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Controllability for a scalar conservation law with nonlocal velocity
J. Differential Equations 252 (2012) 181–201
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Journal of Differential Equations www.elsevier.com/locate/jde
Controllability for a scalar conservation law with nonlocal velocity Jean-Michel Coron a , Zhiqiang Wang b,∗ a
Institut universitaire de France and Université Pierre et Marie Curie-Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, 4 place Jussieu, F-75005 Paris, France b LMNS, Shanghai Key Laboratory for Contemporary Applied Mathematics and School of Mathematical Sciences, Fudan University, Shanghai 200433, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 September 2010 Revised 13 August 2011 Available online 7 September 2011
In this paper, we study the state controllability and nodal profile controllability for a scalar conservation law, with a nonlocal velocity, that models a highly re-entrant manufacturing system as encountered in semi-conductor production. We first prove a local state controllability result, i.e., there exists a control that drives the solution from any given initial data to any desired final data in a certain time period, provided that the initial and final data are both close to a given equilibrium ρ 0. We also obtain a global state controllability result for the same system, where there is no limitation on the distance between the initial and final data. Finally, we prove a nodal profile controllability result, i.e., there exists a control under which the solution starts from any initial data reaches exactly any given out-flux over a fixed time period. © 2011 Elsevier Inc. All rights reserved.
MSC: 35L65 93B05 93C20 Keywords: State controllability Nodal profile controllability Conservation law Nonlocal velocity Re-entrant manufacturing system
1. Introduction and main results In this paper, we study the scalar conservation law
ρt (t , x) + ρ (t , x)λ W (t )
*
x
= 0,
t ∈ (0, T ), x ∈ (0, 1),
Corresponding author. E-mail addresses: [email protected] (J.-M. Coron), [email protected] (Z. Wang).
0022-0396/$ – see front matter doi:10.1016/j.jde.2011.08.042
© 2011
Elsevier Inc. All rights reserved.
(1.1)
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where
1 W (t ) :=
ρ (t , x) dx.
(1.2)
0
We assume that the velocity function λ ∈ C 1 ([0, ∞); (0, ∞)). For instance, we recall that the special case of
λ( W ) =
1 1+W
was used in [4,21]. In the manufacturing system, with a given initial data
ρ (0, x) = ρ0 (x), x ∈ (0, 1),
(1.3)
the natural control input is the in-flux, which suggests the boundary condition
ρ (t , 0)λ W (t ) = u (t ), t ∈ (0, T ).
(1.4)
Thereupon the evolution of ρ is completely determined by the given ρ0 and u (see [12] or [31]). This work is motivated by problems arising in the control of semiconductor manufacturing systems. These systems are characterized by their highly re-entrant feature with very high volume (number of parts manufactured per unit time) and very large number of consecutive production steps as well. This character is, in particular, described in terms of the velocity function λ in the model: it is a function of the total mass W (t ) (the integral of the density ρ ). This partial differential equation model becomes popular due to their superior analytic properties (compared with the ordinary differential equation models) and the availability of efficient numerical tools for simulation. For more detailed discussions, see e.g. [3,4,19,21]. The hyperbolic conservation laws and related control problems have been widely studied for a long time. For the well-posedness problems, we refer to the works [5,6,22,29] (and the references therein) in the content of weak solutions to systems (including scalar case) in conservation laws, and to [23,28] in the content of classical solutions to general quasi-linear hyperbolic systems. For the controllability of linear hyperbolic systems, one can see the important survey [30]. The controllability of nonlinear hyperbolic equations (or systems) are studied in [9,11,16,18,20,24,26,27,32], while the attainable set of conservation laws can be found in [1,2]. Moreover, [10] provides a comprehensive survey of controllability and stabilization in partial differential equations that also includes nonlinear conservation laws. Recently initiated by [17], [25] studies the nodal profile controllability, as a new control problem, for general quasi-linear hyperbolic systems. The main difficulty of this paper comes from the nonlocal velocity in the model. There are also some other one-dimensional models with a nonlocal velocity, either in divergence form or not, see [33] for a model on sedimentation of particles in a dilute fluid suspension and see [13] (see also the references therein, especially [8]) for models related to the 3D Navier–Stokes equations or the Euler equations in the vorticity formulation. A more related paper [7], which is also motivated in part by [4,21], addressed well-posedness for systems of hyperbolic conservation laws with a nonlocal velocity in Rn . The authors studied the Cauchy problem in the whole space Rn without considering any boundary conditions. The differentiability of the solution ρ with respect to the initial data ρ0 is also shown. In addition, a necessary condition for the possible optimal controls is given in [7]. As for this manufacturing model itself, an optimal control problem related to the Demand Tracking Problem was studied in [12] and originally inspired by [21]. The objective of that optimal control problem is to minimize the L p -norm (p ∈ [1, +∞)) of the difference between the actual out-flux
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183
ρ (t , 1)λ( W (t )) and a given demand forecast yd (t ) over a fixed time period. As a generalization of [12], [31] studies the corresponding well-posedness and optimal control problem for the model ρt + (λ(x, W (t ))ρ (t , x))x = 0. This generalized model has both the local and nonlocal features which can be also regarded as a simplification of the multi-dimensional biological model describing the follicular ovulation [14,15]. In this paper, we study the state controllability and nodal profile controllability of the manufacturing system (1.1), (1.3) and (1.4). The main results that we obtained are Theorem 1.1, Theorem 1.2 and Theorem 1.3. The problem of state controllability that we are interested in can be described as follows: For any given initial data ρ0 and any final data ρ1 , to find suitable T > 0 and suitable control u : (0, T ) → [0, +∞) such that the solution ρ to the following Cauchy problem ⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ W (t ) x = 0, t ∈ (0, T ), x ∈ (0, 1), ρ (0, x) = ρ0 (x), x ∈ (0, 1), ⎪ ⎩ ρ (t , 0)λ W (t ) = u (t ), t ∈ (0, T )
(1.5)
satisfies also
ρ ( T , x) = ρ1 (x), x ∈ (0, 1).
(1.6)
We first consider the local state controllability for this control problem in the case that the initial data ρ0 and final data ρ1 are both close to a given constant equilibrium ρ 0. Theorem 1.1. Let ρ 0 be the given constant equilibrium and let
T 0 :=
1
(1.7)
λ(ρ )
be the critical control time. Then, for any T > T 0 , any ε > 0 and any p ∈ [1, +∞), there exists that, for any ρ0 ∈ L p ((0, 1); [0, ∞)) and any ρ1 ∈ L p ((0, 1); [0, ∞)) with
ρ0 (·) − ρ
L p (0,1)
ν,
ρ1 (·) − ρ
L p (0,1)
ν,
ν > 0 such
(1.8)
there exists u ∈ L p ((0, T ); [0, ∞)) with
u (·) − ρ λ(ρ ) such that the weak solution (1.6) and
L p ( 0, T )
ε,
(1.9)
ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies the final condition
ρ (t , x) 0, t ∈ (0, T ), x ∈ (0, 1), ρ (t , ·) − ρ p ε , ∀t ∈ [0, T ]. L (0,1)
(1.10) (1.11)
Since the well-posedness of the Cauchy problem (1.5) (see [12] or [31]) does not require that the initial data be close to some equilibrium, it is quite natural to study also the global state controllability of the control problem (1.5)–(1.6) for general initial and final data.
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Theorem 1.2. For any p ∈ [1, +∞), any ρ0 ∈ L p ((0, 1); [0, ∞)) and any ρ1 ∈ L p ((0, 1); [0, ∞)), there exists T 1 > 0 (depending on ρ0 and ρ1 , see in particular (4.6) for its description) such that the following holds: For any T T 1 , there exists u ∈ L p ((0, T ); [0, ∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies the final condition (1.6). Besides the above two results on state controllability, we are also interested in the problem of nodal profile controllability which was originally introduced in [17] for 1-D isothermal Euler equations. This kind of controllability was later named by [25] and generalized for first order quasi-linear hyperbolic systems. It can be described as follows: For any given initial data ρ0 , boundary data yd and any T 1 , T with 0 < T 1 < T , to find suitable control u : (0, T ) → [0, +∞) such that the solution ρ to the following Cauchy problem (1.5) satisfies also the nodal profile condition:
ρ (t , 1) = yd (t ), t ∈ ( T 1 , T ).
(1.12)
However, for practical reasons, we care more about controllability of the out-flux ρ (t , 1)λ( W (t )) rather than the density ρ (t , 1) itself. We finally prove the following theorem on out-flux controllability which is a slight modification of nodal profile controllability. This is a local result in the sense that the solution belongs to a neighborhood of a given equilibrium ρ 0. Theorem 1.3. Let ρ 0 be the given constant equilibrium and let T 0 be given by (1.7). For any p ∈ [1, +∞), any ε > 0 and any T 1 , T with T 0 < T 1 < T , there exists ν > 0 such that the following holds: For any ρ0 ∈ L p ((0, 1); [0, ∞)) and any yd ∈ L p (( T 1 , T ); [0, ∞))
ρ0 (·) − ρ
L p (0,1)
y (·) − ρ λ(ρ )
ν,
L p (T 1 ,T )
ν,
(1.13)
there exists u ∈ L p ((0, T ); [0, ∞)) satisfying (1.9) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies (1.10), (1.11) and the out-flux condition
ρ (t , 1)λ W (t ) = yd (t ), t ∈ ( T 1 , T ).
(1.14)
In order to prove Theorem 1.1 and Theorem 1.3, we first establish the controllability results for a corresponding linear control problem. Then we construct a contraction mapping, relying on solving the linear controllability problem, whose fixed point gives the existence of the desired control u : (0, T ) → [0, +∞) to the original controllability problem (1.5)–(1.6) and controllability problem (1.5)–(1.14). See [10, Section 4.2.1] for an example of this method applied to another scalar conservation law. The idea to prove Theorem 1.2 is to drive first the initial state ρ0 to an intermediate state ρ 0, then, using the reversibility of the system, to drive ρ to the final state ρ1 . In this way, we manage to drive any given initial data ρ0 to any final data ρ1 by suitable control u, without requiring that ρ0 and ρ1 both belong to a neighborhood of a constant equilibrium. Let us emphasize that we need a longer time period (depending on ρ0 and ρ1 ) to realize the global controllability. This is quite natural, as we have encountered in other situations when studying global controllability, see e.g. [27,32]. In this paper, we have required that λ ∈ C 1 ([0, ∞); (0, ∞)) and that all the data (equilibria, initial data, final data, boundary control, solution) be nonnegative almost everywhere. Note that it is important to make such requirements to meet the practical need of the original production model. Note also that, if we assume λ ∈ C 1 (R; (0, ∞)) instead of λ ∈ C 1 ([0, ∞); (0, ∞)), then Theorem 1.1, Theorem 1.2 and Theorem 1.3 hold still without assuming the nonnegative-ness of all the data (equilibria, initial data, final data, in-flux, out-flux, solution, etc.). The organization of this paper is as follows: First in Section 2.1 and Section 2.2, we show the state controllability and nodal profile controllability for a linear control problem. Next in Section 3, Section 4 and Section 5, we give the proofs of our main results — Theorem 1.1, Theorem 1.2 and Theorem 1.3, respectively. The same notations may have different meanings in deferent sections and subsections as well.
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2. Controllability for linear control problem 2.1. State controllability for linear control problem The goal of this subsection is to prove the following proposition on state controllability. Proposition 2.1. Let ρ 0 be the given constant equilibrium and let T 0 be the critical control time given by (1.7). Then, for any p ∈ [1, +∞) and any T > T 0 , there exists η > 0 such that the following holds: For any a ∈ C 0 ([0, T ]; [0, ∞)) with
a(·) − ρ
C 0 ([0, T ])
:= sup a(t ) − ρ η, t ∈[0, T ]
(2.1)
any initial data ρ0 ∈ L p ((0, 1); [0, +∞)) and any final data ρ1 ∈ L p ((0, 1); [0, +∞)), there exists a control u ∈ L p ((0, T ); [0, +∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the linear Cauchy problem
⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ a(t ) x = 0, t ∈ (0, T ), x ∈ (0, 1), ρ (t , 0)λ a(t ) = u (t ), t ∈ (0, T ), ⎪ ⎩ ρ (0, x) = ρ0 (x), x ∈ (0, 1)
(2.2)
satisfies also the final condition
ρ ( T , x) = ρ1 (x), x ∈ (0, 1).
(2.3)
Proof. We prove the existence of the control u by direct construction through characteristic method. Define
M := ρ + 1 > 0,
λ( M ) := λ( M ) := d( M ) :=
(2.4)
inf
λ( W ) > 0,
(2.5)
sup
λ( W ) > 0,
(2.6)
W ∈[0, M ] W ∈[0, M ]
sup λ ( W ) > 0.
W ∈[0, M ]
(2.7)
Let T > T 0 be given. We denote by ξ1 the characteristic that passes through the origin:
d ξ1 ds
= λ a(s) ,
∀s ∈ [0, T ] and ξ1 (0) = 0.
