Exact controllability for first order quasilinear hyperbolic systems with vertical characteristics

Acta Mathematica Scientia 2009,29B(4):980–990 http://actams.wipm.ac.cn

EXACT CONTROLLABILITY FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS WITH VERTICAL CHARACTERISTICS∗ Dedicated to Professor Wu Wenjun on the occasion of his 90th birthday


)

Li Tatsien (

School of Mathematical Sciences, Fudan University, Shanghai 200433, China E-mail: [email protected]

 )

Rao Bopeng (

Institut de Recherche Math´ ematique Avanc´ ee, Universit´ e Louis Pasteur de Strasbourg, 67084 Strasbourg, France E-mail: [email protected]

Abstract We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningful internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls. Key words quasilinear hyperbolic systems; exact controllability; local distributed control; switching controls 2000 MR Subject Classification

1

93B05; 35B37; 35L60

Introduction Consider the first order quasilinear hyperbolic system ∂u ∂u + A(u) = F (u), ∂t ∂x

(1.1)

where u = (u1 , · · · , un )T is the unknown vector function of (t, x), A(u) is an n × n matrix with smooth entries, F (u) = (f1 (u), · · · , fn (u))T is a smooth vector function of u such that F (0) = 0. ∗ Received

January 14, 2009

(1.2)

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By hyperbolicity, for any given u on the domain under consideration, A(u) has n real eigenvalues λ1 (u), · · · , λn (u) and a complete set of left eigenvectors l1 (u), · · · , ln (u). Let us introduce the “diagonal” variable vi corresponding to the i-th eigenvalue λi (u) by vi = li (u)u

(i = 1, · · · , n).

(1.3)

We arbitrarily give the initial condition t=0:

u = φ(x),

0≤x≤L

(1.4)

t=T :

u = Φ(x),

0 ≤ x ≤ L.

(1.5)

and the final condition Suppose that there are no zero eigenvalues: λp (u) < 0 < λr (u) (p = 1, · · · , m;

r = m + 1, · · · , n).

(1.6)

The boundary conditions are given as follows: x=0: x=L:

vr = Gr (t, v1 , · · · , vm ) + Hr (t)

(r = m + 1, · · · , n),

vp = Gp (t, vm+1 , · · · , vn ) + Hp (t)

(p = 1, · · · , m),

(1.7) (1.8)

where L is the length of the interval 0 ≤ x ≤ L and Gi (i = 1, · · · , n) are smooth functions with Gi (t, 0, · · · , 0) ≡ 0

(i = 1, · · · , n).

(1.9)

Using a constructive method based on the theory of semi-global C 1 solution, we have established in [2–3] the local exact boundary controllability for system (1.1) by means of two-sided boundary controls Hi (i = 1, · · · , n), or one-sided boundary controls Hp (p = 1, · · · , m) applied to the side x = L if the number of negative eigenvalues is not less than that of positive ones: n − m ≤ m ⇐⇒ n ≤ 2m.

(1.10)

λp (u) < 0 ≡ λq (u) < λr (u),

(1.11)

If there are zero eigenvalues

in which p = 1, · · · , l; q = l + 1, · · · , m; r = m + 1, · · · , n, the situation is more complicated. In this case, the boundary conditions are given as x=0: x=L:

vr = Gr (t, v1 , · · · , vl , vl+1 , · · · , vm ) + Hr (t) (r = m + 1, · · · , n), vp = Gp (t, vl+1 , · · · , vm , vm+1 , · · · , vn ) + Hp (t)

(p = 1, · · · , l).

(1.12) (1.13)

Since the characteristics corresponding to the variables vq (q = l + 1, · · · , m) are vertical, there is no possibility to modify directly the behavior of these variables by means of the action of boundary controls. So it seems natural to employ suitably internal controls for the variables vq (q = l + 1, · · · , m) to achieve the exact controllability. More precisely, let us write system (1.1) into the following characteristic form  ∂u ∂u  + λp (u) + μp (u) = 0 (p = 1, · · · , l), (1.14) lp (u) ∂t ∂x

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∂u + μq (u) + Hq (t, x) = 0 (q = l + 1, · · · , m), ∂t  ∂u ∂u  + λr (u) + μr (u) = 0 (r = m + 1, · · · , n), lr (u) ∂t ∂x

lq (u)

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(1.15) (1.16)

in which μi (u) = li (u)F (u) (i = 1, · · · , n)

