ON THE EXISTENCE OF TIME OPTIMAL CONTROLS WITH

c 2011 Society for Industrial and Applied Mathematics 

SIAM J. CONTROL OPTIM. Vol. 0, No. 0, pp. 000–000

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS WITH CONSTRAINTS OF THE RECTANGULAR TYPE FOR HEAT EQUATIONS∗ ¨ † AND GENGSHENG WANG‡ QI LU Abstract. This paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations. An existence result of time optimal controls for such a problem is established. The rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations. In the finite dimensional case, there is no difference between such problems with control constraints of the rectangular type and those of the ball type, from the perspective of the study on the existence of optimal controls. Interestingly, in the infinite dimensional case, the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type. For infinite dimensional systems, the existence for time optimal controls with constraints of the ball type has already been discussed in the literature, while the study of the rectangular type has not been touched upon as far as we know. Key words. time optimal control, heat equations, control constraints of the rectangular type AMS subject classifications. 49J20, 93C20, 93B05 DOI. 10.1137/10081277X

1. Introduction. Let Ω be a bounded domain (with a smooth boundary ∂Ω) in Rd . Let ω be a nonempty open subset of Ω. Denote by χω the characteristic function of ω. Let {ai }+∞ i=1 be an element in the set   2  2 = {ci }+∞ l+ i=1 ∈ l : ci > 0, i = 1, 2, · · · . Define an unbounded linear operator A on L2 (Ω) by setting  D(A) = H 2 (Ω) ∩ H01 (Ω), (1.1) Ay = Δy for each y ∈ D(A). Let {λi }+∞ i=1 (with 0 < λ1 < λ2 ≤ · · · ) be the family of eigenvalues of −A. Write {ei }+∞ for such a family of the corresponding eigenfunctions that serves as an ori=1 thonormal basis of L2 (Ω). We define two sets as follows:   ∞   2 U = v ∈ L (Ω) : v = vi ei with |vi | ≤ ai (1.2) i=1

and (1.3)

   Uad = u(·) ∈ L∞ (0, +∞; L2 (Ω)) : u(t) ∈ U, for a.e. t ∈ (0, +∞) .

∗ Received by the editors October 25, 2010; accepted for publication (in revised form) February 24, 2011; published electronically DATE. This work was partially supported by National Basis Research Program of China (973 Program) under grant 2011CB808002 and the National Natural Science Foundation of China under grants 10871154, 10831007, 60821091, and 60974035. http://www.siam.org/journals/sicon/x-x/81277 † School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China ([email protected]). ‡ School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China (wanggs62@ yeah.net).

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¨ AND GENGSHENG WANG QI LU

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We call Uad a control constraint set of the rectangular type. When U is replaced by B(0, ρ) (the closed ball in L2 (Ω), centered at the origin and of radius ρ > 0) in (1.3), we call Uad a control constraint set of the ball type. The system studied in this paper is the following internally controlled heat equation: ⎧ ⎪ ⎪ yt − Δy = χω u in Ω × (0, +∞), ⎨ y=0 on ∂Ω × (0, +∞), (1.4) ⎪ ⎪ ⎩ in Ω, y(0) = y0 where the initial datum y0 is a nonzero function in L2 (Ω) and controls u(·) are taken from L∞ (0, +∞; L2 (Ω)). Throughout this paper, we will treat solutions of (1.4) as functions from [0, +∞) to L2 (Ω) and denote by y(·; u) the solution of (1.4) corresponding to the control u(·). The time optimal control problem studied in this paper is as follows: (P)

  Min t : y(t; u) = 0, u(·) ∈ Uad .

This problem depends on the controller χω , the initial datum y0 and the set Uad . When ω and y0 are fixed, it only depends on the set Uad . On the other hand, there is clearly a one-to-one correspondence between rectangles U (formed as (1.2)) and the +∞ elements {ai }+∞ i=1 . Therefore, the problem (P) only depends on the element {ai }i=1 in the case where ω and y0 are fixed. We will sometimes refer this problem as the +∞ problem (P) with the constraint {ai }+∞ i=1 , or simply the problem (P) with {ai }i=1 . For the problem (P), when    t∗ = inf t : u(·) ∈ Uad , y(t; u) = 0 is a positive number, we call it the optimal time; when t∗ < +∞, a control u∗ (·), which belongs to Uad and satisfies y(t∗ ; u∗ ) = 0, is called a time optimal control (or simply, an optimal control); a pair ( t, u (·)), holding properties u (·) ∈ Uad , t < +∞, and y(t; u ) = 0, is called an admissible pair. This paper aims to study the existence of optimal controls for the problem (P). The main results obtained in this paper can be roughly presented as follows: When {ai }+∞ i=1 satisfies a condition, the problem (P) has admissible pairs and optimal controls. First of all, we would like to stress the following fact: Even in the special case 2 where ω = Ω, the problem (P), with certain elements {ai }+∞ i=1 in l+ , can have no admissible pairs. (This will be seen from Example 2.1 in section 2.) Therefore, it is necessary to impose certain conditions on {ai }+∞ i=1 to ensure the existence of time optimal controls for the problem (P). In section 2, we will provide such a condition, namely Condition 2.1. Furthermore, we prove a proposition, namely Proposition 2.4. It shows that when {ai }+∞ i=1 belongs to the set +∞ 2 2 A = {{ai }+∞ i=1 ∈ l+ : ∃ a polynomial p with {1/p(i)}i=1 ∈ l+ , s.t. ai ≥ 1/p(i) for all i},

it satisfies Condition 2.1. Thus, the statement that {ai }+∞ i=1 ∈ A presents an easily verifiable condition to ensure the existence of optimal controls for the problem (P). In plain language, our results indicate that to ensure the existence of the admissible pairs, the convergence of ai (to zero) cannot be arbitrarily rapid. They illustrate that most of the controls can be imposed in the lower frequency part to gain the

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

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existence of the admissible pairs, while the high frequency part cannot be completely ignored. Our proof on the existence of admissible pairs to the problem (P) is based on an application of the Lebeau–Robbiano type estimate (which will be introduced in section 2), and a modification of the iteration method used in [11] and [13]. The existence on optimal controls to (P) is derived from the existence result on admissible pairs, but is not its trivial consequence. Due to the importance from both perspectives of mathematics and applied sciences, time optimal control problems have been discussed in the literature (see [2, 8, 10, 12, 15, 17, 19, 20, 21] and the references therein). Two of the main subjects of time optimal control problems are to study the existence of optimal controls and to research characteristics of an optimal control. In the field of time optimal control problems of linear ordinary differential equations, two kinds of control constraints are quite important. One is of the ball type, namely, the control constraint sets are as follows:

 1  = u(·) : (0, +∞) → RM measurable : u(t) ∈ B M (0, ρ) for a.e. t ∈ (0, +∞) . Uad Here M ∈ N, and B M (0, ρ) denotes the ball in RM , centered at the origin and of radius ρ. The other is of the rectangular type, namely, the control constraint sets are as follows:

 2  Uad = u(·) : (0, +∞) → RM measurable : u(t) ∈ AM for a.e. t ∈ (0, +∞) . Here, AM is a rectangle in RM . It is defined by AM = {(v1 , v2 , · · · , vM ) ∈ RM : |vi | ≤ ai , i = 1, · · · , M }, with ai > 0 for all i = 1, · · · , M. From the perspective of the existence for admissible pairs, the above-mentioned two types of control constraints do not make any difference in finite dimensional cases (namely, in the cases where the problems are governed by ordinary differential equations). The reason is that each rectangle contains a ball and the converse is also true in a finite dimensional space. Moreover, it has been proved (see, for instance, [1]) that when (A, B) is controllable, where A ∈ RN ×N and B ∈ RN ×M , and all eigenvalues of A belong to {λ ∈ C : Re (λ) ≤ 0}, the time optimal control problem of the following system has optimal controls: x (t) = Ax(t) + Bu(t),

x(0) = x0 ∈ RN ,

2 1 where u(·) ∈ Uad (or Uad ).