(2.8)
Since λ ∈ C 1 ([0, ∞); (0, ∞)), we know that ξ1 is strictly increasing with respect to s and thus #{s ∈ [0, T ]: ξ1 (s) = 1} ∈ {0, 1}. Set
η 1, then by (2.1) and (2.4),
0 a(t ) ρ + η M ,
∀t ∈ [0, T ].
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Direct computations show that for all s ∈ [0, T ]
s
ξ1 (s) − λ(ρ )s =
λ a(θ) − λ(ρ ) dθ d( M ) T a(·) − ρ C 0 ([0,T ]) ηd( M ) T
0
and thus by (1.7)
ξ1 ( T ) λ(ρ ) T − ηd( M ) T = 1 + λ(ρ )( T − T 0 ) − ηd( M ) T . Setting
η min 1,
λ(ρ )( T − T 0 ) , d( M ) T
then ξ1 ( T ) 1, which implies that #{s ∈ [0, T ]: ξ1 (s) = 1} = 1. Denote t 1 ∈ [0, T ] to be such that ξ1 (t 1 ) = 1. Now we denote by ξ2 the characteristic that passes through the point (t , x) = ( T , 1):
d ξ2 ds
= λ a(s) ,
∀s ∈ [0, T ] and ξ2 ( T ) = 1.
(2.9)
Then
ξ2 (s) = 1 − ξ1 ( T ) + ξ1 (s),
∀s ∈ [0, T ],
in particular,
ξ2 (0) = 1 − ξ1 ( T ) 0. Since λ ∈ C 1 ([0, ∞); (0, ∞)) implies the strict monotonicity of ξ2 and noticing that ξ2 ( T ) = 1 > 0, there exists a unique t 2 ∈ [0, T ] such that ξ2 (t 2 ) = 0. Now we define a function u: (0, T ) → [0, ∞) by
u (t ) :=
ρ λ(a(t )), t ∈ (0, t 2 ), ρ1 (1 − ξ2 (t ))λ(a(t )), t ∈ (t 2 , T ),
(2.10)
which obviously belongs to L p ((0, T ); [0, ∞)). It is also easy to verify that, under this control u, the corresponding weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to (2.2) can be expressed explicitly by
⎧ if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1), ⎨ ρ0 (x − ξ1 (t )), ρ (t , x) = ρ1 (x + 1 − ξ2 (t )), if t ∈ (t 2 , T ), x ∈ (0, ξ2 (t )), ⎩ ρ, else. Hence, we have (2.3). This concludes the proof of Proposition 2.1.
2
(2.11)
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Fig. 1. Case t 2 t 1 .
Fig. 2. Case t 2 t 1 .
Remark 2.1. Using the control u given by (2.10), we can obtain the expression of W (t ) := in terms of ξ1 and ξ2 . When t 2 t 1 (see Fig. 1),
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ∀t ∈ [0, t 2 ], ⎪ ⎪ ⎨ 1−ξ1 (t ) 1 (ρ0 (x) − ρ ) dx + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 2 , t 1 ], W (t ) = ρ + 0 ⎪ ⎪ 1 ⎩ ρ + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 1 , T ].
1 0
ρ (t , x) dx
(2.12)
When t 2 t 1 (see Fig. 2),
⎧ 1−ξ1 (t ) ⎪ (ρ0 (x) − ρ ) dx, ∀t ∈ [0, t 1 ], ⎪ ⎨ρ + 0 ∀t ∈ [t 1 , t 2 ], W (t ) = ρ , ⎪ ⎪ ⎩ρ + 1 (ρ (x) − ρ ) dx, ∀t ∈ [t 2 , T ]. 1−ξ2 (t ) 1
(2.13)
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2.2. Nodal profile controllability for linear control problem In Section 2.2, we prove the following proposition on nodal profile controllability, or more precisely, on out-flux controllability. Proposition 2.2. Let ρ 0 be the given constant equilibrium and let T 0 be the critical control time given by (1.7). Then, for any p ∈ [1, +∞) and any T 1 , T with T 0 < T 1 < T , there exists η > 0 such that the following holds: For any a ∈ C 0 ([0, T ]; [0, ∞)) satisfying (2.1) and any initial data ρ0 ∈ L p ((0, 1); [0, +∞)), boundary data y d ∈ L p (( T 1 , T ); [0, +∞)), there exists a control u ∈ L p ((0, T ); [0, +∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the linear Cauchy problem (2.2) satisfies also the out-flux condition
ρ (t , 1)λ W (t ) = yd (t ), t ∈ ( T 1 , T ).
(2.14)
Proof. We prove this proposition by direct construction following again the characteristic method. Let ξ1 , ξ2 be defined by (2.8), (2.9) and correspondingly, t 1 = ξ1−1 (1), t 2 = ξ2−1 (0). In addition, we denote by ξ3 the characteristic that passes through the point (t , x) = ( T 1 , 1):
= λ a(s) ,
d ξ3 ds
∀s ∈ [0, T ] and ξ3 ( T 1 ) = 1.
(2.15)
Then
ξ3 (s) = 1 − ξ1 ( T 1 ) + ξ1 (s),
∀s ∈ [0, T ].
Similarly as in the proof of Proposition 2.1, one can choose η > 0 sufficiently small such that there exists a unique t 3 ∈ [0, T 1 ] such that ξ3 (t 3 ) = 0 and at the same time the following relations hold:
0 < t3 < t2 < T
and
0 < t1 < T 1 < T .
Obviously, there are five various possibilities concerning the order of t 1 , t 2 , t 3 , T 1 and T : (1) (2) (3) (4) (5)
0 < t3 < t2 t1 < T 1 < T ; 0 < t3 < t1 < t2 < T 1 < T ; 0 < t1 t3 < t2 T 1 < T ; 0 < t1 t3 < T 1 < t2 < T ; 0 < t1 < T 1 t3 < t2 < T .
For the simplification of the statements, we only discuss the case that 0 < t 3 < t 2 t 1 < T 1 < T (see Fig. 3). All the other cases can be dealt with in a similar way, so we omit the details of those discussions. Using the property that ρ is constant along the characteristics, we can construct the solution to Cauchy problem (2.2) in the following way. The rectangle domain [0, T ] × [0, 1] is divided into four sub-domains by ξ1 , ξ2 and ξ3 (see Fig. 3). Two of the sub-domains are correspondingly determined by ρ0 and yd , while the other two are filled with the constant ρ . Then it is easy to see that under the control
u (t ) :=
⎧ ⎨ ρ λ(a(t )), ⎩
yd (ξ1−1 (ξ1 (t )+1)) a(ξ1−1 (ξ1 (t )+1))
t ∈ (0, t 3 ) ∪ (t 2 , T ), a(t ), t ∈ (t 3 , t 2 ),
(2.16)
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Fig. 3. The case that 0 < t 3 < t 2 t 1 < T 1 < T .
the unique weak solution to (2.2) is given by
ρ (t , x) =
⎧ ρ0 (x − ξ1 (t )), ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yd (ξ1−1 (ξ1 (t )+1−x)) a(ξ1−1 (ξ1 (t )+1−x))
ρ,
if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1),
, if t ∈ (t 3 , T ), x ∈ (0, 1) ∩ (ξ2 (t ), ξ3 (t )),
(2.17)
else.
It verifies the out-flux condition (2.14). This concludes the proof of Proposition 2.2.
2
Remark 2.2. In the case that 0 < t 3 < t 2 t 1 < T 1 < T and using the control (2.16), we have the 1 expression of W (t ) := 0 ρ (t , x) dx in terms of ξ1 :
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎪ ρ + ξ1 (ξ1 (t )+1) ( y (s) − ρ λ(ρ )) ds + 1−ξ1 (t ) (ρ0 (x) − ρ ) dx ⎪ ⎪ T 0 1 ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎨ + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 (t ) + 1 − λ(ρ )ξ1 (ξ1 (t ) + 1)), W (t ) = ρ + T ( y (s) − ρ λ(ρ )) ds + 1−ξ1 (t ) (ρ (x) − ρ ) dx 0 ⎪ T1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎪ ρ + TT1 ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎪ ⎪ ⎩ ρ + tT ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 (t ) − λ(ρ )t ) − ρ (ξ1 ( T ) − λ(ρ ) T ),
∀t ∈ [0, t 3 ],
∀t ∈ [t 3 , t 2 ], ∀t ∈ [t 2 , t 1 ], ∀t ∈ [t 1 , T 1 ], ∀t ∈ [ T 1 , T ]. (2.18)
3. Proof of Theorem 1.1 In this section, we prove Theorem 1.1 by using linear iteration and a fixed point argument. First, we choose a domain candidate as a closed subset of C 0 ([0, T ]) with respect to the C 0 ([0, T ])-norm. Then, we define a contraction mapping (relying on the controllability problem studied in Section 2.1) on this domain. The existence of the fixed point of this mapping implies the existence of the desired control to the original nonlinear controllability problem. The key point is to prove that the mapping is contracting.
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Let us define the domain candidate as follows:
Ω M ,δ := ξ ∈ C 0 [0, T ] : ξ(0) = 0, ξ(t ) − λ(ρ )t δ, ∀t ∈ [0, T ] λ( M ) and
ξ(s) − ξ(t ) λ( M ), ∀s, t ∈ [0, T ], s = t , s−t
(3.1)
where λ( M ), λ( M ) are correspondingly defined by (2.5), (2.6) and M , δ > 0 are two constants to be determined later. Obviously, Ω M ,δ is not empty and it is a compact and closed subset of the Banach space C 0 ([0, T ]) equipped with the usual C 0 ([0, T ])-norm:
ξ C 0 ([0,T ]) := sup ξ(t ) . t ∈[0, T ]
For any ξ1 ∈ Ω M ,δ , let ξ2 ∈ C 0 ([0, T ]) be defined by
ξ2 (t ) := 1 − ξ1 ( T ) + ξ1 (t ),
∀t ∈ [0, T ].
(3.2)
Hence we know that ξi (i = 1, 2) are both strictly increasing with respect to t and satisfy the following properties:
λ( M ) 1
λ( M )
ξi (s) − ξi (t ) λ( M ), s−t
ξi−1 (x) − ξi−1 ( y ) 1 , x− y λ( M )
∀s, t ∈ [0, T ], s = t , ∀x, y ∈ ξi (0), ξi ( T ) , x = y .
(3.3)
(3.4)
By the definitions of Ω M ,δ , ξ1 , ξ2 and (1.7), we have
ξ1 ( T ) λ(ρ ) T − δ = 1 + λ(ρ )( T − T 0 ) − δ and
ξ2 ( T ) = 1 ,
ξ2 (0) = 1 − ξ1 ( T ) δ − λ(ρ )( T − T 0 ).
Consequently, setting
δ λ(ρ )( T − T 0 )
(3.5)
yields that ξ1 ( T ) 1 and ξ2 (0) 0. By the strict monotonicity of ξ1 and ξ2 , there exists a unique (t 1 , t 2 ) ∈ [0, T ]2 such that ξ1 (t 1 ) = 1 and ξ2 (t 2 ) = 0. We assume that t 2 t 1 (the case that t 1 t 2 can be discussed similarly without essential difficulties). Next, inspired by (2.12), we define W (ξ1 ) ∈ C 0 ([0, T ]; [0, ∞)) as
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ∀t ∈ [0, t 2 ], ⎪ ⎪ ⎨ 1−ξ1 (t ) 1 (ρ0 (x) − ρ ) dx + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 2 , t 1 ], W (ξ1 )(t ) := ρ + 0 ⎪ ⎪ 1 ⎩ ρ + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 1 , T ].
(3.6)
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191
This implies immediately with assumption (1.8) and Hölder’s inequality that
0 W (ξ1 )(t ) ρ + ρ0 (·) − ρ L p (0,1) + ρ1 (·) − ρ L p (0,1) ρ + 2ν ,
∀t ∈ [0, T ].
Let
1
M := ρ + 1 and
ν . 2
Then,
0 W (ξ1 )(t ) M ,
∀t ∈ [0, T ].
(3.7)
Now we define a mapping F : Ω M ,δ ξ1 → F (ξ1 ) ∈ C 0 ([0, T ]) by
t F (ξ1 )(t ) :=
λ W (ξ1 )(s) ds,
∀t ∈ [0, T ],
(3.8)
0
where W (ξ1 ) is defined by (3.6). We emphasize here that the definition of the mapping F is strongly motivated by solving the linear controllability problem (2.2)–(2.3). In order to prove the existence of the fixed point of F by using the contraction mapping theorem, we first prove that F maps into Ω M ,δ for suitably small δ and ν . By definitions (2.5), (2.6), (3.8) and property (3.7), it is easy to see that F (ξ1 )(0) = 0 and for all s, t ∈ [0, T ], s = t
λ( M )
F (ξ1 )(s) − F (ξ1 )(t ) s−t
t =
s
λ( W (ξ1 )(θ)) dθ λ( M ). s−t
(3.9)
On the other hand, by definitions (2.7), (3.8), property (3.7), assumption (1.8) and Hölder’s inequality, we have for all t ∈ [0, T ] that
F (ξ1 )(t ) − λ(ρ )t t d( M ) ρ0 (·) − ρ
L 1 (0,1)
+ ρ1 (·) − ρ L 1 (0,1) 2ν T d( M ).