(1.17)

and we add internal controls Hq (t, x) (q = l + 1, · · · , m) in (1.15), the second part of the system corresponding to zero eigenvalues. It was shown in [4] that the exact controllability can be realized by means of the boundary controls Hp (p = 1, · · · , l), Hr (r = m + 1, · · · , n) and the internal controls Hq (r = l + 1, · · · , m). However, internal controls are very difficult or even impossible to be realized in applications from physical or engineering considerations. Let us first consider the following one-dimensional quasilinear wave equation:  ∂u  ∂ 2 u  ∂u ∂u  ∂2u = 0, − K u, + F u, , ∂t2 ∂x ∂x2 ∂x ∂t

(1.18)

where K is a C 1 function such that K(u, v) > 0. Let v=

∂u , ∂x

w=

∂u . ∂t

(1.19)

(1.18) can be rewritten as the following first order system ⎧ ∂u ⎪ ⎪ − w = 0, ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v ∂w − = 0, ⎪ ∂t ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂w − K(u, v) ∂v + F (u, v, w) = 0. ∂t ∂x

(1.20)

Notice that the first equation of (1.20), corresponding to zero eigenvalue, is introduced as the definition of the variable w, we can not put any internal control in it to realize the exact controllability, then the theory with internal controls in [4] can not be applied to this situation. Nevertheless, the exact boundary controllability of equation (1.18) only by boundary controls was established in [5] by directly using second order equation (1.18) without zero eigenvalues. We next consider the system of one-dimensional adiabatic flow: ⎧ ∂τ ∂v ⎪ − = 0, ⎪ ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎨ ∂v ∂p(τ, S) + = 0, ⎪ ∂t ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂S = 0, ∂t

(1.21)

where τ > 0 is the specific volume, v the velocity, p the pressure and S the entropy. Under the assumption pτ (τ, S) < 0, the system admits three distinct real eigenvalues:   λ1 = − −pτ (τ, S) < λ2 ≡ 0 < λ3 = − −pτ (τ, S).

(1.22)

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Since the third equation of (1.21), corresponding to zero eigenvalue, has a physical meaning, we can not put any internal control in it to get the exact controllability. Once again the use of internal controls in this situation is not possible from physical reason. However, the exact boundary controllability of (1.21) was realized in [7]. We finally notice that the internal controls, even physically reasonable, are in general very difficult to be realized in practice. The goal of this paper is to investigate the exact controllability of hyperbolic systems by means of physically meaningful internal controls. In Section 2, we will establish the exact controllability for a kind of quasilinear hyperbolic systems with zero eigenvalues by means of internal controls applied to equations corresponding to non-zero eigenvalues. In Section 3, we study the exact controllability for a very simplified linear model in various situations. We first give a necessary and sufficient condition on the initial and final values for the exact boundary controllability. We next prove the null exact controllability by means of an internal control with decreasing support. The general case will be treated by using first a local internal control to steer the initial value to a suitable middle state, then a boundary control to steer the middle state to the desired final state.

2 Exact Controllability by Internal Controls Acting on Equations Corresponding to Non-Zero Eigenvalues Let’s consider the following quasilinear hyperbolic system of characteristic form  ∂u ∂u  + λp (u) + μp (u) + Hp (t, x) = 0 (p = 1, · · · , l), lp (u) ∂t ∂x

(2.1)

∂uq + μq (u) = 0 (q = l + 1, · · · , m), (2.2) ∂t  ∂u ∂u  + λr (u) + μr (u) + Hr (t, x) = 0 (r = m + 1, · · · , n), lr (u) (2.3) ∂t ∂x where μi (u) (i = 1, · · · , n), λp (u), lp (u) (p = 1, · · · , l) and λr (u), lr (u) (r = m + 1, · · · , n) are all C 1 functions. Assume furthermore that μq (0) = 0

(q = l + 1, · · · , m)

(2.4)

and λp (u) < 0 < λr (u) (p = 1, · · · , l;

r = m + 1, · · · , n).

(2.5)

In this situation, we have ⎧ ⎨ vp = lp (u)u (p = 1, · · · , l), vq = uq (q = l + 1, · · · , m), ⎩ vr = lr (u)u (r = m + 1, · · · , n).

