Contrôle de l’équation de Burgers avec un unique contrôle scalaire
3.2 Preliminary technical lemmas
However, as a depends linearly on u, and b depends quadratically on a, we expect that we can find a kernelKε(s1, s2) such that :
hρ, b(1,·)i= Z 1
0
Z 1 0
Kε(s1, s2)u(s1)u(s2)ds1ds2. (3.16) Thanks to equation (3.15), we expect that (3.16) actually defines a positive definite kernel acting on u, allowing us to use its coercivity to overwhelm various residues.
In Section3.2, we recall a set of technical well-posedness estimates for heat and Burgers systems.
In Section3.3, we show that formula (3.16) holds and we give an explicit construction of the kernelKε. Moreover, we compute formally its limitK0as ε→0.
In Section 3.4, we prove that the kernelK0 is coercive with respect to the H−5/4(0,1) norm of the controlu, by recognizing a Riesz potential and a fractional laplacian.
In Section3.5, we use weakly singular integral operator estimates to bound the residues betweenKε andK0 and thus deduce thatKε is also coercive, forεsmall enough.
In Section3.6, we use these results to go back to the controllability of Burgers.
In Appendix3.8, we give a short presentation of the theory of weakly singular integral operators and a sketch of proof of the main estimation lemma we use.
3.2.2 Smooth setting for the heat equation
We start by recalling standard estimates in a smooth (strong) setting for one dimensional heat equa- tions that will be useful in the sequel. We state all results for standard forward heat equations, but the same results hold for backwards heat equations with final time conditions.
Lemma 11. Let f ∈L2((0, T)×(0,1))andz0∈H01(0,1). We consider the system :
zt−νzxx=f in (0, T)×(0,1), z(t,0) = 0 in (0, T),
z(t,1) = 0 in (0, T), z(0, x) =z0(x) in (0,1).
(3.22)
There is a unique solutionz∈XT to system (3.22). Moreover, it satisfies the estimate : νkzxxk2+√
νkzxk2+kztk2.kfk2+√
ν|z0x|2. (3.23)
Démonstration. The proof of the existence and uniqueness is standard. Let us recall how we can obtain estimate (3.23). We multiply equation (3.22) byzxx and integrate by parts overx∈(0,1). Thus,
d dt
1 2
Z 1 0
z2x
+ν Z 1
0
zxx2 =− Z 1
0
f zxx. (3.24)
For anyT0 < T, we can integrate (3.24) overt∈(0, T0). Hence, we obtain : 1
2|zx(T0)|22+ν Z T0
0
Z 1 0
zxx2 =− Z T0
0
Z 1 0
f zxx+1 2
z0x
2
2. (3.25)
From (3.25), we easily deduce that :
νkzxxk2.kfkL2+√
ν|z0x|2, (3.26)
√νkzxkL∞(L2).kfkL2+√
ν|z0x|2. (3.27)
Eventually, we obtain estimate (3.23) from estimates (3.26) and (3.27) since we can writeztasf+νzxx. Lemma 12. Let z0 ∈ H01(0,1) and considerz ∈XT the solution to system (3.22) with a null forcing term (f = 0). It satisfies :
kzk∞≤ z0
∞. (3.28)
Démonstration. Although (3.28) is not a direct consequence of the combination of (3.20) and (3.23) (which would yield a weaker conclusion), it can be obtained via a standard application of the maximum principle, which can be applied in this strong setting.
3.2.3 Weaker settings for the heat equation
Let us move on to weaker settings for the heat equation. Moreover, we introduce inhomogeneous boundary data as we will need them in the sequel.
Definition 5. Let f ∈(XT)0,v0, v1∈H−1/4(0, T)andz0∈H−1(0,1). We consider :
zt−νzxx=f in (0, T)×(0,1), z(t,0) =v0(t) in (0, T),
z(t,1) =v1(t) in (0, T), z(0, x) =z0(x) in (0,1).
(3.29)
We say thatz∈L2((0, T)×(0,1))is a weak solution to system (3.29)if, for all g∈L2((0, T)×(0,1)), hz, giL2,L2 =hf, ϕi(XT)0,XT +hz0, ϕ(0,·)iH−1(0,1),H01(0,1)
+νhv0, ϕx(·,0)iH−1/4(0,T),H1/4(0,T)
−νhv1, ϕx(·,1)iH−1/4(0,T),H1/4(0,T),
(3.30)
whereϕ∈XT is the solution to the dual system :
ϕt+νϕxx=−g in(0, T)×(0,1), ϕ(t,0) = 0 in(0, T),
ϕ(t,1) = 0 in(0, T), ϕ(T, x) = 0 in(0,1).
(3.31)
Lemma 13. There exists a unique weak solutionz∈L2((0, T)×(0,1)) to system (3.29). Moreover : kzk2.T−1/2ν−1
kfk(X
T)0+|z0|H−1
+T−1/4(|v0|H−1/4+|v1|H−1/4). (3.32) Démonstration. For any g ∈ L2((0, T)×(0,1)), Lemma 11asserts that system (3.31) admits a unique solutionϕ∈ XT such thatkϕkXT .T−1/2ν−1kgkL2. Moreover, thanks to estimates (3.19) and (3.21), the right-hand side of equation (3.30) defines a continuous linear form onL2. The Riesz representation theorem therefore proves the existence of a uniquez∈L2satisfying estimate (3.32).