Setting
ν min
δ 1 , , 2T d( M ) 2
then F (ξ1 ) ∈ Ω M ,δ . For any given ξ1 , ξ1 ∈ ΩM ,δ , let us define ξ2 : [0, T ] → R by
ξ2 (t ) := 1 − ξ1 ( T ) + ξ1 (t ),
∀t ∈ [0, T ].
Similarly as seen before for ξ1 , ξ2 , we know that under (3.5), there exists a unique ( t 1 , t 2 ) ∈ [0, T ]2 such that ξ1 ( t 1 ) = 1 and ξ2 ( t 2 ) = 0. Without loss of generality, let us assume that the order of t 1 , t 2 , t 1 , t 2 is the following one: 0 t 2 t2 t1 t 1 T . All the other possible cases can be discussed similarly without essential difficulties and we omit them. Then we estimate the pointwise difference | F ( ξ1 )(t ) − F (ξ1 )(t )| on the time interval [0, t 2 ], [t 2 , t 2 ], [ t 2 , t 1 ], [t 1 , t 1 ], [ t 1 , T ] successively. Using the facts (3.3), (3.4), (3.7) and the idea of changing the order of integration when necessary (cf. proof of Theorem 2.3 in [12]), we can reach the following estimate
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J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
F ( ξ1 ) − F (ξ1 )
C 0 ([0, T ])
ρ0 (·) − ρ
+ ρ1 (·) − ρ L 1 (0,1) · ξ1 − ξ1 C 0 ([0,T ]) ,
Cd( M )
λ( M )
L 1 (0,1)
where C is a constant independent of ξ1 , ξ1 . Furthermore by Hölder’s inequality and (1.8),
F ( ξ1 ) − F (ξ1 )
C 0 ([0, T ])
2C ν d( M )
λ( M )
· ξ1 − ξ1 C 0 ([0,T ]) .
Setting
δ λ( M ) 1 , , ν min 4Cd( M ) 2T d( M ) 2
together with (3.5), we get
F ( ξ1 ) − F (ξ1 )
1
C 0 ([0, T ])
ξ1 − ξ1 C 0 ([0,T ]) . 2
Then the contraction mapping theorem implies that F has a unique fixed point ξ1 in Ω M ,δ (see (3.1) for definition): F (ξ1 ) = ξ1 , i.e.,
t ξ1 (t ) =
λ W (ξ1 )(s) ds,
∀t ∈ [0, T ].
0
Let the control function u ∈ L p ((0, T ); [0, ∞)) be defined by
1−ξ1 (t ) ⎧ (ρ0 (x) − ρ ) dx), t ∈ (0, t 2 ), ⎪ ⎪ ρ λ(ρ + 0 ⎨ 1−ξ1 (t ) 1 (ρ0 (x) − ρ ) dx + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx), t ∈ (t 2 , t 1 ), u (t ) := ρ1 (1 − ξ2 (t ))λ(ρ + 0 ⎪ ⎪ 1 ⎩ ρ1 (1 − ξ2 (t ))λ(ρ + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx), t ∈ (t 1 , T ), (3.10) where ξ2 is still given by (3.2), t 1 = ξ1−1 (1), t 2 = ξ2−1 (0). The unique weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) can be expressed by (2.11). It satisfies (1.10) together with the final condition (1.6). Meanwhile, the expression of W obtained from (2.11) is accordingly given by (2.12). Finally, let us turn to proving (1.9) and (1.11). We first obtain from (1.8), (2.7), (3.2), (3.10) and Hölder’s inequality that
u (·) − ρ λ(ρ ) t2
L p ( 0, T )
u (t ) − ρ λ(ρ ) p dt
T
1p +
u (t ) − ρ λ(ρ ) p dt
1p
t2
0
t2 1−ξ p 1p T 1p 1 (t )
p
ρ0 (x) − ρ dx dt
ρ1 1 − ξ2 (t ) ξ (t ) − ρ λ(ρ ) dt ρ d( M ) + 2 0
0
t2
J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
193
Fig. 4. Idea to prove Theorem 1.2.
|t 2 | ρ d( M ) ρ0 (·) − ρ L p (0,1) + 1 p
T
ρ1 1 − ξ2 (t ) − ρ p ξ (t ) p dt
1p
2
t2
T +ρ
ξ (t ) − λ(ρ ) p dt
1p
2
t2
p −1 1 |t 2 | p ρ d( M ) ρ0 (·) − ρ L p (0,1) + d( M ) p ρ1 (·) − ρ L p (0,1) 1 + | T − t 2 | p ρ d( M ) ρ0 (·) − ρ L p (0,1) + ρ1 (·) − ρ L p (0,1)
p −1 1
ν d( M ) p + 3T p ρ d( M ) . On the other hand, we obtain from (1.8) and (2.11) that
⎧ 1−ξ (t ) 1 ⎪ ∀t ∈ [0, t 2 ], ( 0 1 |ρ0 (x) − ρ | p dx) p , ⎪ ⎪ ⎨ 1 1 1 −ξ ( t ) 1 p p ρ (t , ·) − ρ p = ( 0 |ρ0 (x) − ρ | dx + 1−ξ2 (t ) |ρ1 (x) − ρ | dx) p , ∀t ∈ [t 2 , t 1 ], L (0,1) ⎪ ⎪ ⎪ 1 ⎩ 1 ∀t ∈ [t 1 , T ], ( 1−ξ2 (t ) |ρ1 (x) − ρ | p dx) p , ρ0 (·) − ρ L p (0,1) + ρ1 (·) − ρ L p (0,1) 2ν , ∀t ∈ [0, T ]. Clearly, setting
ν min
ε 2
δ 1 λ( M ) , , 1 + 3T p ρ d( M ) 4Cd( M ) 2T d( M ) 2
ε
, |d( M )|
p −1 p
together with (3.5) finishes the proof of Theorem 1.1.
,
2
4. Proof of Theorem 1.2 In this section, we prove Theorem 1.2. The idea is to drive the state from ρ0 to an intermediate equilibrium ρ , then to drive ρ to ρ1 by using the reversibility of the Cauchy problem (1.5) (see Fig. 4). First let us prove the following lemma.
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Lemma 4.1. For any p ∈ [1, +∞), any ρ0 ∈ L p ((0, 1); [0, ∞)) and any constant ρ 0, there exists T 00 > 0 (depending on ρ0 and ρ ) such that the following holds: For any T T 00 , there exists u ∈ L p ((0, T ); [0, ∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies also the constant final condition
ρ ( T , x) = ρ , x ∈ (0, 1). Proof. For any given
ρ0 , ρ , we define T 00 :=
1
λ(ρ + ρ0 L 1 (0,1) )
(4.1)
,
where
λ ρ + ρ0 L 1 (0,1) :=
inf
W ∈[0,ρ + ρ0 L 1 (0,1) ]
λ( W ) > 0.
For any fixed T > T 00 , we first look at the following Cauchy problem
⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ W (t ) x = 0, ρ (0, x) = ρ0 (x), x ∈ (0, 1), ⎪ ⎩ ρ (t , 0) = ρ , t ∈ (0, T ).
t ∈ (0, T ), x ∈ (0, 1), (4.2)
Similarly as in [12], we can prove, by fixed point arguments, the existence of the unique weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (4.2). More precisely, the solution can be expressed as
ρ (t , x) =
ρ0 (x − ξ1 (t )), if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1), ρ, else,
(4.3)
where ξ1 (0) = 0 and
d ξ1 dt
= λ W (t ) ,
∀t ∈ [0, T ],
with
1 W (t ) := 0
ρ ξ1 (t ) + ρ (t , x) dx = ρ,
1−ξ1 (t ) 0
ρ0 (x) dx, if t ∈ [0, t 1 ], if t ∈ [t 1 , T ].
Here t 1 ∈ [0, T 00 ] ⊂ [0, T ] is defined by requiring ξ1 (t 1 ) = 1. The existence of t 1 ∈ [0, T 00 ] can be implied by the facts that ξ1 is continuous and
ξ1 ( T 00 ) =
T 00 λ W (t ) dt T 00 · λ ρ + ρ0 L 1 (0,1) = 1 > 0 = ξ1 (0). 0
On the other hand, the fact that λ( W (t )) > 0 for all t ∈ [0, T ] gives us the strict monotonicity of ξ1 , and thus the uniqueness of t 1 ∈ [0, T 00 ]. Obviously, the solution ρ given by (4.3) is equal to ρ for all t ∈ [t 1 , T ], and in particular for t = T . 2
J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
195
By the reversibility of the Cauchy problem (1.5), we easily get, from Lemma 4.1 and its proof, the following corollary. Corollary 4.1. For any p ∈ [1, +∞), any ρ1 ∈ L p ((0, 1); [0, ∞)) and any constant ρ 0, let
T 01 :=
1
λ(ρ + ρ1 L 1 (0,1) )
(4.4)
,
where
λ ρ + ρ1 L 1 (0,1) :=
inf
W ∈[0,ρ + ρ1 L 1 (0,1) ]
λ( W ) > 0.
Then, for any T T 01 , there exists v ∈ L p ((0, T ); [0, ∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the backward Cauchy problem
⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ W (t ) x = 0, t ∈ (0, T ), x ∈ (0, 1), ρ ( T , x) = ρ1 (x), x ∈ (0, 1), ⎪ ⎩ ρ (t , 1)λ W (t ) = v (t ), t ∈ (0, T ) satisfies also the final condition
ρ (0, x) = ρ , x ∈ (0, 1). Let us now go back to the proof of Theorem 1.2. For any p ∈ [1, +∞), any and any ρ1 ∈ L p ((0, 1); [0, ∞)), we define
ρ :=
1
ρ0 L 1 (0,1) + ρ0 L 1 (0,1)
ρ0 ∈ L p ((0, 1); [0, ∞))
2 T 1 := T 00 + T 01 ,
(4.5) (4.6)
where T 00 and T 01 are defined by (4.1) and (4.4), respectively. Then for any T T 1 , we define a nonnegative function (see Fig. 5)
⎧ if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1), ⎨ ρ0 (x − ξ1 (t )), ρ (t , x) := ρ1 (x + 1 − ξ2 (t )), if t ∈ (t 2 , T ), x ∈ (0, ξ2 (t )), ⎩ ρ, else,
(4.7)
where ξ1 , ξ2 are defined correspondingly by
d ξ1 dt d ξ2 dt with
1 W (t ) := 0
= λ W (t ) ,
∀t ∈ [0, T ] and ξ1 (0) = 0,
= λ W (t ) ,
∀t ∈ [0, T ] and ξ2 ( T ) = 1,
⎧ 1−ξ1 (t ) ⎪ ρ0 (x) dx, if t ∈ [0, t 1 ], ⎪ ⎨ ρ ξ1 (t ) + 0 if t ∈ [t 1 , t 2 ], ρ (t , x) dx = ρ , ⎪ ⎪ ⎩ ρ ξ (t ) + 1 ρ (x) dx, if t ∈ [t 2 , T ] 2 1−ξ2 (t ) 1
and t 1 = ξ1−1 (1), t 2 = ξ2−1 (0).
(4.8)
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Fig. 5. Definition
ρ in the case 0 t 1 T 00 T − T 01 t 2 T .
By the proof of Lemma 4.1 and the fact that T T 1 = T 00 + T 01 , we have the relation that 0 t 1 T 00 T − T 01 t 2 T . Therefore, the above ρ and W are both well defined. Let
⎧ 1−ξ1 (t ) ⎪ ρ0 (x) dx), if t ∈ (0, t 1 ), ⎪ ⎨ ρ λ(ρ ξ1 (t ) + 0 if t ∈ (t 1 , t 2 ), u (t ) := ρ λ(ρ ), ⎪ ⎪ ⎩ ρ (1 − ξ (t ))λ(ρ ξ (t ) + 1 ρ (x) dx), if t ∈ (t 2 , T ). 1 2 2 1−ξ2 (t ) 1
(4.9)
Then, it is easy to check that ρ and W defined by (4.7) and (4.8) satisfy the Cauchy problem (1.5) and the final condition (1.6) simultaneously. This finishes the proof of Theorem 1.2. 2 5. Proof of Theorem 1.3 In this section, we prove Theorem 1.3 by using a fixed point argument. Similarly as in the proof of Theorem 1.1, we choose a closed domain as a subspace of the Banach space C 0 ([0, T ]) and define a contraction mapping (based on Proposition 2.2 in Section 2.2). The fixed point of this mapping establishes the nodal profile controllability (or out-flux controllability) for the original nonlinear problem. Actually, we use the same domain candidate Ω M ,δ as in Section 3, see (3.1) for definition. We also λ( M ), λ( M ), which are still defined by (2.5), (2.6). M , δ > 0 are two constants to be determined. use For any ξ1 ∈ Ω M ,δ , let ξ2 ∈ C 0 ([0, T ]) be still defined by (3.2) and ξ3 ∈ C 0 ([0, T ]) be defined by
ξ3 (t ) = 1 − ξ1 ( T 1 ) + ξ1 (t ),
∀t ∈ [0, T ].