(2.6)

The boundary conditions are given as x=0: x=L:

vr = Gr (t, v1 , · · · , vl , vl+1 , · · · , vm )

(r = m + 1, · · · , n),

(2.7)

vp = Gp (t, vl+1 , · · · , vm , vm+1 , · · · , vn ) (p = 1, · · · , l),

(2.8)

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where Gp (p = 1, · · · , l) and Gr (r = m + 1, · · · , n) are all C 1 functions with respect to their arguments such that Gp (t, 0, · · · , 0) ≡ Gr (t, 0 · · · , 0) ≡ 0

(p = 1, · · · l;

r = m + 1, · · · , n).

(2.9)

The initial condition is given as t=0:

u = φ(x),

0 ≤ x ≤ L.

(2.10)

Under the assumption that equation (2.2) is essentially coupled with equations (2.1) and (2.3) (see below), our object is to find some internal controls Hp (p = 1, · · · , l) and Hr (r = m + 1, · · · , n) applied to equations (2.1) and (2.3), corresponding to non-zero eigenvalues, such that the C 1 solution to the mixed initial-boundary values problem (2.1)–(2.3), (2.7)–(2.8) and (2.10) satisfies exactly the final condition t=T :

u = Φ(x),

0 ≤ x ≤ L.

(2.11)

The difficulty is that we can not put any control in equation (2.2) corresponding to zero eigenvalues. A basic idea consists of constructing a function u = u(t, x) which satisfies simultaneously the equation (2.2), the boundary conditions (2.7)–(2.8), the initial condition (2.10) and the final condition (2.11). Then, inserting this function u = u(t, x) into the equations (2.1) and (2.3) we get the corresponding controls Hp (p = 1, · · · , l) and Hr (r = m + 1, · · · , n). For this purpose, we have to assume that the number of zero eigenvalues is less than or equal to that of non-zero eigenvalues: m − l ≤ n/2.

(2.12)

In fact, let U = (ul+1 , · · · , um )T ,

V = (ul , · · · , ul , um+1 , · · · , un )T .

We consider a special case of equation (2.2): dU = U + BV, dt where B is a (m − l) × (l + n − m) constant matrix. Following Kalman’s Theorem (see [1]), this linear system is controllable if and only if the matrix m−l



 (B, B, · · · , B) is of rank m − l, which could be true only if m − l ≤ l + n − m ⇐⇒ m − l ≤ n/2. In what follows, without loss of generality, we suppose that there exist some integers l, n with 1 ≤ l ≤ l, m + 1 ≤ n ≤ n, such that m − l = l + (n − m).

(2.13)

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We next assume that equation (2.2) provides a real coupling with equations (2.1) and (2.3). This means that in a neighbourhood of u = 0, equation (2.2) can be equivalently rewritten as up = Rp (ul+1 , · · · , ul , ul+1 , · · · , um , ∂t ul+1 , · · · , ∂t um , un+1 , · · · , un ),

(2.14)

ur = Rr (ul+1 , · · · , ul , ul+1 , · · · , um , ∂t ul+1 , · · · , ∂t um , un+1 , · · · , un )

(2.15)

for p = 1, · · · , l and r = m + 1, · · · , n. We have the following result. Theorem 2.1 Assume that μi (u) (i = 1, · · · , n), λp (u), lp (u) (p = 1, · · · , l) and λr (u), lr (u) (r = m + 1, · · · , n) are all C 1 functions with respect to their arguments. Suppose furthermore that (2.4)–(2.5), (2.9) and (2.12)–(2.15) hold. Let T > 0, then for any given initial data φ ∈ C 1 [0, L] and final data Φ ∈ C 1 [0, L] with small C 1 norm, satisfying φ(0) = φ(L) = Φ(0) = Φ(L) = 0,

(2.16)

there exist internal controls Hp (t, x) (p = 1, · · · , l) and Hr (t, x) (r = m + 1, · · · , n) such that the corresponding mixed initial-boundary value problem (2.1)–(2.3), (2.7)–(2.8) and (2.10) admits a unique C 1 solution u = u(t, x) with small C 1 norm on the domain R(T ) = {(t, x)| 0 ≤ t ≤ T, 0 ≤ x ≤ L}, which exactly satisfies the final condition (2.11). Proof We first define the initial values and the final values of ∂t uq (q = l + 1, · · · , m) by means of equation (2.2): t=0:

∂uq = −μq (φ) := ψq ∂t

∂uq = −μq (Φ) := Ψq ∂t Noting (2.4) and (2.16), we have t=T :

(q = l + 1, · · · , m),

0 ≤ x ≤ L,

(2.17)

(q = l + 1, · · · , m),

0 ≤ x ≤ L.