When people generalize the study on time optimal control problems of ordinary differential equations to the case of heat equations (or parabolic equations and evolution equations in Banach spaces), more attention has been paid to the situation with constraints of the ball type (see [2, 12, 15], and see [8, 17, 19] for more recent studies). It has already been shown that such problems, with constraints of the ball type for some linear evolution equations (in a Banach space), which are controllable, have optimal controls (see [17] and [2]). The case of internally controlled heat equations with constraints of the ball type is included. From this point of view, the existence results on time optimal control problems, with control constraints of the ball type, for linear controllable systems have been smoothly extended from finite dimensional cases to infinite dimensional situations. However, as we mentioned above, the problem (P),

¨ AND GENGSHENG WANG QI LU

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with certain elements {ai }+∞ i=1 , can have no admissible pairs. Hence, from the perspective of the existence, the time optimal control problems, with control constraints of the rectangular type, essentially differ from those with control constraints of the ball type, in infinite dimensional cases. The reason for such a difference can be explained as follows: In the infinite dimensional space L2 (Ω), any ball contains a rectangle U formed as (1.2); while such a rectangle U does not contains any ball. Therefore, it is interesting to study the existence of optimal controls for such problems with certain elements {ai }+∞ i=1 . The existence of time optimal control to the problem (P) is a problem of time optimal exact control. A similar approach based upon bounds for the coefficients of the series representation for L∞ -norm minimal exact optimal boundary control problems with the wave equation is presented in [5]. In this paper, the sequence of moment equations that characterizes the successful controls is relaxed to a sequence of moment inequalities. It turns out that this improves the numerical approximations of the optimal controls. Another paper we would like to mention is [4] where a numerical approach for computation of time optimal controls is given. The rest of this paper is organized as follows: section 2 presents a condition on the element {ai }+∞ i=1 and two examples. The first example shows that even in 2 the case where ω = Ω, the problem (P), with certain elements {ai }+∞ i=1 in l+ , can have no admissible pair. The second example, together with the remark following this example, explains that there are many such elements {ai }+∞ i=1 that satisfy the aforementioned condition. Section 3 provides the main results and their proof. Section 4 gives an conclusion of this paper. 2. Main results and conditions. First of all, we would like to point out that 2 the problem (P), with some {ai }+∞ i=1 ∈ l+ , has no admissible pairs. (This will be seen in an example given later in this section.) Thus, it is necessary to impose certain conditions on {ai }+∞ i=1 to ensure the existence of admissible pairs for this problem. Before giving such a condition, we introduce the following lemma, which is one of the bases to prove the existence of admissible pairs to the problem (P), with a certain class of {ai }+∞ i=1 . Lemma 2.1 (see [11, Theorem 3]). There exist two positive constants C1 and C2 such that 2    √   |αi |2 ≤ C1 eC2 r  αi ei (x) dx (2.1) λi ≤r

ω λ ≤r i

for each finite r > 0 and any choice of the coefficients {αi }λi ≤r with αi ∈ C. l Remark 2.1. In [11], the above lemma was established under the condition that ∂Ω is smooth. This condition has been relaxed to the case when ∂Ω ∈ C 2 in [14]. Now, we are going to provide the following condition on {ai }+∞ i=1 . It illustrates, in plain language, that ai has a lower bound depending on λi , C1 , and C2 . +∞ Notation. Let {mk }+∞ k=1 be a sequence of natural numbers, and let {Tk }k=1 be a sequence of positive numbers. Write

(2.2) θk ≡ θ(mk , Tk ) =

⎧ m1  2 √  λi ⎪ −λ1 T1 2 2C2 λm1 ⎪ e C e ⎪ 1 ⎪ λ T ⎨ e i 1 −1 i=1 mk  ⎪ √  ⎪ ⎪ −λ T λ 2 2C ⎪ ⎩ e mk−1 +1 k C1 e 2 mk i=1

λi eλi Tk − 1

if k = 1, 2

if k > 1,

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

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where C1 and C2 are the positive numbers given by Lemma 2.1, and write ⎧ 2e−λ1 T1 ⎪ −4λm1 +1 T1 ⎪ if k = 1, ⎪ ⎨ (λm +1 )2 θ1 + 2e 1 (2.3) ηk ≡ η(mk , Tk ) = ⎪ ⎪ 2e−λmk−1 +1 Tk ⎪ ⎩ θk ηk−1 + 2e−4λmk +1 Tk ηk−1 if k > 1. (λmk +1 )2 +∞ Condition 2.1 (on {ai }+∞ i=1 ). There exist a sequence {mk }k=1 (of natural num+∞ bers) and a sequence {Tk }k=1 (of positive numbers), with the properties that

(2.4)

mk → +∞ as k → +∞, +∞ 

(2.5)

Tk < +∞,

k=1 2 such that {ai }+∞ i=1 belongs to l+ and holds the property that  θ1 ≤ a2i for each i with 1 ≤ i ≤ m1 if k = 1, (2.6) θk ηk−1 ≤ a2i for each i with 1 ≤ i ≤ mk if k > 1.

Here, θk and ηk are given by (2.2) and (2.3), respectively. The main results of this paper are as follows. Theorem 2.2. Suppose that {ai }+∞ i=1 satisfies Condition 2.1. Then, the problem (P) has at least one admissible pair ( t, u (·)). Furthermore, u (·) can be taken as a piecewise D(A)-valued function switching values at most countably many times over (0, +∞), namely, there are at most countably many disjoint open intervals Ik , k = 1, 2, · · · , such  that for each k ∈ N, u(t) ≡ vk ∈ D(A) when t ∈ Ik , and such that [0, +∞) = k∈N Ik . Here Ik denotes to the closure of Ik . Theorem 2.3. Suppose that {ai }+∞ i=1 satisfies Condition 2.1. Then the problem (P) has at least one time optimal control. Condition 2.1 cannot be easily verified. In what follows, we will provide a proposition to show that when {ai }+∞ i=1 satisfies a certain easily verifiable condition, it holds Condition 2.1. Therefore, we provide an easily verifiable condition for {ai }+∞ i=1 to ensure the existence of admissible pairs for the problem (P). 2 , such that for each of its elements, Proposition 2.4. Let A be the subset of l+ +∞ +∞ 2 {ai }i=1 , there is a polynomial p with {1/p(i)}i=1 ∈ l+ , such that ai ≥ 1/p(i) for all +∞ i ∈ N. Then any element {ai }i=1 in A satisfies Condition 2.1. Proof. Let {ai }+∞ i=1 ∈ A. According to the definition of A, there is a polynomial 2 ∈ l+ , such that ai ≥ 1/p(i) for all i. Clearly, it suffices to find p, with {1/p(i)}+∞ i=1 +∞ two sequences {mk }k=1 and {Tk }+∞ k=1 holding properties (2.4) and (2.5), respectively, such that the element {ai }+∞ satisfies (2.6). The argument is as follows. i=1 Let β be the degree of the polynomial p. Then there is a constant K ∈ lN such that Kiβ ≥ p(i) for any i ∈ lN. Hence, it holds that (2.7)

ai ≥

1 Kiβ

for any i = 1, 2, 3, · · ·

We first construct a sequence {mk }+∞ k=1 in the following manner. Let C1 and C2 be the constants given in Lemma 2.1. By Weyl’s asymptotic formula (see [7, p. 45]

¨ AND GENGSHENG WANG QI LU

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for the case when ∂Ω is smooth and [16] for a more general boundary of Ω), we have a positive number C3 such that d

max{i : λi ≤ a, i = 1, 2, · · · } ≤ C3 a 2 for each a > 0.

(2.8)

Clearly, there is a natural number N such that 4

4

3

e−(n−1) C12 e2C2 n C3 n3d+4 ≤ 1,

4 2e−(n−1) 1 + 2e−4n ≤ 2 for all n ≥ N. n12 K (C3 )2 (n + 1)6dβ

(2.9) 1  Let τ be the smallest natural number such that τ ≥ max N, 2, λ16 . Then, for each k ∈ N, we define the following set:



Λk = {λi : λi ≤ (k + τ )6 }.

(2.10) Since

λ1 < (1 + τ )6 ≤ (k + τ )6 for all k ∈ N, the set Λk is not empty for each k ∈ N. Now we construct {mk }+∞ k=1 by setting 

mk = max{i : λi ∈ Λk }, k ∈ N.

(2.11) It is obvious that

λmk ≤ (k + τ )6 and λmk +1 > (k + τ )6 for each k ∈ N.

(2.12) We claim that

mk → +∞ as k → +∞.

(2.13)

By seeking a contradiction, we suppose that (2.13) did not hold. Then we would have +∞ a subsequence {mkj }+∞ j=1 of {mk }k=1 and a positive number C independent of j such that 

mkj = max{i : i ∈ Λkj } ≤ C for each j ∈ N.