Then ξi (i = 1, 2, 3) are all strictly increasing with respect to t and satisfy properties (3.3)–(3.4). By definitions of Ω M ,δ , ξi (i = 1, 2, 3) and (1.7), we have
ξ1 ( T 1 ) λ(ρ ) T 1 − δ = 1 + λ(ρ )( T 1 − T 0 ) − δ and
ξ 2 ( T ) = ξ3 ( T 1 ) = 1 ,
ξ2 (0) = 1 − ξ1 ( T ) 1 − ξ1 ( T 1 ) = ξ3 (0) δ − λ(ρ )( T 1 − T 0 ).
(5.1)
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197
Consequently, setting
δ λ(ρ )( T 1 − T 0 ),
(5.2)
then ξ1 ( T ) 1 and ξ2 (0) ξ3 (0) 0. By the strict monotonicity of ξi (i = 1, 2, 3), there exists a unique (t 1 , t 2 , t 3 ) ∈ [0, T ]3 such that ξ1 (t 1 ) = 1, ξ2 (t 2 ) = ξ3 (t 3 ) = 0 with 0 < t 3 < t 2 < T , 0 < t 1 < T 1 < T . Similarly as in Section 2.2, there are five various possibilities concerning the order of t 1 , t 2 , t 3 , T 1 , T and without loss of generality we discuss only the case that 0 < t 3 < t 2 t 1 < T 1 < T . Inspired by (2.18), we define W (ξ1 ) ∈ C 0 ([0, T ]; [0, ∞)) by
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ⎪ ⎪ ⎪ ⎪ ⎪ ξ −1 (ξ (t )+1) 1−ξ (t ) ⎪ ⎪ ρ + T 11 1 ( y (s) − ρ λ(ρ )) ds + 0 1 (ρ0 (x) − ρ ) dx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 (t ) + 1 − λ(ρ )ξ1−1 (ξ1 (t ) + 1)), ⎪ ⎪ ⎪ ⎨ T 1−ξ1 (t ) (ρ0 (x) − ρ ) dx W (ξ1 )(t ) := ρ + T 1 ( y (s) − ρ λ(ρ )) ds + 0 ⎪ ⎪ ⎪ + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎪ ⎪ ⎪ ρ + TT1 ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎩ ρ + tT ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 (t ) − λ(ρ )t ) − ρ (ξ1 ( T ) − λ(ρ ) T ),
∀t ∈ [0, t 3 ],
∀t ∈ [t 3 , t 2 ], ∀t ∈ [t 2 , t 1 ], ∀t ∈ [t 1 , T 1 ], ∀t ∈ [ T 1 , T ]. (5.3)
This implies immediately with assumption (1.8) and Hölder’s inequality that
0 W (ξ1 )(t ) ρ + ρ0 (·) − ρ L 1 (0,1) + y (·) − ρ λ(ρ ) L 1 ( T , T ) + 2ρ sup ξ1 (t ) − λ(ρ )t
1 t ∈[0, T ]
ρ + 2ν + 2ρ δ,
∀t ∈ [0, T ].
Obviously, by choosing suitable constant M (see (5.7)) and ν , δ small enough, we could get the estimate (3.7) of W (ξ1 ). Motivated by solving the linear controllability problem (1.5)–(1.14), we define a mapping F : Ω M ,δ ξ1 → F (ξ1 ) ∈ C 0 ([0, T ]) by (3.8) but with (5.3) instead of (3.6). We are going to prove that F is contraction mapping on Ω M ,δ for small δ and ν . For any ξ1 ∈ Ω M ,δ , it is clear that F (ξ1 )(0) = 0 and (3.9) holds. Unlike the proof of Theorem 1.1, we don’t have directly
F (ξ1 )(t ) − λ(ρ )t δ, for small δ and
∀t ∈ [0, T ]
ν . Actually we have only the following estimate:
t
F (ξ1 )(t ) − λ(ρ )t = λ W (ξ1 )(s) − λ(ρ ) ds
0
t d( M )
W (ξ1 )(s) − ρ ds
0
2d( M ) T (ν + ρ δ),
∀t ∈ [0, T ],
(5.4)
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J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
which does not infer (5.4) generically. This is the main difficulty that we need to overcome. Nevertheless, the contraction estimate of the mapping F helps to obtain (5.4). Thus the main idea is to prove first that F is a contraction mapping from Ω M ,δ to C 0 ([0, T ]), then to prove that F maps into Ω M ,δ indeed. Similarly as in the proof of Theorem 1.1, for any ξ1 , ξ1 ∈ ΩM ,δ , one can obtain
F ( ξ1 ) − F (ξ1 )
C 0 ([0, T ])
C ν d( M )
λ( M )
· ξ1 − ξ1 C 0 ([0,T ])
for some constant C independent of ξ1 , ξ1 . Setting
ν
λ( M ) , 2Cd( M )
then
F ( ξ1 ) − F (ξ1 )
1
C 0 ([0, T ])
ξ1 − ξ1 C 0 ([0,T ]) .
(5.5)
2
Now it is left to show that (5.4) holds with the help of (5.5). Denote
ξ0 (t ) := λ(ρ )t ,
∀t ∈ [0, T ].
Obviously, ξ0 ∈ Ω M ,δ . Then, for any ξ1 ∈ Ω M ,δ ,
F (ξ1 ) − ξ0
C 0 ([0, T ])
F (ξ1 ) − F (ξ0 ) C 0 ([0,T ]) + F (ξ0 ) − ξ0 C 0 ([0,T ]) 1 ξ1 − ξ0 C 0 ([0,T ]) + F (ξ0 ) − ξ0 C 0 ([0,T ]) 2
δ 2
+ F (ξ0 ) − ξ0 C 0 ([0,T ]) .
It suffices to show that for any fixed δ > 0 there exists
F (ξ0 ) − ξ0
ν > 0 sufficiently small such that
C 0 ([0, T ])
δ . 2
In fact, by (5.3)
⎧ ρ + 01−λ(ρ )t (ρ0 (x) − ρ ) dx, ⎪ ⎪ ⎪ ⎪ ⎪ t + λ(1ρ ) 1−λ(ρ )t ⎪ ⎪ ⎪ ρ + ( y (s) − ρ λ(ρ )) ds + 0 (ρ0 (x) − ρ ) dx, ⎪ T1 ⎨ T 1−λ(ρ )t W (ξ0 )(t ) = ρ + ( y (s) − ρ λ(ρ )) ds + 0 (ρ0 (x) − ρ ) dx, T1 ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ρ + T 1 ( y (s) − ρ λ(ρ )) ds, ⎪ ⎪ ⎪ ⎩ ρ + tT ( y (s) − ρ λ(ρ )) ds,
∀t ∈ [0, t 3 ], ∀t ∈ [t 3 , t 2 ], ∀t ∈ [t 2 , t 1 ], ∀t ∈ [t 1 , T 1 ], ∀t ∈ [ T 1 , T ].
(5.6)
J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
199
Thus
t
F (ξ0 )(t ) − ξ0 (t ) =
λ W (ξ0 )(s) − λ(ρ )s ds
0
t d( M )
W (ξ0 )(s) − λ(ρ )s ds
0
td( M ) ρ0 (·) − ρ L 1 (0,1) + y (·) − ρ λ(ρ ) L 1 (T 2T d( M )ν ,
1 ,T )
∀t ∈ [0, T ].
Finally setting
M := 2ρ + 1
(5.7)
and
1 δ min λ(ρ )( T 1 − T 0 ), , 2
δ λ( M ) 1 , , ν min , 2Cd( M ) 4T d( M ) 2 we get (3.7) and (5.4). Applying the contraction mapping theorem, we conclude that F has a unique fixed point in Ω M ,δ : F (ξ1 ) = ξ1 . Under the control
u (t ) :=
⎧ ⎨ ρ λ( W (t )), ⎩
t ∈ (0, t 3 ) ∪ (t 2 , T ),
yd (ξ1−1 (ξ1 (t )+1))
λ( W (ξ1−1 (ξ1 (t )+1)))
λ( W (t )), t ∈ (t 3 , t 2 ),
(5.8)
where W (t ) is given by (2.18), t 1 = ξ1−1 (1), t 2 = ξ2−1 (0), t 3 = ξ3−1 (0), ξ2 , ξ3 are given by (3.2), (5.1) respectively, the unique weak solution to (1.5) is given by
ρ (t , x) =
⎧ ρ0 (x − ξ1 (t )), ⎪ ⎪ ⎨ −1
if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1),
yd (ξ1 (ξ1 (t )+1−x))
λ( W (ξ1−1 (ξ1 (t )+1−x))) ⎪ ⎪ ⎩ ρ,
, if t ∈ (t 3 , T ), x ∈ (0, 1) ∩ (ξ2 (t ), ξ3 (t )),
(5.9)
else.
1
It verifies (1.10) and the out-flux condition (1.14), and 0 ρ (t , x) dx = W (t ) for all t ∈ [0, T ] as well. It remains to deal with (1.9) and (1.11) for ν > 0 small enough. Let us deal with (1.9) (the proof of (1.11) is similar and in fact (1.11) follows from (1.9)). Let τ : [t 3 , t 2 ] → [ T 1 , T ], t → ξ1−1 (ξ1 (t ) + 1). Then τ is a C 1 diffeomorphism and
0<
dt dτ
λ( M ) . λ( M )
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Hence
t2
−1
p
yd ξ ξ1 (t ) + 1 − ρ λ(ρ ) dt =
T
1
t3
yd (τ ) − ρ λ(ρ ) p dt dτ λ( M ) ν p . dτ λ( M )
(5.10)
T1
From (5.10) and the fact that
0 W (t ) − ρ 2ν + 2ρ δ, one easily gets (1.9) by letting first δ then rem 1.3. 2
∀t ∈ [0, T ],
ν sufficiently small. This concludes the proof of Theo-
Acknowledgments The authors are grateful to the referees for their valuable suggestions and important remarks. This work was mainly finished when Zhiqiang Wang did his post-doc at the Laboratoire JacquesLouis Lions of Université Pierre et Marie Curie-Paris 6. The authors would like to thank the Fondation Sciences Mathématiques de Paris for generous support during this period. Jean-Michel Coron was partially supported by the ERC Advanced Grant CPDENL. Zhiqiang Wang was partially supported by Shanghai Pujiang Program (No. 11PJ1401200) and by the Natural Science Foundation of Shanghai (No. 11ZR1402500). References [1] Fabio Ancona, Andrea Marson, On the attainable set for scalar nonlinear conservation laws with boundary control, SIAM J. Control Optim. 36 (1) (1998) 290–312 (electronic). [2] Fabio Ancona, Andrea Marson, Asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point, in: Control Methods in PDE-Dynamical Systems, in: Contemp. Math., vol. 426, Amer. Math. Soc., Providence, RI, 2007, pp. 1–43. [3] Dieter Armbruster, Pierre Degond, Christian Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math. 66 (3) (2006) 896–920 (electronic). [4] Dieter Armbruster, Dan Marthaler, Christian Ringhofer, Karl Kempf, Tae-Chang Jo, A continuum model for a re-entrant factory, Oper. Res. 54 (5) (2006) 933–950. [5] Stefano Bianchini, Alberto Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (1) (2005) 223–342. [6] Alberto Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford Lecture Ser. Math. Appl., vol. 20, Oxford University Press, Oxford, 2000. [7] Rinaldo M. Colombo, Michael Herty, Magali Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var. 17 (5) (2011) 353–379. [8] Peter Constantin, Peter D. Lax, Andrew J. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 (6) (1985) 715–724. [9] Jean-Michel Coron, Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, ESAIM Control Optim. Calc. Var. 8 (2002) 513–554 (electronic). A tribute to J.-L. Lions. [10] Jean-Michel Coron, Control and Nonlinearity, Math. Surveys Monogr., vol. 136, Amer. Math. Soc., Providence, RI, 2007. [11] Jean-Michel Coron, Oliver Glass, Zhiqiang Wang, Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed, SIAM J. Control Optim. 48 (5) (2009/2010) 3105–3122. [12] Jean-Michel Coron, Matthias Kawski, Zhiqiang Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B 14 (4) (2010) 1337–1359. [13] Hongjie Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal. 255 (11) (2008) 3070–3097. [14] Nki Echenim, Frederique Clément, Michel Sorine, Multiscale modeling of follicular ovulation as a reachability problem, Multiscale Model. Simul. 6 (3) (2007) 895–912. [15] Nki Echenim, Danielle Monniaux, Michel Sorine, Frédérique Clément, Multi-scale modeling of the follicle selection process in the ovary, Math. Biosci. 198 (1) (2005) 57–79. [16] Olivier Glass, On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. (JEMS) 9 (3) (2007) 427–486. [17] Martin Gugat, Michael Herty, Veronika Schleper, Flow control in gas networks: exact controllability to a given demand, Math. Methods Appl. Sci. 34 (7) (2011) 745–757. [18] Martin Gugat, Günter Leugering, Global boundary controllability of the de St. Venant equations between steady states, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (1) (2003) 1–11.