(2.18)

ψq (0) = ψq (L) = Ψq (0) = Ψq (L) = 0

(q = l + 1, · · · , m).

(2.19)

h0 , h1 and h1 be the basis functions of Hermite interpolation on the interval [0, T ]. Let h0 , For q = l + 1, · · · , m, we define uq (t, x) = h0 (t)φq (x) + h0 (t)ψq (x) + h1 (t)Φq (x) + h1 (t)Ψq (x).

(2.20)

It is clear that the functions uq ∈ C 1 (q = l + 1, · · · , m) satisfy the following initial and final conditions: t=0:

uq = φq ,

∂uq = ψq ∂t

(q = l + 1, · · · , m),

0 ≤ x ≤ L,

(2.21)

∂uq = Ψq (q = l + 1, · · · , m), 0 ≤ x ≤ L. (2.22) ∂t Similarly, let l0 and l1 be the basis functions of Lagrange interpolation on the interval [0, T ]. We define t=T :

u q = Φq ,

up (t, x) = l0 (t)φp (x) + l1 (t)Φp (x)

(p = l + 1, · · · , l),

(2.23)

ur (t, x) = l0 (t)φr (x) + l1 (t)Φr (x)

(r = n + 1, · · · , n).

(2.24)

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It is clear that up , ur ∈ C 1 satisfy the following initial and final conditions: t=0:

up = φp ,

t=T :

u p = Φp ,

ur = φr , u r = Φr ,

0 ≤ x ≤ L,

(2.25)

0≤x≤L

(2.26)

for p = l + 1, · · · , l and r = n + 1, · · · , n. Once the components uq (q = l + 1, · · · , m), up (p = l + 1, · · · , l) and ur (r = n + 1, · · · , n) are chosen by (2.20), and (2.23)–(2.24), we define the other components up (p = 1, · · · , l) and ur (r = m + 1, · · · , n) by means of (2.14)–(2.15). After having determined all the components ui (i = 1, · · · , n) of u, we now define the internal controls as  ∂u ∂u  + λp (u) − μp (u) Hp (t, x) := −lp (u) (p = 1, · · · , l), (2.27) ∂t ∂x  ∂u ∂u  − μr (u) Hr (t, x) := −lr (u) (r = m + 1, · · · , n). (2.28) + λr (u) ∂t ∂x Since (2.2) is equivalent to (2.14)–(2.15), the function u = u(t, x) satisfies naturally (2.2). Moreover, the choice of ψq , Ψq (q = l + 1, · · · , m) by (2.17)–(2.18) together with (2.14)–(2.15) implies that t=0: up = φp , ur = φr , 0 ≤ x ≤ L, (2.29) t=T :

u p = Φp ,

u r = Φr ,

0≤x≤L

(2.30)

for p = 1, · · · , l and r = m + 1, · · · , n. Then, noting (2.21)–(2.22) and (2.25)–(2.26), u = u(t, x) satisfies the initial condition (2.9) and the final condition (2.10). Moreover, noting (2.16) and (2.19), it is easy to see from (2.20), (2.14)–(2.15) and (2.23)–(2.24) that x=0: u=0

then vi = 0

(i = 1, · · · , n),

(2.31)

x = L : u = 0 then vi = 0 (i = 1, · · · , n).

(2.32)

Thus, noting (2.9), u = u(t, x) also satisfies the boundary conditions (2.7)–(2.8). The proof is then complete.

3 Exact Controllability by Local Internal Controls Acting on Equations Corresponding to Non-Zero Eigenvalues and Boundary Controls Let’s consider the following linear model  rt − rx = 0, st − r = 0.

(3.1)

We want to find a boundary control h applied at the end x = L x=L:

r = h(t)

(3.2)

such that the solution of (3.1)–(3.2) with the initial condition t=0:

(r, s) = (r0 , s0 ),

0≤x≤L

(3.3)

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satisfies also the final condition t=T :

(r, s) = (rT , sT ),

0 ≤ x ≤ L,

(3.4)

provided that T > 0 is large enough. Proposition 3.1 Let T ≥ L. For any given initial data (r0 , s0 ) and final data (rT , sT ) of class C 1 [0, L] × C 2 [0, L], verifying the condition rT (x) − sT (x) = r0 (x) − s0 (x),