(2.14)

By Weyl’s asymptotic formula, we also have that λi = O(i2/d ) for each i ∈ N. Thus, it holds that (2.15)

 k = Λ {λi : k 1/d ≤ λi ≤ (k + τ )6 } = ∅, for each k sufficiently large.

Since it is clear that kj ⊂ Λkj , Λ we necessarily have that kj } ≤ mkj for each j sufficiently large. max{i : i ∈ Λ

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This, together with (2.15), yields that (kj )1/d ≤ mkj for each j sufficiently large. This contradicts to (2.14). Therefore, (2.13) has been proved. Then, we are going to construct a sequence {Tk }+∞ k=1 . To serve such a purpose, we first observe from (2.11), (2.10), and (2.8) that (2.16)

mk = max{i : λi ≤ (k + τ )6 } ≤ C3 (k + τ )3d .

Since it holds for any T > 0 and for each i ∈ N that eλi T − 1 ≥ λi T, namely,

1 λi ≤ , eλi T − 1 T

we deduce from (2.16), where k = 1, that e−λ1 T C12 e2C2

√

λm1

m1   i=1

(2.17)

2 √ λi −λ1 T 2 2C2 λm1 m1 ≤ e C e 1 eλi T − 1 T2 e−λ1 T 2 2C2 √λm1 ≤ C1 e C3 (1 + τ )3d . T2

√ On one hand, because C12 e2C2 λm1 C3 (1 + τ )3d is a positive constant, there is clearly a positive number σ1 such that for any T ≥ σ1 , it holds that (2.18)

e−λ1 T 2 2C2 √λm1 1 C1 e C3 (1 + τ )3d ≤ 2 . T2 K (C3 )2 (1 + τ )6dβ

On the other hand, there exists obviously a positive number σ2 > 0 such that for any T ≥ σ2 , we have that (2.19)

6 1 2e−λ1 T + 2e−4(1+τ ) T ≤ . 6dβ 2 2 (1 + τ )12 K (C3 ) (2 + τ )

Let σ = max{σ1 , σ2 }. Then, by (2.17), (2.18), and (2.19), we can conclude that (2.20)

e−λ1 σ C12 e2C2

√

λm1

m1   i=1

λi 2 1 ≤ 2 λ σ 2 i e −1 K (C3 ) (1 + τ )6dβ

and (2.21)

6 2e−λ1 σ 1 + 2e−4(1+τ ) σ ≤ . 6dβ 2 2 (1 + τ )12 K (C3 ) (2 + τ )

Now we construct {Tk }+∞ k=1 by setting ⎧ ⎪ ⎨ σ  Tk = (2.22) 1 ⎪ ⎩ (k − 1)2

if k = 1, if k = 2, 3, · · · .

¨ AND GENGSHENG WANG QI LU

8 It is clear that

+∞ 

(2.23)

Tk < +∞.

k=1 +∞ Finally, we will prove that the element {ai }+∞ i=1 satisfies (2.6), where {θk }k=1 and +∞ +∞ +∞ {ηk }k=1 are, respectively, defined by (2.2) and (2.3), with {mk }k=1 and {Tk }k=1 being given by (2.11) and (2.22), respectively. The proof will be carried out in three steps as follows. Step 1. The estimate on θk . In the case where k = 1, we derive from (2.22), (2.2), and (2.20) that

θ1 ≤

(2.24)

1 K 2 (C3 )2 (1

+ τ )6dβ

.

In the case where k ≥ 2, by (2.2), (2.22), and (2.12) we obtain that θk = e ≤

(2.25)

−λmk−1 +1 Tk

C12 e2C2

√

λmk

mk  

λi

2

eλi Tk − 1 i=1 mk  4 3  λi e−K(k−1+τ ) C12 e2C2 K(k+τ ) λ T i e k− i=1

2 1

.

Since eλi Tk − 1 ≥ λi Tk , namely,

1 λi ≤ , eλi Tk − 1 Tk

it follows from (2.16) and (2.22) that mk   i=1

2

λi eλi Tk −1

≤

mk   1 2 ≤ mk (k − 1)4 ≤ C3 (k + τ )3d (k − 1)4 < C3 (k + τ )3d+4 . T k i=1

Along with (2.25) and the first estimate in (2.9) where n = k + τ , this indicates that (2.26)

4

3

θk ≤ e−(k−1+τ ) C12 e2C2 (k+τ ) C3 (k + τ )3d+4 ≤ 1 for k ≥ 2.

Step 2. The estimate about ηk . We are going to show, through utilizing the mathematical induction, the following estimate: (2.27)

ηk ≤

1 for all k ∈ N. K 2 (C3 )2 (k + 1 + τ )6dβ

Here is the argument: When k = 1, it follows from (2.3) and (2.22) that (2.28)

η1 =

2e−λ1 σ θ1 + 2e−4λm1 +1 σ . (λm1 +1 )2

By the second inequality of (2.12) and (2.24), we obtain that (2.29)

2e−λ1 σ 2e−λ1 σ θ ≤ . 1 (λm1 +1 )2 (1 + τ )12

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By making use of the second inequality of (2.12) again, we deduce that 6

2e−4λm1 +1 σ ≤ 2e−4(1+τ ) σ . This, together with (2.28), (2.29), and (2.21), indicates that η1 ≤

1 . K 2 (C3 )2 (2 + τ )6dβ

We next show that (2.27) stands when k = n + 1, provided that it holds for the case where k = n, namely, it holds that ηn ≤

(2.30)

1 . K 2 (C3 )2 (n + 1 + τ )6dβ

To serve such a purpose, we first recall that ηn+1 =

(2.31)

2e−λmn +1 Tn+1 θn+1 ηn + 2e−4λmn+1 +1 Tn+1 ηn . λ2mn+1 +1

By (2.22), (2.26), (2.30) and the second inequality of (2.12), we find that (n+τ )6

4

2e−λmn +1 Tn+1 2e− n2 2e−(n+τ ) θ η ≤ ≤ n+1 n (λmn+1 +1 )2 (n + 1 + τ )12 (n + 1 + τ )12

(2.32) and

2e−4λmn+1 +1 Tn+1 ηn ≤ 2e−

(2.33)

4(n+1+τ )6 n2

4

≤ 2e−4(n+1+τ ) .

It follows from (2.31), (2.32), (2.33) and the second inequality of (2.9) that 4

ηn+1

4 2e−(n+τ ) 1 ≤ + 2e−4(n+1+τ ) ≤ . 2 2 (n + 1 + τ )12 K (C3 ) (n + 2 + τ )6dβ

Hence, we have proved (2.27). Furthermore, it follows at once from (2.26) and (2.27) that (2.34)

θk ηk−1 ≤

1 K 2 (C3 )2 (k

+ 1 + τ)

6dβ

for all k ≥ 2.

Step 3. The conclusion. By (2.7), (2.16), (2.24), and (2.34), we find that θk ηk−1 ≤

1 6dβ

≤

1 K 2 m2β k

≤

1 K 2 i2β

 2 ≤ ai

K 2 (C3 )2 (k + 1 + τ ) for all k ≥ 2 and all i with 1 ≤ i ≤ mk ,

and θ1 ≤

1 K 2 (C3 )2 (1

+ 1 + τ)

6dβ

≤

1 K 2 m2β 1

≤

1 ≤ (ai )2 for all 1 ≤ i ≤ m1 . K 2 i2β

+∞ +∞ Hence, the element {ai }+∞ i=1 satisfies (2.6) where {mk }k=1 and {Tk }k=1 are provided by (2.11) and (2.22), respectively. Along with (2.13) and (2.23), this indicates that the element {ai }+∞ i=1 satisfies Condition 2.1. This completes the proof.

¨ AND GENGSHENG WANG QI LU

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Now, we are going to present the following example. It shows that even in the case where ω = Ω, the problem (P), with some {ai }+∞ i=1 , has no admissible pairs. +∞ Example 2.1. Let ω = Ω and y0 = i=1 21i ei . Set λi i 2 eiλi

ai =

, i = 1, 2, · · · .

2 Clearly, the element { ai }+∞ ai }+∞ i=1 belongs to l+ . However, the problem (P), with { i=1 , has no any admissible pair. We next explain why the problem (P) in the above example does not have any admissible pair. By seeking a contradiction, we suppose that it had an admissible pair ( t, u (·)). Then we would have

y( t; u ) = 0.