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[19] Michael Herty, Axel Klar, Benedetto Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal. 39 (1) (2007) 160–173. [20] Thierry Horsin, On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var. 3 (1998) 83–95 (electronic). [21] Michael La Marca, Dieter Armbruster, Michael Herty, Christian Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Control 55 (11) (2010) 2511–2526. [22] Philippe G. LeFloch, Hyperbolic systems of conservation laws, in: The Theory of Classical and Nonclassical Shock Waves, in: Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2002. [23] Tatsien Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, RAM: Res. in Appl. Math., vol. 32, Masson, Paris, 1994. [24] Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math., vol. 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010. [25] Tatsien Li, Exact boundary controllability of nodal profile for quasilinear hyperbolic systems, Math. Methods Appl. Sci. 33 (17) (2010) 2101–2106. [26] Tatsien Li, Bopeng Rao, Exact boundary controllability for quasi-linear hyperbolic systems, SIAM J. Control Optim. 41 (6) (2003) 1748–1755 (electronic). [27] Tatsien Li, Zhiqiang Wang, Global exact boundary controllability for first order quasilinear hyperbolic systems of diagonal form, Int. J. Dyn. Syst. Differ. Equ. 1 (1) (2007) 12–19. [28] Tatsien Li, Yu Wenci, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Math. Ser., vol. V, Duke University Mathematics Department, Durham, NC, 1985. [29] Tai-Ping Liu, Tong Yang, Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (12) (1999) 1553–1586. [30] David L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (4) (1978) 639–739. [31] Peipei Shang, Zhiqiang Wang, Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system, J. Differential Equations 250 (2) (2011) 949–982. [32] Zhiqiang Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems, Chinese Ann. Math. Ser. B 27 (6) (2006) 643–656. [33] Kevin Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Quart. Appl. Math. 57 (3) (1999) 573– 600.
Contents lists available at SciVerse ScienceDirect
Journal of Differential Equations www.elsevier.com/locate/jde
Controllability for a scalar conservation law with nonlocal velocity Jean-Michel Coron a , Zhiqiang Wang b,∗ a
Institut universitaire de France and Université Pierre et Marie Curie-Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, 4 place Jussieu, F-75005 Paris, France b LMNS, Shanghai Key Laboratory for Contemporary Applied Mathematics and School of Mathematical Sciences, Fudan University, Shanghai 200433, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 15 September 2010 Revised 13 August 2011 Available online 7 September 2011
In this paper, we study the state controllability and nodal profile controllability for a scalar conservation law, with a nonlocal velocity, that models a highly re-entrant manufacturing system as encountered in semi-conductor production. We first prove a local state controllability result, i.e., there exists a control that drives the solution from any given initial data to any desired final data in a certain time period, provided that the initial and final data are both close to a given equilibrium ρ 0. We also obtain a global state controllability result for the same system, where there is no limitation on the distance between the initial and final data. Finally, we prove a nodal profile controllability result, i.e., there exists a control under which the solution starts from any initial data reaches exactly any given out-flux over a fixed time period. © 2011 Elsevier Inc. All rights reserved.
MSC: 35L65 93B05 93C20 Keywords: State controllability Nodal profile controllability Conservation law Nonlocal velocity Re-entrant manufacturing system
1. Introduction and main results In this paper, we study the scalar conservation law
ρt (t , x) + ρ (t , x)λ W (t )
*
x
= 0,
t ∈ (0, T ), x ∈ (0, 1),
Corresponding author. E-mail addresses: [email protected] (J.-M. Coron), [email protected] (Z. Wang).
0022-0396/$ – see front matter doi:10.1016/j.jde.2011.08.042
© 2011
Elsevier Inc. All rights reserved.
(1.1)
182
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where
1 W (t ) :=
ρ (t , x) dx.
(1.2)
0
We assume that the velocity function λ ∈ C 1 ([0, ∞); (0, ∞)). For instance, we recall that the special case of
λ( W ) =
1 1+W
was used in [4,21]. In the manufacturing system, with a given initial data
ρ (0, x) = ρ0 (x), x ∈ (0, 1),
(1.3)
the natural control input is the in-flux, which suggests the boundary condition
ρ (t , 0)λ W (t ) = u (t ), t ∈ (0, T ).
(1.4)
Thereupon the evolution of ρ is completely determined by the given ρ0 and u (see [12] or [31]). This work is motivated by problems arising in the control of semiconductor manufacturing systems. These systems are characterized by their highly re-entrant feature with very high volume (number of parts manufactured per unit time) and very large number of consecutive production steps as well. This character is, in particular, described in terms of the velocity function λ in the model: it is a function of the total mass W (t ) (the integral of the density ρ ). This partial differential equation model becomes popular due to their superior analytic properties (compared with the ordinary differential equation models) and the availability of efficient numerical tools for simulation. For more detailed discussions, see e.g. [3,4,19,21]. The hyperbolic conservation laws and related control problems have been widely studied for a long time. For the well-posedness problems, we refer to the works [5,6,22,29] (and the references therein) in the content of weak solutions to systems (including scalar case) in conservation laws, and to [23,28] in the content of classical solutions to general quasi-linear hyperbolic systems. For the controllability of linear hyperbolic systems, one can see the important survey [30]. The controllability of nonlinear hyperbolic equations (or systems) are studied in [9,11,16,18,20,24,26,27,32], while the attainable set of conservation laws can be found in [1,2]. Moreover, [10] provides a comprehensive survey of controllability and stabilization in partial differential equations that also includes nonlinear conservation laws. Recently initiated by [17], [25] studies the nodal profile controllability, as a new control problem, for general quasi-linear hyperbolic systems. The main difficulty of this paper comes from the nonlocal velocity in the model. There are also some other one-dimensional models with a nonlocal velocity, either in divergence form or not, see [33] for a model on sedimentation of particles in a dilute fluid suspension and see [13] (see also the references therein, especially [8]) for models related to the 3D Navier–Stokes equations or the Euler equations in the vorticity formulation. A more related paper [7], which is also motivated in part by [4,21], addressed well-posedness for systems of hyperbolic conservation laws with a nonlocal velocity in Rn . The authors studied the Cauchy problem in the whole space Rn without considering any boundary conditions. The differentiability of the solution ρ with respect to the initial data ρ0 is also shown. In addition, a necessary condition for the possible optimal controls is given in [7]. As for this manufacturing model itself, an optimal control problem related to the Demand Tracking Problem was studied in [12] and originally inspired by [21]. The objective of that optimal control problem is to minimize the L p -norm (p ∈ [1, +∞)) of the difference between the actual out-flux
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183
ρ (t , 1)λ( W (t )) and a given demand forecast yd (t ) over a fixed time period. As a generalization of [12], [31] studies the corresponding well-posedness and optimal control problem for the model ρt + (λ(x, W (t ))ρ (t , x))x = 0. This generalized model has both the local and nonlocal features which can be also regarded as a simplification of the multi-dimensional biological model describing the follicular ovulation [14,15]. In this paper, we study the state controllability and nodal profile controllability of the manufacturing system (1.1), (1.3) and (1.4). The main results that we obtained are Theorem 1.1, Theorem 1.2 and Theorem 1.3. The problem of state controllability that we are interested in can be described as follows: For any given initial data ρ0 and any final data ρ1 , to find suitable T > 0 and suitable control u : (0, T ) → [0, +∞) such that the solution ρ to the following Cauchy problem ⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ W (t ) x = 0, t ∈ (0, T ), x ∈ (0, 1), ρ (0, x) = ρ0 (x), x ∈ (0, 1), ⎪ ⎩ ρ (t , 0)λ W (t ) = u (t ), t ∈ (0, T )
(1.5)
satisfies also
ρ ( T , x) = ρ1 (x), x ∈ (0, 1).
(1.6)
We first consider the local state controllability for this control problem in the case that the initial data ρ0 and final data ρ1 are both close to a given constant equilibrium ρ 0. Theorem 1.1. Let ρ 0 be the given constant equilibrium and let
T 0 :=
1
(1.7)
λ(ρ )
be the critical control time. Then, for any T > T 0 , any ε > 0 and any p ∈ [1, +∞), there exists that, for any ρ0 ∈ L p ((0, 1); [0, ∞)) and any ρ1 ∈ L p ((0, 1); [0, ∞)) with
ρ0 (·) − ρ
L p (0,1)
ν,
ρ1 (·) − ρ
L p (0,1)
ν,
ν > 0 such
(1.8)
there exists u ∈ L p ((0, T ); [0, ∞)) with
u (·) − ρ λ(ρ ) such that the weak solution (1.6) and
L p ( 0, T )
ε,
(1.9)
ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies the final condition
ρ (t , x) 0, t ∈ (0, T ), x ∈ (0, 1), ρ (t , ·) − ρ p ε , ∀t ∈ [0, T ]. L (0,1)
(1.10) (1.11)
Since the well-posedness of the Cauchy problem (1.5) (see [12] or [31]) does not require that the initial data be close to some equilibrium, it is quite natural to study also the global state controllability of the control problem (1.5)–(1.6) for general initial and final data.
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Theorem 1.2. For any p ∈ [1, +∞), any ρ0 ∈ L p ((0, 1); [0, ∞)) and any ρ1 ∈ L p ((0, 1); [0, ∞)), there exists T 1 > 0 (depending on ρ0 and ρ1 , see in particular (4.6) for its description) such that the following holds: For any T T 1 , there exists u ∈ L p ((0, T ); [0, ∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies the final condition (1.6). Besides the above two results on state controllability, we are also interested in the problem of nodal profile controllability which was originally introduced in [17] for 1-D isothermal Euler equations. This kind of controllability was later named by [25] and generalized for first order quasi-linear hyperbolic systems. It can be described as follows: For any given initial data ρ0 , boundary data yd and any T 1 , T with 0 < T 1 < T , to find suitable control u : (0, T ) → [0, +∞) such that the solution ρ to the following Cauchy problem (1.5) satisfies also the nodal profile condition:
ρ (t , 1) = yd (t ), t ∈ ( T 1 , T ).
(1.12)
However, for practical reasons, we care more about controllability of the out-flux ρ (t , 1)λ( W (t )) rather than the density ρ (t , 1) itself. We finally prove the following theorem on out-flux controllability which is a slight modification of nodal profile controllability. This is a local result in the sense that the solution belongs to a neighborhood of a given equilibrium ρ 0. Theorem 1.3. Let ρ 0 be the given constant equilibrium and let T 0 be given by (1.7). For any p ∈ [1, +∞), any ε > 0 and any T 1 , T with T 0 < T 1 < T , there exists ν > 0 such that the following holds: For any ρ0 ∈ L p ((0, 1); [0, ∞)) and any yd ∈ L p (( T 1 , T ); [0, ∞))
ρ0 (·) − ρ
L p (0,1)
y (·) − ρ λ(ρ )
ν,
L p (T 1 ,T )
ν,
(1.13)
there exists u ∈ L p ((0, T ); [0, ∞)) satisfying (1.9) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies (1.10), (1.11) and the out-flux condition
ρ (t , 1)λ W (t ) = yd (t ), t ∈ ( T 1 , T ).
(1.14)
In order to prove Theorem 1.1 and Theorem 1.3, we first establish the controllability results for a corresponding linear control problem. Then we construct a contraction mapping, relying on solving the linear controllability problem, whose fixed point gives the existence of the desired control u : (0, T ) → [0, +∞) to the original controllability problem (1.5)–(1.6) and controllability problem (1.5)–(1.14). See [10, Section 4.2.1] for an example of this method applied to another scalar conservation law. The idea to prove Theorem 1.2 is to drive first the initial state ρ0 to an intermediate state ρ 0, then, using the reversibility of the system, to drive ρ to the final state ρ1 . In this way, we manage to drive any given initial data ρ0 to any final data ρ1 by suitable control u, without requiring that ρ0 and ρ1 both belong to a neighborhood of a constant equilibrium. Let us emphasize that we need a longer time period (depending on ρ0 and ρ1 ) to realize the global controllability. This is quite natural, as we have encountered in other situations when studying global controllability, see e.g. [27,32]. In this paper, we have required that λ ∈ C 1 ([0, ∞); (0, ∞)) and that all the data (equilibria, initial data, final data, boundary control, solution) be nonnegative almost everywhere. Note that it is important to make such requirements to meet the practical need of the original production model. Note also that, if we assume λ ∈ C 1 (R; (0, ∞)) instead of λ ∈ C 1 ([0, ∞); (0, ∞)), then Theorem 1.1, Theorem 1.2 and Theorem 1.3 hold still without assuming the nonnegative-ness of all the data (equilibria, initial data, final data, in-flux, out-flux, solution, etc.). The organization of this paper is as follows: First in Section 2.1 and Section 2.2, we show the state controllability and nodal profile controllability for a linear control problem. Next in Section 3, Section 4 and Section 5, we give the proofs of our main results — Theorem 1.1, Theorem 1.2 and Theorem 1.3, respectively. The same notations may have different meanings in deferent sections and subsections as well.