0 ≤ x ≤ L,

(3.5)

there exists a boundary control h ∈ C 1 [0, T ] such that on the domain {(t, x)| 0 ≤ t ≤ T, 0 ≤ x ≤ L}, the solution (r, s) = (r(t, x), s(t, x)) ∈ C 1 × C 2 to the mixed initial-boundary value problem (3.1)–(3.3) exactly satisfies the final condition (3.4). Proof First of all, assume that the solution (r, s) to the mixed initial-boundary value problem (3.1)–(3.3) satisfies the final condition (3.4). Then, inserting the second equation of (3.1) into the first one, we obtain stt − sxt = 0. (3.6) Integrating (3.6) over the time interval [0, T ] gives st (T, x) − sx (T, x) = st (0, x) − sx (0, x),

0 ≤ x ≤ L,

which, together with the second equation of (3.1), implies the condition (3.5). We now assume that (3.5) is satisfied. Let s = s(t, x) be the C 2 solution of the backward problem ⎧  ⎨ st − sx = r0 − s0 , (3.7) x = 0 : s = 0, ⎩ t = T : s = sT . By [6], for any T ≥ L there exists a boundary control h of class C 2 such that the C 2 solution s = s (t, x) to the forward problem ⎧ ⎨ s t − s x = 0, x = L : s = h, ⎩ t=0: s = s0 − s(0)

(3.8)

satisfies the null final condition t=T :

s = 0.

It follows from (3.7)–(3.9) that s = s + s ∈ C 2 satisfies ⎧ st − sx = r0 − s0 , ⎪ ⎪ ⎨x = L : s = h + s, ⎪ s = s0 , ⎪ ⎩t = 0 : t = T : s = sT .

(3.9)

(3.10)

Then, let r = st .

(3.11)

It follows from (3.10) that 

rt − rx = (st − sx )t = 0, x=L: r= ht + st := h ∈ C 1 [0, T ];

(3.12)

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and, noting the condition (3.5), that  r(0, x) = st (0, x) = sx (0, x) + r0 (x) − s0 (x) = r0 (x), r(T, x) = st (T, x) = sx (T, x) + r0 (x) − s0 (x) = rT (x).

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(3.13)

Thus, (r, s) = (r(t, x), s(t, x)) is a C 1 × C 2 solution to the mixed initial-boundary value problem (3.1)–(3.3). The proof of Proposition 3.1 is then complete. If the condition (3.5) is not satisfied, we will look for a local internal control q = q(t, x), acting on the first equation corresponding to non-zero eigenvalue such that the solution (r, s) to the following mixed initial-boundary value problem ⎧ r − r + q = 0, t x ⎪ ⎪ ⎨ s − r = 0, t (3.14) ⎪ x=L: r = 0, ⎪ ⎩ t=0: (r, s) = (r0 , s0 ), 0 ≤ x ≤ L satisfies the null final condition t=T :

(r, s) = (0, 0),

0 ≤ x ≤ L.

(3.15)

Proposition 3.2 Let T ≥ 2L. For any given initial data (r0 , s0 ) ∈ C 1 [0, L] × C 2 [0, L], there exists a local internal control q = q(t, x) such that the C 1 × C 2 solution (r, s) to the mixed initial-boundary value problem (3.14) exactly satisfies the null final condition (3.15). Proof Let s be a C 2 solution to the mixed initial-boundary values problem ⎧ s − s = 0, tt xx ⎪ ⎪ ⎨x = 0 : st − sx = 0, (3.16) ⎪ x = L : s = 0, ⎪ ⎩ t=0: (s, st ) = (s0 , r0 ), 0 ≤ x ≤ L. Let u = s t − sx . u is of class C 1 and (3.16) implies that 

(3.17)

ut + ux = 0, x = 0 : u = 0.

(3.18)

Then, it follows that u ≡ 0, Then, we obtain



∀ t ≥ L,

st − sx = 0, t ≥ L, x = L : s = 0.

0 ≤ x ≤ L. 0 ≤ x ≤ L,

(3.19)

(3.20)

Hence, we have s ≡ 0,

∀ t ≥ 2L,

0 ≤ x ≤ L.

(3.21)

Finally, putting r = st ,

q = rx − rt = −ut ,

(3.22)

it is easy to see that (r, s) satisfies (3.14) and (3.15) with T ≥ 2L. This proves the Proposition 3.2.

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Remark 3.1 From (3.18) and (3.19), we see that q ≡ 0,

∀ 0 ≤ t ≤ L,

0≤x≤t

or t ≥ L,

0 ≤ x ≤ L.