Lemma 14. Let f ∈L2((0, T)×(0,1)). We consider the following heat system :
zt−νzxx=fx in(0,1)×(0,1), z(t,0) = 0 in(0,1),
z(t,1) = 0 in(0,1), z(0, x) = 0 in(0,1).
(3.33)
There is a unique solution z∈L2((0, T)×(0,1))to system (3.33). Moreover, it satisfies the estimate : ν1/2kzkL∞(L2)+νkzxkL2 .kfkL2. (3.34) Démonstration. For f ∈ L2, it is easy to check that fx ∈ XT0 . Hence, we can apply Lemma 13 and system (3.33) has a unique solution z ∈ L2. In fact, this solution is even smoother. Estimate (3.34) is obtained as usual by multiplying equation (3.33) byz and integration by parts.
3.2.4 Burgers and forced Burgers systems
We move on to Burgers-like systems. For the sake of completeness, we provide a short proof of the existence of a solution to system (3.1) and a precise estimate for forced Burgers-like systems that will be necessary in the sequel.
Lemma 15. Let w∈XT,g∈L2((0, T), H1(0,1))andy0∈H01(0,1). We considery∈XT a solution to the following forced Burgers-like system :
yt−νyxx=−yyx+ (wy)x+gx in(0, T)×(0,1),
y(t,0) = 0 in(0, T),
y(t,1) = 0 in(0, T),
y(0, x) =y0(x) in(0,1).
(3.35)
Then,
νkyxxk2+√
νkyxk2+kytk2.kgxk2+eγkwxkL2(L∞)
ν−1/2kgk2+ y0
2 2
+ (1 +√
γeγ)kwk∞
ν−1kgk2+ν−1/2 y0
2 2
+ 1 +√ γe6γ
eγkgkL2(L∞)
ν−3/2kgk2+ν−1 y0
2
+ 1 +√ γe6γ
ν−1/2 y0
2
4+ν1/2 y0x
2.
(3.36)
where we introduce γ= 1νkwk2L2(L∞).
Démonstration. L2estimates foryandyx.We start by multiplying equation (3.35) byy, and integrate by parts over (0,1) :
1 2
d dt
Z 1 0
y2+ν Z 1
0
y2x=− Z 1
0
wyyx− Z 1
0
gyx
≤ 2 2ν
Z 1 0
w2y2+ν 4
Z 1 0
y2x+ 2 2ν
Z 1 0
g2+ν 4
Z 1 0
yx2.
(3.37)
From (3.37), we deduce : d dt
Z 1 0
y2+ν Z 1
0
yx2≤ 2
ν|w(t,·)|2∞ Z 1
0
y2+2 ν
Z 1 0
g2. (3.38)
We apply Grönwall’s lemma to (3.38) to obtain : kyk2L∞(L2)≤e2γ
2
ν kgk22+ y0
2 2
. (3.39)
Plugging (3.39) into (3.38) yields :
νkyxk22≤ 1 + 2γe2γ 2
ν kgk22+ y0
2 2
. (3.40)
L2 estimate for yyx.We repeat a similar technique, multiplying this time equation (3.35) byy3. Using the same approach yields :
d dt
Z 1 0
y4+ 6ν Z 1
0
y2yx2≤ 12
ν |w(t,·)|2∞ Z 1
0
y4+12
ν |g(t,·)|2∞ Z 1
0
y2. (3.41)
We apply Grönwall’s lemma to (3.41) to obtain : kyk4L∞(L4)≤e12γ
12
ν kgk2L2(L∞)kyk2L∞(L2)+ y0
4 4
. (3.42)
Once again, plugging back estimate (3.42) into (3.41) gives : 6νkyyxk22≤ 1 + 12γe12γ
12
ν kgk2L2(L∞)kyk2L∞(L2)+ y0
4 4
. (3.43)
Conclusion.To conclude the proof, we use Lemma11, with a source term f =gx+wxy+wyx−yyx. Estimate (3.36) comes from the combination of (3.23) with equations (3.39), (3.40) and (3.43).
Lemma 16. For any initial data y0∈H01(0,1) and any controlu∈L2(0, T), system (3.1)has a unique solutiony∈XT. Moreover :
kyxxk2+kytk2.|u|2+|u|22+|y0|24+|y0x|2, (3.44)
kyk∞≤ |y0|∞+|u|L1. (3.45)
Démonstration. This type of existence result relies on standard a priori estimates and the use of a fixed point theorem. Such techniques are described in [119]. One can also use a semi-group method as in [139]. The quantitative estimate is obtained by applying Lemma 15 with w = 0 (hence γ = 0) andg(t, x) = xu(t). Equation (3.36) yields (3.44). The second estimate (3.45) is a consequence of the maximum principle, which can be applied in this strong setting.