(2.35) Write u (t) =

+∞  

u (t), ei



e L2 (Ω) i

i=1



=

+∞ 

ui (t)ei .

i=1

    Here and throughout this paper, ·, · L2 (Ω) and  · L2 (Ω) stand for the inner product (·) ∈ Uad , it holds that and the norm of L2 (Ω), respectively. Since u (2.36)

ai for a.e. t ∈ (0, +∞), and for all i ∈ N. |ui (t)| ≤

Because ω = Ω and y0 =

+∞  1 e , we derive from (2.35) that i i 2 i=1  0 = y( t; u ) =

+∞ 

yi ( t)ei ,

i=1

where  t) = e−λi t /2i + yi (



 t 0



e−λi (t−t) ui (t)dt for all i ∈ N.

Therefore, it holds that e

−λi  t

 i

/2 =

 t

0



e−λi (t−t) (−ui (t))dt for all i ∈ N.

This, together with (2.36), indicates that 

ai e−λi t /2i ≤

1 ai  (1 − e−λi t ) ≤ for all i ∈ N. λi λi

Hence, it stands that (2.37)

λi  λi = ai ≥ for all i ∈ N, 2i eiλi 2i eλi t

which clearly leads to a contradiction since t < +∞. Thus, the problem (P) in the above example has no admissible pairs.

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

11

Remark 2.2. (i) The inequality (2.37) implies that when the problem (P), with {ai }+∞ i=1 , has admissible pairs, the convergence of ai (to zero) cannot be arbitrarily rapid. It accurately shows that the element {ai }+∞ i=1 should be bounded from below 2 by a given element {bi }+∞ ∈ l in the sense that ai ≥ bi for each i ∈ N. The element + i=1 2 {bi }+∞ ∈ l may depend on the operator A, the initial datum y0 , the subset ω, and + i=1 so on. When ω is a proper subset of Ω, it is reasonable for us to put certain stronger conditions on {ai }+∞ i=1 to ensure the existence of admissible pairs for the problem (P). (ii) The statement that an element {ai }+∞ i=1 satisfies Condition 2.1 means that 2 belongs to l , and there are two sequences {mk }+∞ {ai }+∞ + i=1 k=1 (of natural numbers) +∞ and {Tk }k=1 (of positive numbers), with properties (2.4) and (2.5), respectively, such that (2.6) stands. Thus it follows that when {ai }+∞ i=1 satisfies Condition 2.1, the +∞ and {η } have the following property: corresponding {θk }+∞ k k=1 k=1 (θ1 )2 +

+∞ 

(θk ηk−1 )2 < +∞.

k=2 +∞ However, this never means that for all sequences {mk }+∞ k=1 and {Tk }k=1 with (2.4) +∞ +∞ and (2.5), respectively, the sequences {θk }k=1 and {ηk }k=1 defined by (2.2) and (2.3), respectively, must hold the above property. Indeed, we can construct {mk }+∞ k=1 and satisfying (2.4) and (2.5), respectively, such that {Tk }+∞ k=1

θk ηk−1 → +∞ as k → +∞. Such sequences are structured as follows. Since λi → +∞ as i → +∞, we can take 2 {mk }+∞ k=1 such that λmk ≥ (8k/C2 ) for each k ∈ N. Then we set Tk = 1/λmk +1 . +∞ Clearly, such sequences {mk }k=1 and {Tk }+∞ k=1 satisfy (2.4) and (2.5), respectively. In this case, it holds that θk = e−λmk−1 +1 Tk C12 e2C2

√

λmk

mk   i=1

λi eλi Tk −

2 1

for k ≥ 2

and ηk =

2e−λmk−1 +1 Tk θk ηk−1 + 2e−4λmk +1 Tk ηk−1 for k ≥ 2. (λmk +1 )2

Through direct computations, one can easily check that θk ηk−1 → +∞ as k → +∞. (iii) The inequality (2.6) in Condition 2.1 is indeed analogous to the inequality (2.37) (though it looks much more complicated), since the left-hand term of (2.6) precisely gives a lower bound for ai . The complicated expression of the left-hand term in the inequality (2.6) is due to the fact that the control region ω is an open (nonempty) subset arbitrarily given in Ω. In addition, we would like to emphasize that the left-hand term in the condition (2.6) is independent of y0 . 3. The proof of main results. Throughout this section, C1 and C2 stand for the positive constants given by Lemma 2.1, | · |Rm denotes the Euclid norm in Rm , and {eAt }t≥0 stands for the C0 -semigroup generated by A. We begin with studying the following two lemmas.

¨ AND GENGSHENG WANG QI LU

12

Lemma 3.1. Let Bm be the matrix

 ω

 ei ej dx

1≤i,j≤m m

, where m ∈ N. Then

−1 2 positive definite. Furthermore, for any γ ∈ R , it holds that |Bm γ|Rm ≤ Bm is √ 2 2C2 λm 2 C1 e |γ|Rm . Proof. According to Lemma 2.1, for each β = (β1 , β2 , · · · , βm )T ∈ Rm , it holds that 2    m m  √ √    |β|2Rm = βi2 ≤ C1 eC2 λm βi ei  dx = C1 eC2 λm β T Bm β.   ω i=1

i=1

This shows that Bm is a positive definite matrix and √  |β|2Rm ≤ C1 eC2 λm | Bm β|2Rm for all β = (β1 , β2 , · · · , βm )T ∈ Rm . −1 Let β = Bm γ. Then, we can easily derive from the above-mentioned estimate that √ √ √  −1 2 −1 2 −1 2 |Bm γ|Rm ≤ C1 eC2 λm | Bm Bm γ|Rm ≤ C12 e2C2 λm |Bm Bm γ|Rm = C12 e2C2 λm |γ|2Rm .

Thus, we complete the proof. Let t1 and t2 be such that 0 ≤ t1 < t2 < +∞. Consider the following controlled system of ordinary differential equations:  zt = Am z + Bm f in [t1 , t2 ], (3.1) z(t1 ) = z0 . Here, Bm is the matrix given in Lemma 3.1, f (·) is a control taken from L∞ (t1 , t2 ; Rm ), z0 ∈ Rm , and Am is the following matrix: ⎛ ⎞ −λ1 0 ··· 0 ⎜ 0 −λ2 · · · 0 ⎟ ⎜ ⎟ ⎜ .. .. ⎟ . .. .. ⎝ . . . ⎠ . 0

···

0

−λm

We write z(·; f ) for the solution of (3.1) corresponding to the control f (·). Lemma 3.2. Let m ∈ N. Then for each z0 ∈ Rm , the control f (·) defined by  t2 −1 −1 f (t) ≡ −Bm e−Am (s−t1 ) ds z0 for all t ∈ (t1 , t2 ), t1

drives the solution z(·; f ) from z0 at time t1 to the origin at time t2 . Namely, z(t2 ; f ) = 0. Furthermore, this control satisfies the estimate that  2 −1   t2 √   2 2 2C λ −A (s−t ) 2 m m 1 e ds z0  . |f |L∞ (t1 ,t2 ;Rm ) ≤ C1 e   t1  m R

Proof. On one hand, we can easily check that  t2 Am (t2 −t1 ) z(t2 ; f ) = e z0 + eAm (t2 −s) Bm f (s)ds t1 !   t2 Am (t2 −t1 ) Am (t2 −s) −1 z0 + e Bm −Bm =e t1

=e

Am (t2 −t1 )

z0 − e

Am (t2 −t1 )



t2

e t1

=0

Am (t1 −s)

t2

e

−Am (s−t1 )

−1

"

ds

z0 ds

t1



t2

ds t1

e−Am (s−t1 ) ds

−1

z0

13

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

On the other hand, we can apply Lemma 3.1 to get that  $−1 2 #  T   −1 |f |2L∞ (t1 ,t2 ;Rm ) = −Bm e−Am (s−t1 ) ds z0  0  m  R  2 −1   t2 √   ≤ C12 e2C2 λm  e−Am (s−t1 ) ds z0  .  t1  m R