(3.23)

Therefore the support of q is included in the triangle  (t, x)| 0 ≤ t ≤ L,

t≤x≤L



and will disappear for t ≥ L. Such an internal control is indeed (even better than) a local control. Remark 3.2 Let sm ∈ C 2 [0, L]. (0, sm ) obviously satisfies the two equations of (3.14) with q ≡ 0. Therefore, by superposition, the null final condition (3.15) in Proposition 3.2 can be replaced by a more general final condition t=T :

(r, s) = (0, sm ).

(3.24)

Now we return to the general case for the control problem of system (3.1). Theorem 3.1 Let T ≥ 3L. For any given initial data (r0 , s0 ) and final data (rT , sT ) of class C 1 [0, L] × C 2 [0, L], there exists a local internal control q = q(t, x) ∈ C 0 ([0, T ] × [0, L]) and a boundary control h = h(t) ∈ C 1 [0, T ] such that the C 1 × C 2 solution (r, s) to the following mixed initial-boundary values problem ⎧ r − r + q = 0, t x ⎪ ⎪ ⎨ s − r = 0, t (3.25) ⎪ x = L : r = h, ⎪ ⎩ t = 0 : (r, s) = (r0 , s0 ), 0 ≤ x ≤ L satisfies the final condition t=T :

(r, s) = (rT , sT ),

0 ≤ x ≤ L.

(3.26)

Proof Let sm ∈ C 2 [0, L]. By Proposition 3.2 and Remark 3.2, for any T0 ≥ 2L there exists a local internal control q = q(t, x) on the domain {(t, x)| 0 ≤ t ≤ T0 , 0 ≤ x ≤ L} such that the C 1 × C 2 solution (r, s) to the following mixed initial-boundary value problem ⎧ r − r + q = 0, t x ⎪ ⎪ ⎨ s − r = 0, t ⎪ x = L : r = 0, ⎪ ⎩ t = 0 : (r, s) = (r0 , s0 ),

(3.27) 0≤x≤L

satisfies the condition t = T0 :

(r, s) = (0, sm ),

0 ≤ x ≤ L.

(3.28)

Moreover, by Remark 3.1, the support of q is contained in the triangle supp{q} ⊆ {0 ≤ t ≤ L,

t ≤ x ≤ L}.

(3.29)

Now we particularly assume that sm = sm (x) satisfies 0 − sm = rT − sT ,

0 ≤ x ≤ L.

(3.30)

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ACTA MATHEMATICA SCIENTIA

Vol.29 Ser.B

Then, by Proposition 3.1, for any T with T − T0 ≥ 1, there exists a boundary control h ∈ C 1 [T0 , T ] such that the C 1 × C 2 solution (r, s) to the mixed initial-boundary value problem ⎧ r − r = 0, t x ⎪ ⎪ ⎨ s − r = 0, t

⎪ x=L: ⎪ ⎩ t = T0 :

r = h, (r, s) = (0, sm ),

(3.31) 0≤x≤L

satisfies the desired final condition t=T :

(r, s) = (rT , sT ),

0 ≤ x ≤ L.

(3.32)

The proof is then complete. Remark 3.3 The strategy consists of using a local internal control q to steer the initial state (r0 , s0 ) to a middle state (0, sm ) on the period [0, T0 ], then switching to a boundary control h to steer the middle state (0, sm ) to the desired final state (rT , sT ) on the period [T0 , T ]. We notice that the support of q is reduced to the right side x = L at the moment T0 , where the boundary control h takes over from the internal control q. References [1] Lee E B, Markus L. Foundations of optimal control theory. Melbourne, FL: Robert E Krieger Publishing Co Inc, 1986 [2] Li Tatsien, Rao Bopeng. Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chin Ann Math, Ser B, 2002, 23: 209–218 [3] Li Tatsien, Rao Bopeng. Exact boundary controllability for quasilinear hyperbolic systems. SIAM J Control Optim, 2003, 41: 1748–1755 [4] Li Tatsien, Yu Lixin. Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues. Chin Ann Math, Ser B, 2003, 24: 415–422 [5] Li Tatsien, Yu Lixin. Exact boundary controllability for 1-D quasilinear wave equations. SIAM J Control Optim, 2006, 45: 1074–1083 [6] Russell D L. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev, 1978, 20: 639–739 [7] Wang Zhiqiang, Yu Lixin. Exact boundary controllability for a one-dimensional adiabatic flow system (in Chinese). Appl Math J Chinese Univ, Ser A, 2008, 23: 35–40