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2. Controllability for linear control problem 2.1. State controllability for linear control problem The goal of this subsection is to prove the following proposition on state controllability. Proposition 2.1. Let ρ 0 be the given constant equilibrium and let T 0 be the critical control time given by (1.7). Then, for any p ∈ [1, +∞) and any T > T 0 , there exists η > 0 such that the following holds: For any a ∈ C 0 ([0, T ]; [0, ∞)) with
a(·) − ρ
C 0 ([0, T ])
:= sup a(t ) − ρ η, t ∈[0, T ]
(2.1)
any initial data ρ0 ∈ L p ((0, 1); [0, +∞)) and any final data ρ1 ∈ L p ((0, 1); [0, +∞)), there exists a control u ∈ L p ((0, T ); [0, +∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the linear Cauchy problem
⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ a(t ) x = 0, t ∈ (0, T ), x ∈ (0, 1), ρ (t , 0)λ a(t ) = u (t ), t ∈ (0, T ), ⎪ ⎩ ρ (0, x) = ρ0 (x), x ∈ (0, 1)
(2.2)
satisfies also the final condition
ρ ( T , x) = ρ1 (x), x ∈ (0, 1).
(2.3)
Proof. We prove the existence of the control u by direct construction through characteristic method. Define
M := ρ + 1 > 0,
λ( M ) := λ( M ) := d( M ) :=
(2.4)
inf
λ( W ) > 0,
(2.5)
sup
λ( W ) > 0,
(2.6)
W ∈[0, M ] W ∈[0, M ]
sup λ ( W ) > 0.
W ∈[0, M ]
(2.7)
Let T > T 0 be given. We denote by ξ1 the characteristic that passes through the origin:
d ξ1 ds
= λ a(s) ,
∀s ∈ [0, T ] and ξ1 (0) = 0.
(2.8)
Since λ ∈ C 1 ([0, ∞); (0, ∞)), we know that ξ1 is strictly increasing with respect to s and thus #{s ∈ [0, T ]: ξ1 (s) = 1} ∈ {0, 1}. Set
η 1, then by (2.1) and (2.4),
0 a(t ) ρ + η M ,
∀t ∈ [0, T ].
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Direct computations show that for all s ∈ [0, T ]
s
ξ1 (s) − λ(ρ )s =
λ a(θ) − λ(ρ ) dθ d( M ) T a(·) − ρ C 0 ([0,T ]) ηd( M ) T
0
and thus by (1.7)
ξ1 ( T ) λ(ρ ) T − ηd( M ) T = 1 + λ(ρ )( T − T 0 ) − ηd( M ) T . Setting
η min 1,
λ(ρ )( T − T 0 ) , d( M ) T
then ξ1 ( T ) 1, which implies that #{s ∈ [0, T ]: ξ1 (s) = 1} = 1. Denote t 1 ∈ [0, T ] to be such that ξ1 (t 1 ) = 1. Now we denote by ξ2 the characteristic that passes through the point (t , x) = ( T , 1):
d ξ2 ds
= λ a(s) ,
∀s ∈ [0, T ] and ξ2 ( T ) = 1.
(2.9)
Then
ξ2 (s) = 1 − ξ1 ( T ) + ξ1 (s),
∀s ∈ [0, T ],
in particular,
ξ2 (0) = 1 − ξ1 ( T ) 0. Since λ ∈ C 1 ([0, ∞); (0, ∞)) implies the strict monotonicity of ξ2 and noticing that ξ2 ( T ) = 1 > 0, there exists a unique t 2 ∈ [0, T ] such that ξ2 (t 2 ) = 0. Now we define a function u: (0, T ) → [0, ∞) by
u (t ) :=
ρ λ(a(t )), t ∈ (0, t 2 ), ρ1 (1 − ξ2 (t ))λ(a(t )), t ∈ (t 2 , T ),
(2.10)
which obviously belongs to L p ((0, T ); [0, ∞)). It is also easy to verify that, under this control u, the corresponding weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to (2.2) can be expressed explicitly by
⎧ if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1), ⎨ ρ0 (x − ξ1 (t )), ρ (t , x) = ρ1 (x + 1 − ξ2 (t )), if t ∈ (t 2 , T ), x ∈ (0, ξ2 (t )), ⎩ ρ, else. Hence, we have (2.3). This concludes the proof of Proposition 2.1.
2
(2.11)
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187
Fig. 1. Case t 2 t 1 .
Fig. 2. Case t 2 t 1 .
Remark 2.1. Using the control u given by (2.10), we can obtain the expression of W (t ) := in terms of ξ1 and ξ2 . When t 2 t 1 (see Fig. 1),
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ∀t ∈ [0, t 2 ], ⎪ ⎪ ⎨ 1−ξ1 (t ) 1 (ρ0 (x) − ρ ) dx + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 2 , t 1 ], W (t ) = ρ + 0 ⎪ ⎪ 1 ⎩ ρ + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 1 , T ].
1 0
ρ (t , x) dx
(2.12)
When t 2 t 1 (see Fig. 2),
⎧ 1−ξ1 (t ) ⎪ (ρ0 (x) − ρ ) dx, ∀t ∈ [0, t 1 ], ⎪ ⎨ρ + 0 ∀t ∈ [t 1 , t 2 ], W (t ) = ρ , ⎪ ⎪ ⎩ρ + 1 (ρ (x) − ρ ) dx, ∀t ∈ [t 2 , T ]. 1−ξ2 (t ) 1
(2.13)
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2.2. Nodal profile controllability for linear control problem In Section 2.2, we prove the following proposition on nodal profile controllability, or more precisely, on out-flux controllability. Proposition 2.2. Let ρ 0 be the given constant equilibrium and let T 0 be the critical control time given by (1.7). Then, for any p ∈ [1, +∞) and any T 1 , T with T 0 < T 1 < T , there exists η > 0 such that the following holds: For any a ∈ C 0 ([0, T ]; [0, ∞)) satisfying (2.1) and any initial data ρ0 ∈ L p ((0, 1); [0, +∞)), boundary data y d ∈ L p (( T 1 , T ); [0, +∞)), there exists a control u ∈ L p ((0, T ); [0, +∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the linear Cauchy problem (2.2) satisfies also the out-flux condition
ρ (t , 1)λ W (t ) = yd (t ), t ∈ ( T 1 , T ).
(2.14)
Proof. We prove this proposition by direct construction following again the characteristic method. Let ξ1 , ξ2 be defined by (2.8), (2.9) and correspondingly, t 1 = ξ1−1 (1), t 2 = ξ2−1 (0). In addition, we denote by ξ3 the characteristic that passes through the point (t , x) = ( T 1 , 1):
= λ a(s) ,
d ξ3 ds
∀s ∈ [0, T ] and ξ3 ( T 1 ) = 1.
(2.15)
Then
ξ3 (s) = 1 − ξ1 ( T 1 ) + ξ1 (s),
∀s ∈ [0, T ].
Similarly as in the proof of Proposition 2.1, one can choose η > 0 sufficiently small such that there exists a unique t 3 ∈ [0, T 1 ] such that ξ3 (t 3 ) = 0 and at the same time the following relations hold:
0 < t3 < t2 < T
and
0 < t1 < T 1 < T .
Obviously, there are five various possibilities concerning the order of t 1 , t 2 , t 3 , T 1 and T : (1) (2) (3) (4) (5)
0 < t3 < t2 t1 < T 1 < T ; 0 < t3 < t1 < t2 < T 1 < T ; 0 < t1 t3 < t2 T 1 < T ; 0 < t1 t3 < T 1 < t2 < T ; 0 < t1 < T 1 t3 < t2 < T .
For the simplification of the statements, we only discuss the case that 0 < t 3 < t 2 t 1 < T 1 < T (see Fig. 3). All the other cases can be dealt with in a similar way, so we omit the details of those discussions. Using the property that ρ is constant along the characteristics, we can construct the solution to Cauchy problem (2.2) in the following way. The rectangle domain [0, T ] × [0, 1] is divided into four sub-domains by ξ1 , ξ2 and ξ3 (see Fig. 3). Two of the sub-domains are correspondingly determined by ρ0 and yd , while the other two are filled with the constant ρ . Then it is easy to see that under the control
u (t ) :=
⎧ ⎨ ρ λ(a(t )), ⎩
yd (ξ1−1 (ξ1 (t )+1)) a(ξ1−1 (ξ1 (t )+1))
t ∈ (0, t 3 ) ∪ (t 2 , T ), a(t ), t ∈ (t 3 , t 2 ),
(2.16)
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189
Fig. 3. The case that 0 < t 3 < t 2 t 1 < T 1 < T .
the unique weak solution to (2.2) is given by
ρ (t , x) =
⎧ ρ0 (x − ξ1 (t )), ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
yd (ξ1−1 (ξ1 (t )+1−x)) a(ξ1−1 (ξ1 (t )+1−x))
ρ,
if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1),
, if t ∈ (t 3 , T ), x ∈ (0, 1) ∩ (ξ2 (t ), ξ3 (t )),
(2.17)
else.
It verifies the out-flux condition (2.14). This concludes the proof of Proposition 2.2.
2
Remark 2.2. In the case that 0 < t 3 < t 2 t 1 < T 1 < T and using the control (2.16), we have the 1 expression of W (t ) := 0 ρ (t , x) dx in terms of ξ1 :
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎪ ρ + ξ1 (ξ1 (t )+1) ( y (s) − ρ λ(ρ )) ds + 1−ξ1 (t ) (ρ0 (x) − ρ ) dx ⎪ ⎪ T 0 1 ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎨ + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 (t ) + 1 − λ(ρ )ξ1 (ξ1 (t ) + 1)), W (t ) = ρ + T ( y (s) − ρ λ(ρ )) ds + 1−ξ1 (t ) (ρ (x) − ρ ) dx 0 ⎪ T1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎪ ρ + TT1 ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎪ ⎪ ⎩ ρ + tT ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 (t ) − λ(ρ )t ) − ρ (ξ1 ( T ) − λ(ρ ) T ),
∀t ∈ [0, t 3 ],
∀t ∈ [t 3 , t 2 ], ∀t ∈ [t 2 , t 1 ], ∀t ∈ [t 1 , T 1 ], ∀t ∈ [ T 1 , T ]. (2.18)
3. Proof of Theorem 1.1 In this section, we prove Theorem 1.1 by using linear iteration and a fixed point argument. First, we choose a domain candidate as a closed subset of C 0 ([0, T ]) with respect to the C 0 ([0, T ])-norm. Then, we define a contraction mapping (relying on the controllability problem studied in Section 2.1) on this domain. The existence of the fixed point of this mapping implies the existence of the desired control to the original nonlinear controllability problem. The key point is to prove that the mapping is contracting.
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Let us define the domain candidate as follows:
Ω M ,δ := ξ ∈ C 0 [0, T ] : ξ(0) = 0, ξ(t ) − λ(ρ )t δ, ∀t ∈ [0, T ] λ( M ) and
ξ(s) − ξ(t ) λ( M ), ∀s, t ∈ [0, T ], s = t , s−t
(3.1)
where λ( M ), λ( M ) are correspondingly defined by (2.5), (2.6) and M , δ > 0 are two constants to be determined later. Obviously, Ω M ,δ is not empty and it is a compact and closed subset of the Banach space C 0 ([0, T ]) equipped with the usual C 0 ([0, T ])-norm:
ξ C 0 ([0,T ]) := sup ξ(t ) . t ∈[0, T ]
For any ξ1 ∈ Ω M ,δ , let ξ2 ∈ C 0 ([0, T ]) be defined by
ξ2 (t ) := 1 − ξ1 ( T ) + ξ1 (t ),
∀t ∈ [0, T ].
(3.2)
Hence we know that ξi (i = 1, 2) are both strictly increasing with respect to t and satisfy the following properties:
λ( M ) 1
λ( M )
ξi (s) − ξi (t ) λ( M ), s−t
ξi−1 (x) − ξi−1 ( y ) 1 , x− y λ( M )
∀s, t ∈ [0, T ], s = t , ∀x, y ∈ ξi (0), ξi ( T ) , x = y .
(3.3)
(3.4)
By the definitions of Ω M ,δ , ξ1 , ξ2 and (1.7), we have
ξ1 ( T ) λ(ρ ) T − δ = 1 + λ(ρ )( T − T 0 ) − δ and
ξ2 ( T ) = 1 ,
ξ2 (0) = 1 − ξ1 ( T ) δ − λ(ρ )( T − T 0 ).