Thus, we complete the proof. Before proceeding with the proof of Theorem 2.2, we briefly introduce our main +∞ strategy to construct an admissible pair to the problem (P). Let {mk }+∞ k=1 and {Tk }k=1 be the sequences given by Condition 2.1. For each k ∈ N, write Pmk for the orthogonal projection from L2 (Ω) to Span{e1 , · · · , emk }. We construct the following sequences of time intervals: ⎧ (0, T1 ) if k = 1, ⎪ ⎪ ⎪ ⎨ ⎛ ⎞ k−1 k−1   Ik = (3.2) ⎪ ⎝ 2 T , 2 Tj + Tk ⎠ if k > 1, ⎪ j ⎪ ⎩ j=1

and

(3.3)

j=1

⎧ (T1 , 2T1 ) ⎪ ⎪ ⎪ ⎨ ⎛ ⎞ k−1 k   Jk = ⎪ ⎝ Tj + Tk , 2 Tj ⎠ ⎪ ⎪ 2 ⎩ j=1

if k = 1, if k > 1.

j=1

On each interval Ik , we let the heat equation freely evolve. On each interval Jk , we control the heat equation with a control restricted over ω. By Lemma 3.2, we can take a control f (k) from the finite dimensional space Span{e1 , e2 , · · · , emk } such that the corresponding solution y (k) (·) to the equation on Jk holds the property k Pmk (y (k) (2 j=1 Tj )) = 0. We start by having the initial value for the equation on I1 to be y0 . For the initial value on Ik , k = 2, 3, · · · , we define it to be the ending value of the solution to the equation on Jk−1 . The initial value of the equation on Jk , k = 1, 2, · · · , is given by the ending value of the solution for the equation on Ik . Notice that for each k ∈ N, the control f (k) is independent of time t. On one hand, Lemma 3.2 provides an estimate for the control f (k) . On the other hand, we can have a L2 (Ω)-norm estimate for the ending value of the solution to the equation on Ik . These two estimates, together with Condition 2.1, yields that f (k) belongs to Uad . Finally, we prove that ( t, u (·)) is an admissible pair. Here,   0 if t ∈ (0, +∞) \ +∞ k=1 Jk , u (t) = (3.4) (k) f if t ∈ Jk , with k ∈ N and (3.5)

 t=2

+∞  j=1

Tj .

¨ AND GENGSHENG WANG QI LU

14

Proof of Theorem 2.2. Without loss of generality, we can assume that |y0 |L2 (Ω) ≤ 1. Otherwise, we let the heat equation freely evolve on the time interval (0, T ), where  T is large enough such that |eAT y0 |L2 (Ω) ≤ 1, and then, we consider the controlled  heat equation with the initial datum eAT y0 , on the time interval (T , +∞). +∞ +∞ Let {mk }k=1 and {Tk }k=1 be the sequences given by Condition 2.1. Write {Ik }+∞ k=1 and {Jk }+∞ k=1 for the sequences of time intervals defined by (3.2) and (3.3), respectively. Let z (1) (·) be the solution to the freely evolved heat equation: ⎧ (1) z − Δz (1) = 0 in Ω × I1 , ⎪ ⎪ ⎨ t z (1) = 0 on ∂Ω × I1 , ⎪ ⎪ ⎩ (1) in Ω. z (0) = y0 Due to the energy decay property of the solution to the above equation, we have that  (1)    z (T1 ) 2 (3.6) ≤ e−λ1 T1 y0 L2 (Ω) . L (Ω) Let Pm1 (z

(3.7)

(1)

(T1 )) =

m1 

(1)

zi ei ,

i=1 (1)

where the Fourier coefficients zi , i = 1, · · · , m1 , are real numbers. Set  −1 (1) (1) (1) T −1 −Am1 (s−T1 ) (1) T (3.8) ) = −B e ds (z1 , · · · , zm ) . (f1 , · · · , fm m1 1 1 J1

Define a control f

(1)

∈ D(A) by setting f (1) =

(3.9)

m1 

(1)

fi ei .

i=1

Let y (1) (·) be the solution to the controlled ⎧ (1) y − Δy (1) = χω f (1) ⎪ ⎪ ⎨ t (3.10) y (1) = 0 ⎪ ⎪ ⎩ (1) y (T1 ) = z (1) (T1 )

heat equation: in Ω × J1 , on ∂Ω × J1 , in Ω.

Write y (1) (t) =

∞ 

(1)

yj (t)ej for all t ∈ J1 ,

j=1 (1)

where yj (·), j = 1, 2, · · · , are scalar functions over J1 . Then, by (3.9) and (3.10), (1)

(1)

we find that (y1 (·), · · · , ym1 (·)) is the solution to the following system of ordinary differential equations: ⎧  m1  ⎪ (1) ⎪ ⎨ (yj(1) )t + λj yj(1) = fi ei ej over J1 for all j = 1, 2, · · · , m1 , ⎪ ⎪ ⎩

i=1 (1)

(1)

ω

yj (T1 ) = zj (T1 ) for all j = 1, 2, · · · , m1 .

15

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

Taking into account (3.8), we can apply Lemma 3.2 to the above system to get that   (1) (1) y1 (2T1 ), · · · , ym (2T ) = 0, 1 1 which leads to Pm1 (y (1) (2T1 )) =

(3.11)

m1 

(1)

yi (2T1 )ei = 0.

i=1

Moreover, it holds that m1 

(1) (fi )2

≤

C12 e2C2

√

λm1

i=1

= C12 e2C2 ≤ C12 e2C2

√

λm1

√

λm1

   e−Am1 (s−T1 ) ds   J1 m 2 1   λi  (1)  z    eλi T1 − 1 i  m i=1 m1   i=1

R

λi eλi T1 − 1

−1

(1) (z1 , · · ·

2 

(1) T  , zm )  1



R m1

1

m1 2   (1) 2 . zi i=1

Along with (3.7), (3.6) and (2.2), this indicates that m1 

(1)

√

m1  

√

i=1 m1  

(fi )2 ≤ C12 e2C2

i=1

≤ C12 e2C2

λm1

λm1

λi eλi T1 −

1

2   z(T1 )2 2

L (Ω)

2

λi eλi T1 − 1

i=1

 2 = e−λ1 T1 θ1 y0  2

 2 e−2λ1 T1 y0 L2 (Ω)

L (Ω)

≤e

(3.12)

−λ1 T1

θ1 .

Besides, it follows from (3.11) that

(3.13)

|y (1) (2T1 )|2L2 (Ω)  # $2  m1      (1) 2AT1 A(2T1 −s) = (I − Pm1 ) e y0 + e χω fi ei ds    2 J1 i=1 L (Ω)  2   2AT1  ≤ 2 I − Pm1 e y0  L2 (Ω) m 2  2  1        (1) A(2T −s) 1 + 2  I − Pm1 e ds fi ei     2 J1 L(L2 (Ω);L2 (Ω)) i=1

.

L (Ω)

  Here and in what follows,  · L(L2 (Ω);L2 (Ω)) denotes the operator norm on L2 (Ω). Write 2      Π1 = 2 I −Pm1 e2AT1 y0  2

L (Ω)

and

m 2 1    (1)  fi ei     2 L(L2 (Ω);L2 (Ω))

 2    A(2T1 −s)   e ds Π2 = 2  I −Pm1 

J1

i=1

L (Ω)

.

¨ AND GENGSHENG WANG QI LU

16

  Since y0 L2 (Ω) ≤ 1, it holds that Π1 ≤ 2e−4λm1 +1 T1 .

(3.14) Because

 2     A(2T1 −s)   I −Pm1 e ds ≤ e−λm1 +1 (2T1 −s) ds  2 2 J1 J 1 L(L (Ω);L (Ω))   1 1 −λm1 +1 T1 1−e ≤ = , (λm1 +1 )2 (λm1 +1 )2

2

it follows from (3.12) that Π2 ≤

(3.15)

2e−λ1 T1 θ1 . (λm1 +1 )2

By (3.13), (3.14), (3.15), and (2.3), we deduce that (3.16)

 (1)  y (2T1 )2 2

L (Ω)

≤

2e−λ1 T1 θ1 + 2e−4λm1 +1 T1 = η1 . (λm1 +1 )2

Next, we let z (2) (·) be the solution to the freely evolved equation ⎧ (2) z − Δz (2) = 0 in Ω × I2 , ⎪ ⎪ ⎨ t ⎪ ⎪ ⎩

z (2) = 0

z

(2)

on ∂Ω × I2 ,

(2T1 ) = y

(1)

(2T1 )

in Ω.

