stabilize the problem for a reaction parameter µ ≥ π2 we have to choose the control (3.27) and we can use the results from Theorem 3.15.
With the same arguments as for the Dirichlet controlled system we have C →ξ(K) for λ → 0. Thus, for µ ≥ π2 we cannot guarantee stability with a horizon N = 2 even for arbitrarily small λ.
An interesting question is whether the bound π2 is actually observable in the nu-merical simulation. To answer the question we have to keep the influence of the decay rateσ small. One possibility to do this is to reduce the sampling timeT. For T →0 we obtainσ →1 and, thus, we can observe the pure influence ofC. In Figure 3.1 the maximum value of µ is displayed where stability is obtained with N = 2 depending on the sampling time T (blue line). Furthermore, we see the bound π2 derived from the theory (dashed black line). It is observable that the curve tends to the theoretical bound for small values of T. For even smaller values of T, the numerical errors become predominant.
10−3 10−2 10−1
10 12 14 16 18 20
T
µ
Figure 3.1: Maximum value of µ where stability with N = 2 is observed in the numerical simulation, in dependence of the sampling timeT.
Furthermore, we have seen in Section 3.3.2 that in the Dirichlet case the stage cost, where the gradient of the state is penalized, leads to a much longer horizon than the stage cost, where the state is penalized. In the case of distributed control we observe the contrary behaviour for L = 1 (see Table 3.2). In the next example we investigate this problem for the Neumann controlled case. We consider the one dimensional heat equation (3.49) with stage cost
l(y(n), u(n)) = 1
2ky(·, nT)k2L2(Ω)+ λ
2|u(nT)|2 (3.54)
3 Minimal Stabilizing Horizons and
l(y(n), u(n)) = 1
2kyx(·, nT)k2L2(Ω)+λ
2|u(nT)|2 (3.55) Following our analysis we expect that stage cost (3.55) yields shorter stabilizing horizons than (3.54) forµ < π2 (’distributed control like behaviour’). This property changes to the opposite for µ ≥ π2 (’Dirichlet control like behaviour’). In the numerical simulation we choose the parameters T = λ = 0.01. The results are presented in Table 3.4. We denote the minimal stabilizing horizon for stage cost (3.54) and (3.55) withNkyk andNk∇yk, respectively. Obviously, the required horizon for the cost (3.55) is much smaller than for (3.54) in the case of a reaction parameter up toµ= 9. This behaviour changes drastically for values aboveµ= 10. Thus, the numerical results perfectly match our theoretical findings. Once again we observe that our theoretical bound µ=π2 is actually tight.
µ Nkyk Nk∇yk
5 3 2
6 4 2
7 4 2
8 5 2
9 6 2
10 6 16
11 7 33
Table 3.4: Comparison of the minimal stabilizing horizon for stage costs (3.54) Nkyk
and (3.55)Nk∇yk depending on the reaction parameterµ. The parameters for the numerical MPC simulation are given by T =λ= 0.01.