Consequently, setting
δ λ(ρ )( T − T 0 )
(3.5)
yields that ξ1 ( T ) 1 and ξ2 (0) 0. By the strict monotonicity of ξ1 and ξ2 , there exists a unique (t 1 , t 2 ) ∈ [0, T ]2 such that ξ1 (t 1 ) = 1 and ξ2 (t 2 ) = 0. We assume that t 2 t 1 (the case that t 1 t 2 can be discussed similarly without essential difficulties). Next, inspired by (2.12), we define W (ξ1 ) ∈ C 0 ([0, T ]; [0, ∞)) as
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ∀t ∈ [0, t 2 ], ⎪ ⎪ ⎨ 1−ξ1 (t ) 1 (ρ0 (x) − ρ ) dx + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 2 , t 1 ], W (ξ1 )(t ) := ρ + 0 ⎪ ⎪ 1 ⎩ ρ + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx, ∀t ∈ [t 1 , T ].
(3.6)
J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
191
This implies immediately with assumption (1.8) and Hölder’s inequality that
0 W (ξ1 )(t ) ρ + ρ0 (·) − ρ L p (0,1) + ρ1 (·) − ρ L p (0,1) ρ + 2ν ,
∀t ∈ [0, T ].
Let
1
M := ρ + 1 and
ν . 2
Then,
0 W (ξ1 )(t ) M ,
∀t ∈ [0, T ].
(3.7)
Now we define a mapping F : Ω M ,δ ξ1 → F (ξ1 ) ∈ C 0 ([0, T ]) by
t F (ξ1 )(t ) :=
λ W (ξ1 )(s) ds,
∀t ∈ [0, T ],
(3.8)
0
where W (ξ1 ) is defined by (3.6). We emphasize here that the definition of the mapping F is strongly motivated by solving the linear controllability problem (2.2)–(2.3). In order to prove the existence of the fixed point of F by using the contraction mapping theorem, we first prove that F maps into Ω M ,δ for suitably small δ and ν . By definitions (2.5), (2.6), (3.8) and property (3.7), it is easy to see that F (ξ1 )(0) = 0 and for all s, t ∈ [0, T ], s = t
λ( M )
F (ξ1 )(s) − F (ξ1 )(t ) s−t
t =
s
λ( W (ξ1 )(θ)) dθ λ( M ). s−t
(3.9)
On the other hand, by definitions (2.7), (3.8), property (3.7), assumption (1.8) and Hölder’s inequality, we have for all t ∈ [0, T ] that
F (ξ1 )(t ) − λ(ρ )t t d( M ) ρ0 (·) − ρ
L 1 (0,1)
+ ρ1 (·) − ρ L 1 (0,1) 2ν T d( M ).
Setting
ν min
δ 1 , , 2T d( M ) 2
then F (ξ1 ) ∈ Ω M ,δ . For any given ξ1 , ξ1 ∈ ΩM ,δ , let us define ξ2 : [0, T ] → R by
ξ2 (t ) := 1 − ξ1 ( T ) + ξ1 (t ),
∀t ∈ [0, T ].
Similarly as seen before for ξ1 , ξ2 , we know that under (3.5), there exists a unique ( t 1 , t 2 ) ∈ [0, T ]2 such that ξ1 ( t 1 ) = 1 and ξ2 ( t 2 ) = 0. Without loss of generality, let us assume that the order of t 1 , t 2 , t 1 , t 2 is the following one: 0 t 2 t2 t1 t 1 T . All the other possible cases can be discussed similarly without essential difficulties and we omit them. Then we estimate the pointwise difference | F ( ξ1 )(t ) − F (ξ1 )(t )| on the time interval [0, t 2 ], [t 2 , t 2 ], [ t 2 , t 1 ], [t 1 , t 1 ], [ t 1 , T ] successively. Using the facts (3.3), (3.4), (3.7) and the idea of changing the order of integration when necessary (cf. proof of Theorem 2.3 in [12]), we can reach the following estimate
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F ( ξ1 ) − F (ξ1 )
C 0 ([0, T ])
ρ0 (·) − ρ
+ ρ1 (·) − ρ L 1 (0,1) · ξ1 − ξ1 C 0 ([0,T ]) ,
Cd( M )
λ( M )
L 1 (0,1)
where C is a constant independent of ξ1 , ξ1 . Furthermore by Hölder’s inequality and (1.8),
F ( ξ1 ) − F (ξ1 )
C 0 ([0, T ])
2C ν d( M )
λ( M )
· ξ1 − ξ1 C 0 ([0,T ]) .
Setting
δ λ( M ) 1 , , ν min 4Cd( M ) 2T d( M ) 2
together with (3.5), we get
F ( ξ1 ) − F (ξ1 )
1
C 0 ([0, T ])
ξ1 − ξ1 C 0 ([0,T ]) . 2
Then the contraction mapping theorem implies that F has a unique fixed point ξ1 in Ω M ,δ (see (3.1) for definition): F (ξ1 ) = ξ1 , i.e.,
t ξ1 (t ) =
λ W (ξ1 )(s) ds,
∀t ∈ [0, T ].
0
Let the control function u ∈ L p ((0, T ); [0, ∞)) be defined by
1−ξ1 (t ) ⎧ (ρ0 (x) − ρ ) dx), t ∈ (0, t 2 ), ⎪ ⎪ ρ λ(ρ + 0 ⎨ 1−ξ1 (t ) 1 (ρ0 (x) − ρ ) dx + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx), t ∈ (t 2 , t 1 ), u (t ) := ρ1 (1 − ξ2 (t ))λ(ρ + 0 ⎪ ⎪ 1 ⎩ ρ1 (1 − ξ2 (t ))λ(ρ + 1−ξ2 (t ) (ρ1 (x) − ρ ) dx), t ∈ (t 1 , T ), (3.10) where ξ2 is still given by (3.2), t 1 = ξ1−1 (1), t 2 = ξ2−1 (0). The unique weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) can be expressed by (2.11). It satisfies (1.10) together with the final condition (1.6). Meanwhile, the expression of W obtained from (2.11) is accordingly given by (2.12). Finally, let us turn to proving (1.9) and (1.11). We first obtain from (1.8), (2.7), (3.2), (3.10) and Hölder’s inequality that
u (·) − ρ λ(ρ ) t2
L p ( 0, T )
u (t ) − ρ λ(ρ ) p dt
T
1p +
u (t ) − ρ λ(ρ ) p dt
1p
t2
0
t2 1−ξ p 1p T 1p 1 (t )
p
ρ0 (x) − ρ dx dt
ρ1 1 − ξ2 (t ) ξ (t ) − ρ λ(ρ ) dt ρ d( M ) + 2 0
0
t2
J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
193
Fig. 4. Idea to prove Theorem 1.2.
|t 2 | ρ d( M ) ρ0 (·) − ρ L p (0,1) + 1 p
T
ρ1 1 − ξ2 (t ) − ρ p ξ (t ) p dt
1p
2
t2
T +ρ
ξ (t ) − λ(ρ ) p dt
1p
2
t2
p −1 1 |t 2 | p ρ d( M ) ρ0 (·) − ρ L p (0,1) + d( M ) p ρ1 (·) − ρ L p (0,1) 1 + | T − t 2 | p ρ d( M ) ρ0 (·) − ρ L p (0,1) + ρ1 (·) − ρ L p (0,1)
p −1 1
ν d( M ) p + 3T p ρ d( M ) . On the other hand, we obtain from (1.8) and (2.11) that
⎧ 1−ξ (t ) 1 ⎪ ∀t ∈ [0, t 2 ], ( 0 1 |ρ0 (x) − ρ | p dx) p , ⎪ ⎪ ⎨ 1 1 1 −ξ ( t ) 1 p p ρ (t , ·) − ρ p = ( 0 |ρ0 (x) − ρ | dx + 1−ξ2 (t ) |ρ1 (x) − ρ | dx) p , ∀t ∈ [t 2 , t 1 ], L (0,1) ⎪ ⎪ ⎪ 1 ⎩ 1 ∀t ∈ [t 1 , T ], ( 1−ξ2 (t ) |ρ1 (x) − ρ | p dx) p , ρ0 (·) − ρ L p (0,1) + ρ1 (·) − ρ L p (0,1) 2ν , ∀t ∈ [0, T ]. Clearly, setting
ν min
ε 2
δ 1 λ( M ) , , 1 + 3T p ρ d( M ) 4Cd( M ) 2T d( M ) 2
ε
, |d( M )|
p −1 p
together with (3.5) finishes the proof of Theorem 1.1.
,
2
4. Proof of Theorem 1.2 In this section, we prove Theorem 1.2. The idea is to drive the state from ρ0 to an intermediate equilibrium ρ , then to drive ρ to ρ1 by using the reversibility of the Cauchy problem (1.5) (see Fig. 4). First let us prove the following lemma.
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Lemma 4.1. For any p ∈ [1, +∞), any ρ0 ∈ L p ((0, 1); [0, ∞)) and any constant ρ 0, there exists T 00 > 0 (depending on ρ0 and ρ ) such that the following holds: For any T T 00 , there exists u ∈ L p ((0, T ); [0, ∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (1.5) satisfies also the constant final condition
ρ ( T , x) = ρ , x ∈ (0, 1). Proof. For any given
ρ0 , ρ , we define T 00 :=
1
λ(ρ + ρ0 L 1 (0,1) )
(4.1)
,
where
λ ρ + ρ0 L 1 (0,1) :=
inf
W ∈[0,ρ + ρ0 L 1 (0,1) ]
λ( W ) > 0.
For any fixed T > T 00 , we first look at the following Cauchy problem
⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ W (t ) x = 0, ρ (0, x) = ρ0 (x), x ∈ (0, 1), ⎪ ⎩ ρ (t , 0) = ρ , t ∈ (0, T ).
t ∈ (0, T ), x ∈ (0, 1), (4.2)
Similarly as in [12], we can prove, by fixed point arguments, the existence of the unique weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the Cauchy problem (4.2). More precisely, the solution can be expressed as
ρ (t , x) =
ρ0 (x − ξ1 (t )), if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1), ρ, else,
(4.3)
where ξ1 (0) = 0 and
d ξ1 dt
= λ W (t ) ,
∀t ∈ [0, T ],
with
1 W (t ) := 0
ρ ξ1 (t ) + ρ (t , x) dx = ρ,
1−ξ1 (t ) 0
ρ0 (x) dx, if t ∈ [0, t 1 ], if t ∈ [t 1 , T ].
Here t 1 ∈ [0, T 00 ] ⊂ [0, T ] is defined by requiring ξ1 (t 1 ) = 1. The existence of t 1 ∈ [0, T 00 ] can be implied by the facts that ξ1 is continuous and
ξ1 ( T 00 ) =
T 00 λ W (t ) dt T 00 · λ ρ + ρ0 L 1 (0,1) = 1 > 0 = ξ1 (0). 0
On the other hand, the fact that λ( W (t )) > 0 for all t ∈ [0, T ] gives us the strict monotonicity of ξ1 , and thus the uniqueness of t 1 ∈ [0, T 00 ]. Obviously, the solution ρ given by (4.3) is equal to ρ for all t ∈ [t 1 , T ], and in particular for t = T . 2
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195
By the reversibility of the Cauchy problem (1.5), we easily get, from Lemma 4.1 and its proof, the following corollary. Corollary 4.1. For any p ∈ [1, +∞), any ρ1 ∈ L p ((0, 1); [0, ∞)) and any constant ρ 0, let
T 01 :=
1
λ(ρ + ρ1 L 1 (0,1) )
(4.4)
,
where
λ ρ + ρ1 L 1 (0,1) :=
inf
W ∈[0,ρ + ρ1 L 1 (0,1) ]
λ( W ) > 0.
Then, for any T T 01 , there exists v ∈ L p ((0, T ); [0, ∞)) such that the weak solution ρ ∈ C 0 ([0, T ]; L p (0, 1)) to the backward Cauchy problem
⎧ ⎪ ⎨ ρt (t , x) + ρ (t , x)λ W (t ) x = 0, t ∈ (0, T ), x ∈ (0, 1), ρ ( T , x) = ρ1 (x), x ∈ (0, 1), ⎪ ⎩ ρ (t , 1)λ W (t ) = v (t ), t ∈ (0, T ) satisfies also the final condition
ρ (0, x) = ρ , x ∈ (0, 1). Let us now go back to the proof of Theorem 1.2. For any p ∈ [1, +∞), any and any ρ1 ∈ L p ((0, 1); [0, ∞)), we define
ρ :=
1
ρ0 L 1 (0,1) + ρ0 L 1 (0,1)
ρ0 ∈ L p ((0, 1); [0, ∞))
2 T 1 := T 00 + T 01 ,
(4.5) (4.6)
where T 00 and T 01 are defined by (4.1) and (4.4), respectively. Then for any T T 1 , we define a nonnegative function (see Fig. 5)
⎧ if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1), ⎨ ρ0 (x − ξ1 (t )), ρ (t , x) := ρ1 (x + 1 − ξ2 (t )), if t ∈ (t 2 , T ), x ∈ (0, ξ2 (t )), ⎩ ρ, else,
(4.7)
where ξ1 , ξ2 are defined correspondingly by
d ξ1 dt d ξ2 dt with
1 W (t ) := 0
= λ W (t ) ,
∀t ∈ [0, T ] and ξ1 (0) = 0,
= λ W (t ) ,
∀t ∈ [0, T ] and ξ2 ( T ) = 1,
⎧ 1−ξ1 (t ) ⎪ ρ0 (x) dx, if t ∈ [0, t 1 ], ⎪ ⎨ ρ ξ1 (t ) + 0 if t ∈ [t 1 , t 2 ], ρ (t , x) dx = ρ , ⎪ ⎪ ⎩ ρ ξ (t ) + 1 ρ (x) dx, if t ∈ [t 2 , T ] 2 1−ξ2 (t ) 1
and t 1 = ξ1−1 (1), t 2 = ξ2−1 (0).