By (3.11), we have that  (2)    z (2T1 + T2 ) 2 (3.17) ≤ e−λm1 +1 T2 y (1) (2T1 )L2 (Ω) . L (Ω) Write (3.18)

Pm2 (z

(2)

(2T1 + T2 )) =

m2 

(2)

zi ei ,

i=1 (2)

where the coefficients zi , i = 1, · · · , m2 , are real numbers. Set  −1 (2) (2) (2) T −1 −Am2 (s−2T1 −T2 ) (2) T (3.19) (f1 , · · · , fm ) = −B e ds (z1 , · · · , zm ) . m2 2 2 J2

Define a control f (3.20)

(2)

∈ D(A) by setting f (2) =

m2 

(2)

fi ei .

i=1

Let y (2) (·) be the solution to the controlled equation ⎧ (2) y − Δy (2) = χω f (2) in Ω × J2 , ⎪ ⎪ ⎨ t (3.21) on ∂Ω × J2 , y (2) = 0 ⎪ ⎪ ⎩ (2) (2) y (2T1 + T2 ) = z (2T1 + T2 ) in Ω.

17

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

Write y (2) (t) =

∞ 

(2)

yj (t)ej for all t ∈ J2 ,

j=1 (2)

where yj (·), j = 1, 2, · · · , are scalar functions over J2 . Then, by (3.20) and (3.21), (2)

(2)

we find that (y1 (·), · · · , ym2 (·)) is the solution to the following system of ordinary differential equations: ⎧  m2  ⎪ (2) ⎪ ⎨ (yj(2) )t + λj yj(2) = fi ei ej over J2 for all j = 1, 2, · · · , m2 , ⎪ ⎪ ⎩

i=1 (2)

ω

(2)

yj (2T1 + T2 ) = zj (2T1 + T2 ) for all j = 1, 2, · · · , m2 .

Taking into account (3.19), we can apply Lemma 3.2 to the above system to get that Pm2 (y (2) (2T1 + 2T2 )) =

(3.22)

m2 

(2)

yi (2T1 + 2T2 )ei = 0.

i=1

Moreover, it stands that m2 

(2) (fi )2

≤

C12 e2C2

√

λm2

i=1

= C12 e2C2 ≤ C12 e2C2

√

λm2

√

λm2

   e−Am2 (s−2T1 −T2 ) ds   J2 m 2 2   λi  (2)  z    eλi T2 − 1 i  m i=1 m2   i=1

R

λi λ T i e 2−

1

−1

(2) (z1 , · · ·

2 

(2) T  , zm )  2



R m2

2

m2 2   (2) 2 zi . i=1

This, together with (3.18), (3.17), (3.16), and (2.2), indicates that m2 

(2)

(fi )2 ≤ e−2λm1 +1 T2 C12 e2C2

i=1

(3.23)

√

λm2

m2   i=1

λi eλi T2 − 1

2

η1

= e−λm1 +1 T2 θ2 η1 .

Besides, because of (3.22), we see that  (2)  y (2T1 + 2T2 )2 2 L (Ω)  $2 #  m2       (2) AT2 (2) A(2T1 +2T2 −s) =  I − Pm2 y (2T1 + T2 ) + e χω fi ei ds  e   2 J2 i=1 L (Ω)  2    ≤ 2 I − Pm2 e2AT2 y (1) (2T1 ) 2 L (Ω) m 2  2  2        (2) (3.24) +2 I − Pm2 eA(2T1 +2T2 −s) ds 2 f e .   i i 2 (Ω))   2 L(L (Ω);L J2 i=1

L (Ω)

¨ AND GENGSHENG WANG QI LU

18 Write

 2     Γ1 = 2 I − Pm2 e2AT2 y (1) (2T1 )

L2 (Ω)

and     Γ2 = 2  I − Pm2

J2

m 2 2    (2)  fi ei     2 L(L2 (Ω);L2 (Ω))

2  A(2T1 +2T2 −s)  e ds

i=1

.

L (Ω)

By (3.16), we find that Γ1 ≤ 2e−4λm2 +1 T2 η1 .

(3.25) Since     I − Pm2 

J2

2  eA(2T1 +2T2 −s) ds

 ≤

L(L2 (Ω);L2 (Ω))

e−λm2 +1 (2T1 +2T2 −s) ds

2

J2

≤

1 , (λm2 +1 )2

it follows from (3.23) that Γ2 ≤

(3.26)

2e−λm1 +1 θ2 η1 . λ2m2 +1

By (3.24), (3.25), (3.26), and (2.3), we obtain that (3.27)

  (2) y (2T1 + 2T2 )2 2

L (Ω)

≤ 2e−4λm2 +1 T2 η1 +

2e−λm1 +1 T2 θ2 η1 = η2 . λ2m2 +1

Now, we are ready to prove, by the induction, Statement A: For each k ≥ 2, mk (k) (k) there exist functions z (k) (·), y (k) (·) and a control f (k) = i=1 fi ei , where fi , i = 1, · · · , mk , are real numbers, such that ⎧ (k) z − Δz (k) = 0 ⎪ in Ω × Ik , ⎪ ⎪ t ⎪ ⎪ ⎪ ⎨ z (k) = 0 on ∂Ω × Ik , ⎛ ⎞ ⎛ ⎞ (3.28) ⎪ k−1 k−1 ⎪  ⎪ (k) ⎝  ⎠ (k−1) ⎝ ⎪ ⎪ 2 = y 2 T Tj ⎠ in Ω, z j ⎪ ⎩ j=1

(3.29)

j=1

⎧ (k) yt − Δy (k) = χω f (k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y (k) = 0 ⎛ ⎞ ⎛ ⎞ ⎪ k−1 k−1 ⎪   ⎪ (k) ⎝ ⎪ ⎪ 2 Tj + Tk ⎠ = z (k) ⎝2 Tj + Tk ⎠ ⎪ ⎩ y j=1

j=1

in Ω × Jk , on ∂Ω × Jk , in Ω,

19

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

and such that ⎛

⎛

Pmk ⎝y (k) ⎝2

(3.30)

k 

⎞⎞ Tj ⎠⎠ = 0,

j=1

 ⎛ ⎞   k    (k)  y ⎝2 ⎠ T j     j=1

(3.31)

≤ ηk , L2 (Ω)

and mk  

(3.32)

(k)

fi

2

≤ e−λmk−1 +1 Tk θk ηk−1 .

i=1

In fact, we have proved that Statement A stands when k = 2. We next prove that it holds when k = n + 1, provided that it is true for k = n. Here is the argument: Since there are y (n) (·) and f (n) which satisfy (3.29), (3.30), and (3.31), where k = n, (3.28), where k = n + 1, has a unique solution z (n+1) (·) satisfying that  ⎛ ⎞   n   (n+1)  ⎝2 (3.33) z Tj + Tn+1 ⎠   j=1

L2 (Ω)

 ⎛ ⎞   n    ≤ e−λmn +1 Tn+1 y n ⎝2 Tj ⎠   j=1

≤ ηn .

L2 (Ω)

Write ⎛

⎛

Pmn+1 ⎝z (n+1) ⎝2

(3.34)

n 

⎞⎞ Tj + Tn+1 ⎠⎠ =

j=1 (n+1)

where the coefficients zi 

(n+1)

f1 (3.35)

=

e

(n+1)

zi

ei ,

i=1

, i = 1, · · · , mn+1 , are real numbers. Set

(n+1) , · · · , fm n+1 #

−1 −Bm n+1



mn+1

T

−An+1 (s−2

$−1

n j=1

Tj −Tn+1 )

ds

Jn+1

 T (n+1) (n+1) z1 , · · · , zm . n+1

Define 

mn+1

(3.36)

f (n+1) =

(n+1)

fi

ei .

i=1

Let y (n+1) (·) be the solution to (3.29), where k = n + 1 and f (k) is replaced by f (n+1) , which is given by (3.36). Write y (n+1) (t) =

∞  j=1

(n+1)

yj

(t)ej for all t ∈ Jn+1 ,

¨ AND GENGSHENG WANG QI LU

20 (n+1)

where yj (·), j = 1, 2, · · · , are scalar functions. Then, from (3.36) and (3.29) (where k = n + 1 and f (k) is replaced by f (n+1) , which is given by (3.36)), we observe that (n+1) (n+1) (y1 (·), · · · , ymn+1 (·)) solves the following system of ordinary differential equations: ⎧  mn+1   ⎪ ⎪ y (n+1) + λ y (n+1) =  f (n+1) ⎪ ei ej over Jn+1 for all j = 1, · · · , mn+1 , ⎪ j j j i ⎪ ⎨ t ω i=1 n n       ⎪ ⎪ (n+1) (n+1) ⎪ ⎪ 2 = z 2 for all j = 1, 2, · · · , mn+1 . T + T T + T y j n+1 j n+1 ⎪ j j ⎩ j=1

j=1

Taking into account (3.35), we can apply Lemma 3.2 to the above system to derive that ⎛ ⎛ ⎞⎞ ⎛ ⎞ mn+1 n+1 n+1   (n+1)  ⎝2 Pmn+1 ⎝y (n+1) ⎝2 (3.37) T j ⎠⎠ = yi Tj ⎠ ej = 0. j=1

i=1

j=1

Namely, (3.30) holds when k = n + 1. Moreover, it stands that mn+1 



(n+1)

fi

2

i=1

≤ C12 e2C2 = C12 e2C2 ≤ C12 e2C2

√

λmn+1

√

λmn+1

√

λmn+1

# 2 $−1    n   (n+1) −Amn+1 (s−2 j=1 Tj −Tn+1 ) (n+1) T   e ds (z , · · · , z ) mn+1 1    Jn+1  mn+1 R m 2 n+1   λi  (n+1)  z   λ T i n+1   m e −1 i i=1 mn+1 

 i=1

R

λi eλi Tn+1 − 1

n+1 2 m

n+1

(n+1) 2

(zi

) .

i=1

Along with (3.34), (3.33), and (2.2), indicates that (3.32) holds for k = n + 1. In addition, by (3.37), (3.31) with k = n, and (2.3), we can use exactly the same argument in the proof of (3.27) to show that (3.31) holds when k = n + 1. Thus we have proved Statement A. Finally, we are going to construct an admissible pair for the problem (P). To serve  k (k) (k) such a purpose, we let, for each k ∈ N, f (k) = m i=1 fi ei (where fi , i = 1, · · · , mk , are real numbers), y (k) (·), and z (k) (·), satisfy Statement A. We define t and u (·) by mk (k) (3.5) and (3.4) (where f (k) = i=1 fi ei ), respectively. Clearly, t is finite (by (2.5)) and u (·) is a piecewise D(A)-valued function switching values at most countably many times over (0, +∞). We next claim that ( t, u (·)) is an admissible pair to the problem (P). We first show that u (·) belongs to Uad . In fact, because {ai }+∞ i=1 satisfies Condition 2.1, it follows immediately from (2.6) and (3.32) that for each k ∈ N, (3.38)

(k)

|fi | ≤ ai for all i = 1, 2, · · · , mk .

On the other hand, by (3.4) we see that for each t ∈ (0, +∞), u (t) is either 0 or f (k) mk (k) (k) for some k ∈ N. In the second case, since f = i=1 fi ei , it follows from (3.38)

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

21

that     u (t), ei L2 (Ω)  =

  (k)   f  ≤ ai

when i = 1, · · · , mk ,

i

when i > mk .

0

We conclude that for each t ∈ (0, +∞),     u (t), ei L2 (Ω)  ≤ ai for all i ∈ N, which leads to that u (·) ∈ Uad . Then we prove that y( t; u ) = 0.

(3.39)

Indeed, because y (k) (·), z (k) (·), and f (k) , k = 2, 3, · · · , satisfy (3.28) and (3.29), we can make use of (3.4) to obtain that ⎛ ⎞ ⎞ ⎛ n n   Tj ; u ⎠ = y (n) ⎝2 Tj ⎠ for each n ≥ 2. y ⎝2 j=1

j=1

This, together with (3.30), yields that ⎛ ⎛ ⎞⎞ n  Pmn ⎝y ⎝2 (3.40) Tj ; u ⎠⎠ = 0 for all n ≥ 2. j=1

We arbitrarily fix a k ∈ N. Then it follows from (3.40) that ⎛ ⎛ ⎞⎞ n  Pmk ⎝y ⎝2 Tj ; u ⎠⎠ = 0 for all n ≥ k. j=1

This implies that ⎛ ⎛ (3.41)

0 = lim Pmk ⎝y ⎝2 n→+∞

n 

⎞⎞

   t; u . Tj ; u ⎠⎠ = Pmk y

j=1

Because of (2.4) and (3.5), we can pass to the limit for k → +∞ in (3.41) to get (3.39). This completes the proof. Next, we present the proof of Theorem 2.3. Proof of Theorem 2.3. First of all, we recall that t∗ stands for the optimal time to the problem (P). According to Theorem 2.2, there exists a sequence of admissible pairs {(tm , um (·))}+∞ m=1 such that (3.42)

tm  t∗ as m → +∞,

(3.43)

um (·) → u∗ (·) weakly star in L∞ (0, +∞; L2 (Ω)) as m → +∞,

(3.44)

um (t) = 0 for a.e.. t ∈ (tm , +∞),

¨ AND GENGSHENG WANG QI LU

22 and

ym (tm ; um ) = 0 for all m ∈ N.

(3.45)

It follows at once from (3.42), (3.43), and (3.44) that t∗ < +∞ and u∗ (t) = 0 for a.e. t ≥ t1 .

(3.46) Write um (t) = that

+∞ j=1

umj (t)ej . Since each um (·) with m ∈ N belongs to Uad , we see

|umj (t)| ≤ aj for almost every t ∈ (0, +∞) and for all m, j ∈ N. Thus, it holds that  2   um (t) 2

L (Ω)

=

+∞ 

|umj (t)|2 ≤

j=1

+∞ 

a2j for a.e. t ∈ (0, +∞).

j=1

From this, it follows that for all m ∈ N, (3.47)

  um (·)

L∞ (0,+∞;L2 (Ω))

    = esssupt∈(0,+∞) um (t)

L2 (Ω)

≤

#+∞ 

$ 12 a2i

< +∞.

i=1

In what follows, ym (·) and y ∗ (·) denote to the solutions ym (·; um ) and y(·; u∗ ), respectively. We are going to show that the above-mentioned u∗ (·) is an optimal control to the problem (P). The proof will be carried out by two steps as follows. Step 1: We claim that y ∗ (t∗ ) = 0. Because of (3.45), it suffices to show that ym (tm ) → y ∗ (t∗ ) weakly in L2 (Ω) as m → +∞.

(3.48)

For this purpose, we first prove that ym (t∗ ) → y ∗ (t∗ ) weakly in L2 (Ω) as m → +∞.

(3.49)

In fact, for any ψ ∈ L2 (Ω), we have that     ∗ ∗ ∗ ym (t ) − y (t ) ψdx = Ω

t∗



0

Ω

  ∗ χω um (s) − u∗ (s) eA(t −s) ψdxds.

Since it follows from (3.43) that   χω um (·) − u∗ (·) → 0 weakly-star in L∞ (0, t1 ; L2 (Ω)) as m → +∞, we have that     ∗ ∗ ∗ ym (t )−y (t ) ψdx = lim lim

m→+∞

Ω

m→+∞

0

t∗

 Ω

  ∗ χω um (s)−u∗ (s) eA(t −s) ψdxds = 0.

Because the above equalities hold for all ψ ∈ L2 (Ω), (3.49) follows. We next prove that   (3.50) ym (tm ) − ym (t∗ ) → 0 weakly in L2 (Ω) as m → +∞.

23

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

Indeed, it is clear that ym (tm ) − ym (t∗ )   ∗ = eAtm − eAt y0 +

tm

t∗

eA(tm −s) χω um (s)ds +

 0

t∗ 

∗

eA(tm −s) − eA(t

−s)

 χω um (s)ds.

(3.51) The three terms on the right-hand side of (3.51) can be studied as follows: By the strong continuity of the semigroup {eAt }t≥0 , we see that   ∗ eAtm − eAt y0 = 0. (3.52) lim m→+∞

Because of (3.42), it follows from (3.47) that  (3.53)

tm

lim

m→+∞

t∗

eA(tm −s) χω um (s)ds = 0.