(4.8)
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Fig. 5. Definition
ρ in the case 0 t 1 T 00 T − T 01 t 2 T .
By the proof of Lemma 4.1 and the fact that T T 1 = T 00 + T 01 , we have the relation that 0 t 1 T 00 T − T 01 t 2 T . Therefore, the above ρ and W are both well defined. Let
⎧ 1−ξ1 (t ) ⎪ ρ0 (x) dx), if t ∈ (0, t 1 ), ⎪ ⎨ ρ λ(ρ ξ1 (t ) + 0 if t ∈ (t 1 , t 2 ), u (t ) := ρ λ(ρ ), ⎪ ⎪ ⎩ ρ (1 − ξ (t ))λ(ρ ξ (t ) + 1 ρ (x) dx), if t ∈ (t 2 , T ). 1 2 2 1−ξ2 (t ) 1
(4.9)
Then, it is easy to check that ρ and W defined by (4.7) and (4.8) satisfy the Cauchy problem (1.5) and the final condition (1.6) simultaneously. This finishes the proof of Theorem 1.2. 2 5. Proof of Theorem 1.3 In this section, we prove Theorem 1.3 by using a fixed point argument. Similarly as in the proof of Theorem 1.1, we choose a closed domain as a subspace of the Banach space C 0 ([0, T ]) and define a contraction mapping (based on Proposition 2.2 in Section 2.2). The fixed point of this mapping establishes the nodal profile controllability (or out-flux controllability) for the original nonlinear problem. Actually, we use the same domain candidate Ω M ,δ as in Section 3, see (3.1) for definition. We also λ( M ), λ( M ), which are still defined by (2.5), (2.6). M , δ > 0 are two constants to be determined. use For any ξ1 ∈ Ω M ,δ , let ξ2 ∈ C 0 ([0, T ]) be still defined by (3.2) and ξ3 ∈ C 0 ([0, T ]) be defined by
ξ3 (t ) = 1 − ξ1 ( T 1 ) + ξ1 (t ),
∀t ∈ [0, T ].
Then ξi (i = 1, 2, 3) are all strictly increasing with respect to t and satisfy properties (3.3)–(3.4). By definitions of Ω M ,δ , ξi (i = 1, 2, 3) and (1.7), we have
ξ1 ( T 1 ) λ(ρ ) T 1 − δ = 1 + λ(ρ )( T 1 − T 0 ) − δ and
ξ 2 ( T ) = ξ3 ( T 1 ) = 1 ,
ξ2 (0) = 1 − ξ1 ( T ) 1 − ξ1 ( T 1 ) = ξ3 (0) δ − λ(ρ )( T 1 − T 0 ).
(5.1)
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Consequently, setting
δ λ(ρ )( T 1 − T 0 ),
(5.2)
then ξ1 ( T ) 1 and ξ2 (0) ξ3 (0) 0. By the strict monotonicity of ξi (i = 1, 2, 3), there exists a unique (t 1 , t 2 , t 3 ) ∈ [0, T ]3 such that ξ1 (t 1 ) = 1, ξ2 (t 2 ) = ξ3 (t 3 ) = 0 with 0 < t 3 < t 2 < T , 0 < t 1 < T 1 < T . Similarly as in Section 2.2, there are five various possibilities concerning the order of t 1 , t 2 , t 3 , T 1 , T and without loss of generality we discuss only the case that 0 < t 3 < t 2 t 1 < T 1 < T . Inspired by (2.18), we define W (ξ1 ) ∈ C 0 ([0, T ]; [0, ∞)) by
⎧ ρ + 01−ξ1 (t ) (ρ0 (x) − ρ ) dx, ⎪ ⎪ ⎪ ⎪ ⎪ ξ −1 (ξ (t )+1) 1−ξ (t ) ⎪ ⎪ ρ + T 11 1 ( y (s) − ρ λ(ρ )) ds + 0 1 (ρ0 (x) − ρ ) dx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 (t ) + 1 − λ(ρ )ξ1−1 (ξ1 (t ) + 1)), ⎪ ⎪ ⎪ ⎨ T 1−ξ1 (t ) (ρ0 (x) − ρ ) dx W (ξ1 )(t ) := ρ + T 1 ( y (s) − ρ λ(ρ )) ds + 0 ⎪ ⎪ ⎪ + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎪ ⎪ ⎪ ρ + TT1 ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 ( T 1 ) − λ(ρ ) T 1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ρ (ξ1 ( T ) − λ(ρ ) T ), ⎪ ⎪ ⎩ ρ + tT ( y (s) − ρ λ(ρ )) ds + ρ (ξ1 (t ) − λ(ρ )t ) − ρ (ξ1 ( T ) − λ(ρ ) T ),
∀t ∈ [0, t 3 ],
∀t ∈ [t 3 , t 2 ], ∀t ∈ [t 2 , t 1 ], ∀t ∈ [t 1 , T 1 ], ∀t ∈ [ T 1 , T ]. (5.3)
This implies immediately with assumption (1.8) and Hölder’s inequality that
0 W (ξ1 )(t ) ρ + ρ0 (·) − ρ L 1 (0,1) + y (·) − ρ λ(ρ ) L 1 ( T , T ) + 2ρ sup ξ1 (t ) − λ(ρ )t
1 t ∈[0, T ]
ρ + 2ν + 2ρ δ,
∀t ∈ [0, T ].
Obviously, by choosing suitable constant M (see (5.7)) and ν , δ small enough, we could get the estimate (3.7) of W (ξ1 ). Motivated by solving the linear controllability problem (1.5)–(1.14), we define a mapping F : Ω M ,δ ξ1 → F (ξ1 ) ∈ C 0 ([0, T ]) by (3.8) but with (5.3) instead of (3.6). We are going to prove that F is contraction mapping on Ω M ,δ for small δ and ν . For any ξ1 ∈ Ω M ,δ , it is clear that F (ξ1 )(0) = 0 and (3.9) holds. Unlike the proof of Theorem 1.1, we don’t have directly
F (ξ1 )(t ) − λ(ρ )t δ, for small δ and
∀t ∈ [0, T ]
ν . Actually we have only the following estimate:
t
F (ξ1 )(t ) − λ(ρ )t = λ W (ξ1 )(s) − λ(ρ ) ds
0
t d( M )
W (ξ1 )(s) − ρ ds
0
2d( M ) T (ν + ρ δ),
∀t ∈ [0, T ],
(5.4)
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J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
which does not infer (5.4) generically. This is the main difficulty that we need to overcome. Nevertheless, the contraction estimate of the mapping F helps to obtain (5.4). Thus the main idea is to prove first that F is a contraction mapping from Ω M ,δ to C 0 ([0, T ]), then to prove that F maps into Ω M ,δ indeed. Similarly as in the proof of Theorem 1.1, for any ξ1 , ξ1 ∈ ΩM ,δ , one can obtain
F ( ξ1 ) − F (ξ1 )
C 0 ([0, T ])
C ν d( M )
λ( M )
· ξ1 − ξ1 C 0 ([0,T ])
for some constant C independent of ξ1 , ξ1 . Setting
ν
λ( M ) , 2Cd( M )
then
F ( ξ1 ) − F (ξ1 )
1
C 0 ([0, T ])
ξ1 − ξ1 C 0 ([0,T ]) .
(5.5)
2
Now it is left to show that (5.4) holds with the help of (5.5). Denote
ξ0 (t ) := λ(ρ )t ,
∀t ∈ [0, T ].
Obviously, ξ0 ∈ Ω M ,δ . Then, for any ξ1 ∈ Ω M ,δ ,
F (ξ1 ) − ξ0
C 0 ([0, T ])
F (ξ1 ) − F (ξ0 ) C 0 ([0,T ]) + F (ξ0 ) − ξ0 C 0 ([0,T ]) 1 ξ1 − ξ0 C 0 ([0,T ]) + F (ξ0 ) − ξ0 C 0 ([0,T ]) 2
δ 2
+ F (ξ0 ) − ξ0 C 0 ([0,T ]) .
It suffices to show that for any fixed δ > 0 there exists
F (ξ0 ) − ξ0
ν > 0 sufficiently small such that
C 0 ([0, T ])
δ . 2
In fact, by (5.3)
⎧ ρ + 01−λ(ρ )t (ρ0 (x) − ρ ) dx, ⎪ ⎪ ⎪ ⎪ ⎪ t + λ(1ρ ) 1−λ(ρ )t ⎪ ⎪ ⎪ ρ + ( y (s) − ρ λ(ρ )) ds + 0 (ρ0 (x) − ρ ) dx, ⎪ T1 ⎨ T 1−λ(ρ )t W (ξ0 )(t ) = ρ + ( y (s) − ρ λ(ρ )) ds + 0 (ρ0 (x) − ρ ) dx, T1 ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ρ + T 1 ( y (s) − ρ λ(ρ )) ds, ⎪ ⎪ ⎪ ⎩ ρ + tT ( y (s) − ρ λ(ρ )) ds,
∀t ∈ [0, t 3 ], ∀t ∈ [t 3 , t 2 ], ∀t ∈ [t 2 , t 1 ], ∀t ∈ [t 1 , T 1 ], ∀t ∈ [ T 1 , T ].
(5.6)
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199
Thus
t
F (ξ0 )(t ) − ξ0 (t ) =
λ W (ξ0 )(s) − λ(ρ )s ds
0
t d( M )
W (ξ0 )(s) − λ(ρ )s ds
0
td( M ) ρ0 (·) − ρ L 1 (0,1) + y (·) − ρ λ(ρ ) L 1 (T 2T d( M )ν ,
1 ,T )
∀t ∈ [0, T ].
Finally setting
M := 2ρ + 1
(5.7)
and
1 δ min λ(ρ )( T 1 − T 0 ), , 2
δ λ( M ) 1 , , ν min , 2Cd( M ) 4T d( M ) 2 we get (3.7) and (5.4). Applying the contraction mapping theorem, we conclude that F has a unique fixed point in Ω M ,δ : F (ξ1 ) = ξ1 . Under the control
u (t ) :=
⎧ ⎨ ρ λ( W (t )), ⎩
t ∈ (0, t 3 ) ∪ (t 2 , T ),
yd (ξ1−1 (ξ1 (t )+1))
λ( W (ξ1−1 (ξ1 (t )+1)))
λ( W (t )), t ∈ (t 3 , t 2 ),
(5.8)
where W (t ) is given by (2.18), t 1 = ξ1−1 (1), t 2 = ξ2−1 (0), t 3 = ξ3−1 (0), ξ2 , ξ3 are given by (3.2), (5.1) respectively, the unique weak solution to (1.5) is given by
ρ (t , x) =
⎧ ρ0 (x − ξ1 (t )), ⎪ ⎪ ⎨ −1
if t ∈ (0, t 1 ), x ∈ (ξ1 (t ), 1),
yd (ξ1 (ξ1 (t )+1−x))
λ( W (ξ1−1 (ξ1 (t )+1−x))) ⎪ ⎪ ⎩ ρ,
, if t ∈ (t 3 , T ), x ∈ (0, 1) ∩ (ξ2 (t ), ξ3 (t )),
(5.9)
else.
1
It verifies (1.10) and the out-flux condition (1.14), and 0 ρ (t , x) dx = W (t ) for all t ∈ [0, T ] as well. It remains to deal with (1.9) and (1.11) for ν > 0 small enough. Let us deal with (1.9) (the proof of (1.11) is similar and in fact (1.11) follows from (1.9)). Let τ : [t 3 , t 2 ] → [ T 1 , T ], t → ξ1−1 (ξ1 (t ) + 1). Then τ is a C 1 diffeomorphism and
0<
dt dτ
λ( M ) . λ( M )
200
J.-M. Coron, Z. Wang / J. Differential Equations 252 (2012) 181–201
Hence
t2
−1
p
yd ξ ξ1 (t ) + 1 − ρ λ(ρ ) dt =
T
1
t3
yd (τ ) − ρ λ(ρ ) p dt dτ λ( M ) ν p . dτ λ( M )
(5.10)
T1
From (5.10) and the fact that
0 W (t ) − ρ 2ν + 2ρ δ, one easily gets (1.9) by letting first δ then rem 1.3. 2
∀t ∈ [0, T ],
ν sufficiently small. This concludes the proof of Theo-
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