We now show the weak convergence of the last right hand term of (3.51), in the weak topology of L2 (Ω). To this end, we arbitrarily take a function ψ ∈ L2 (Ω). By (3.42) and the strong continuity of the semigroup {eAt }t≥0 , it holds that ∗

lim eA(tm −s) ψ = eA(t

m→+∞

−s)

ψ for any s ∈ (0, t∗ ),

which, together with (3.47), yields that     A(t −s)   A(t∗ −s) m  (3.54) lim  e ψχω um (s)dx = 0 for a.e. s ∈ (0, t∗ ). −e m→+∞

Ω

Since {eAt }t≥0 is a contractive semigroup, we have the following two estimates:      A(tm −s)  ψ ≤ ψ L2 (Ω) for any s ∈ [0, tm ] e L2 (Ω)

and

   A(t∗ −s)  ψ e

L2 (Ω)

  ≤ ψ L2 (Ω) for any s ∈ [0, t∗ ],

from which it follows that     A(t −s)   A(t∗ −s) m  e ψχω um (s)dx −e  Ω       ∗   A(tm −s) − eA(t −s) ψ dx e ≤ um L∞ (0,t ;L2 (Ω))  1     Ω  ∗     (3.55) ≤ 2{ai }+∞ i=1 l2 Ω ψ L2 (Ω) for a.e. s ∈ (0, t ). Here |Ω| stands for the Lebesgue measure of Ω. By (3.54) and (3.55), we can apply the Lebesgue dominated theorem to derive that   t∗      A(tm −s) A(t∗ −s)  e ψχω um (s)dx ds = 0. −e lim  m→+∞

0

Ω

¨ AND GENGSHENG WANG QI LU

24

Since ψ was arbitrarily taken from L2 (Ω), we derive from the above equality that  (3.56) 0

t∗

  ∗ eA(tm −s) − eA(t −s) χω um (s)ds → 0 weakly in L2 (Ω) as m → +∞.

Now, (3.50) follows immediately from (3.51), (3.52), (3.53), and (3.56). By (3.49) and (3.50), we obtain (3.48). This completes the proof of the claim of Step 1. Step 2: We claim that u∗ (·) belongs to Uad . In fact, the restriction of the function ∗ u (·) over (0, t1 ), still denoted in the same manner, can be expressed as u∗ (t) =

+∞ 

u∗i (t)ei for a.e. t ∈ (0, t1 ).

i=1

Here, all scalar functions u∗i (·), i = 1, 2, · · · , belong to L∞ (0, t1 ). By the second equality of (3.46), we see that in order to prove the above claim, it suffices to show that  ∗  ui (·) ∞ ≤ ai for each i ∈ N. (3.57) L (0,t1 ) In fact, for each i ∈ N, there exists a scalar function vi (·), with   vi (·) 1 = 1, L (0,t1 ) such that 

t1

(3.58) 0

  u∗i (t)vi (t)dt = u∗i (·)L∞ (0,t1 ) .

Define a function v i (·) : (0, t1 ) → L2 (Ω) by setting v i (t) = vi (t)ei for a.e. t ∈ (0, t1 ).

(3.59)

It is obvious that v i (·) ∈ L1 (0, t1 ; L2 (Ω)). From (3.42) and (3.43), we observe that the restrictions of functions um , with m ∈ N, and u∗ over (0, t1 ), still denoted in the same way, satisfy that um (·) → u∗ (·) weakly-star in L∞ (0, t1 ; L2 (Ω)) as m → +∞. Therefore, it holds for each i ∈ N that   t1  (3.60) lim um (t)v i (t)dxdt = m→+∞

0

Ω

0

t1

 Ω

u∗ (t)v i (t)dxdt =

 0

t1

u∗i (t)vi (t)dt.

On the other hand, for each m ∈ N, the restriction of the function um (·) over (0, t1 ), still denoted in the same manner, has the form um (t) =

+∞ 

umj (t)ej for almost every t ∈ (0, t1 ).

j=1

Since um (·) ∈ Uad , the scalar functions umj (·), j ∈ N, enjoy the property that   umj (·) ∞ (3.61) ≤ aj for all j ∈ N. L (0,t1 )

ON THE EXISTENCE OF TIME OPTIMAL CONTROLS

25

Besides, it follows from (3.59) that  um (t)v i (t)dx = umi (t)vi (t) for a.e. t ∈ (0, t1 ). Ω

This, together with (3.60) and (3.58), yields that  t1  t1 (3.62) lim umi (t)vi (t)dt = u∗i (t)vi (t)dt = |u∗i (·)|L∞ (0,t1 ) . m→+∞

0

0

Another simple observation is that  t1 umi (t)vi (t)dt ≤ |umi (·)|L∞ (0,t1 ) . (3.63) 0

Now, it follows from (3.62), (3.63), and (3.61) that    ∗  umi (·) ∞ ui (·) ∞ ≤ lim ≤ ai . (3.64) L (0,t1 ) L (0,t1 ) m→+∞

Because i was arbitrarily taken from N, (3.57) follows immediately from (3.64). Finally, we obtain from the two claims in Steps 1 and 2 that u∗ (·) is an optimal control to the problem (P). This completes the proof. The following results are direct consequences of Proposition 2.4 and Theorems 2.2 and 2.3. 2 , such that for each of its elements, Corollary 3.3. Let A be the subset of l+ +∞ +∞ 2 {ai }i=1 , there is a polynomial p with {1/p(i)}i=1 ∈ l+ , such that ai ≥ 1/p(i) for all +∞ i ∈ N. Then, the problem (P), with {ai }i=1 ∈ A, has admissible pairs and optimal controls. 4. Conclusions. In summary, we conclude that the time optimal control problem (P) essentially differs from the problem with the ball type from the perspective of the existence of optimal controls. Certain conditions on the constraint {ai }+∞ i=1 are necessary to be imposed to ensure the existence. Condition 2.1 is one of them. Proposition 2.4 provides an easily verifiable condition to ensure the existence. With the aid of Condition 2.1, we make use of the Lebeau–Robbiano type estimate and a modification of the iteration method used in [11] and [13] to derive the existence of admissible pairs and optimal controls for the problem studied. REFERENCES [1] L. C. Evans, An Introduction to Mathematical Optimal Control Theory, available online at http://math.berkeley. edu/∼evans/control.course.pdf. [2] H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, North–Holland Math. Stud., 201, Elsevier, Amsterdam, 2005. [3] M. Goossens, F. Mittelbach, and A. Samarin, The LATEX Companion, Addison-Wesley, Reading, MA, 1994. [4] M. Gugat, A Newton method for the computation of time-optimal boundary controls of onedimensional vibrating systems, J. Comput. Appl. Math., 114 (2000), pp. 103–119. [5] M. Gugat and G. Leugering, Regularization of L∞ -optimal control problems for distributed parameter systems, Comput. Optim. Appl., 22 (2002), pp. 151–192. [6] N. J. Higham, Handbook of Writing for the Mathematical Sciences, SIAM, Philadelphia, 1993. ¨ rmander, The Analysis of Linear Partial Differential Operators, III, Pseudo-differential [7] L. Ho Operations, Springer-Verlag, Berlin, 1994. [8] C. Jia and D. Feng, The time optimal control of a nonlinear parabolic equation with a nonlocal lower order term, Acta Anal. Funct. Appl., 10 (2008), pp. 123–138.

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¨ AND GENGSHENG WANG QI LU

[9] L. Lamport, LATEX: A Document Preparation System, Addison-Wesley, Reading, MA, 1986. [10] J. P. LaSalle, The time optimal control problem, in Contributions to the Theory of Nonlinear Oscillations, Vol. 5, Princeton University Press, Princeton, NJ, 1960, pp. 1–24. [11] G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal., 141 (1998), pp. 297–329. [12] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkh¨ auser Boston, Boston, 1995. ´ pez, X. Zhang, and E. Zuazua, Null controllability of the heat equation as singular [13] A. Lo limit of the exact controllability of dissipative wave equations, J. Math. Pures Appl. (9), 79 (2000), pp. 741–808. ¨ , A Lower Bound on Local Energy of Partial Sum of Eigenfunctions for Laplace–Beltrami [14] Q. Lu Operators, submitted. [15] V. J. Mizel and T. I. Seidman, An abstract bang-bang principle and time optimal boundary control of the heat equation, SIAM J. Control Optim., 35 (1997), pp. 1204–1216. [16] Yu. Netrusov and Yu. Safarov, Weyl asymptotic formula for the Laplacian on domains with rough boundaries, Commun. Math. Phys., 253 (2005), pp. 481–509. [17] K. D. Phung, G. Wang, and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), pp. 925–941. [18] R. Seroul and S. Levy, A Beginner’s Book of TEX, Springer-Verlag, Berlin, New York, 1991. [19] G. Wang, The existence of time optimal control of semilinear parabolic equations, Systems Control Lett., 53 (2004), pp. 171–175. [20] G. Wang, L∞ -null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), pp. 1701–1720. [21] L. Wang and G. Wang, The optimal time control of a phase-field system, SIAM J. Control Optim., 42 (2003), pp. 1483–1508.