Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation

Remark 1.1

According to the value of two parameters, it is not difficult to see that (1.8) is a generalization of (1.7) and (1.8) includes (1.5) and (1.6). In fact, let a = 1 − θ , b = θ , then a + b = 1 . If b = θ = 1 3 , a = 2 3 ( a = 2 b ), then by using transformation t → θ t , (1.8) becomes (1.5). If b = θ = 1 4 , a = 3 4 ( a = 3 b ), then by using transformation t → θ t , (1.8) becomes (1.6).

Definition 2.1

A function y ( x , t ) ∈ W ( 0 , T ; V ) is called a weak solution to (2.1), if

for all ψ ∈ V , a.e. t ∈ [ 0 , T ] and y 0 ( x ) = ϕ ∈ H .

Theorem 2.1

Let ϕ ∈ H , f + B ∗ ω ∈ L 2 ( 0 , T ; V ∗ ) . Then there exists a unique weak solution to (1.1) in the interval [ 0 , T ] .

Proof

Let { φ j } j = 1 ∞ be an orthonormal basis in the space H consisting of eigenfunctions of the operator − ∂ x 2 , where φ j is the eigenfunction subject to the Dirichlet condition:

We also normalize φ j such that ∥ φ j ∥ H = 1 [53]. By the elliptic operator theory, φ j forms base functions in V . For n ∈ ℕ , we define the discrete ansatz space by V n = s p a n { φ 1 , φ 2 , … , φ n } ⊂ V . Let y n ( t ) = y n ( x , t ) = ∑ j = 1 n y j n φ j ( x ) with y n ( ⋅ , 0 ) → y 0 in H .

Next, we prove the existence of a unique weak solution to (2.1) by analyzing the limiting behavior of sequences of smooth functions u n and y n , where u n and y n are the solutions of the Cauchy problem as follows:

By standard methods of differential equations [27], we prove the existence of a solution (2.2) on some interval [ 0 , t n ] , then, this solution can be extended to the whole interval [ 0 , T ] by using a priori estimates. That is, we will show that the solution is uniformly bounded as t n → T . Thus, we divide our proof into four steps.

Step 1. We prove a uniform L 2 ( 0 , T ; V ) bound on a sequence { u n } . Taking the inner product of (2.2) with u n in Ω , we have

That is,

First, we estimate the first term of the right hand side of (2.3) as follows:

By the Sobolev embedding theorem ( H 1 ↪ L ∞ , H 2 ↪ H 1 ) and the Poincaré inequality, we get

where C , C 1 are constants.

By (2.4) and (2.5), we obtain

Next, we estimate the second term of the right hand side of (2.3). Owing to f + B ∗ ω ∈ L 2 ( 0 , T ; V ∗ ) is a control item, there exists a constant M > 0 such that ∥ f + B ∗ ω ∥ L 2 ( 0 , T ; V ∗ ) ≤ M . Thus, we deduce

It follows from (2.3), (2.6), (2.7) and Young’s inequality that

That is,

Therefore, by (2.8), we have

where t ∈ [ 0 , T ] , T < π ε 2 | a − 2 b | C 1 C 2 M and

Based on the above analysis, we obtain ∥ u n ∥ ≤ M 1 and ∥ u n , x ∥ ≤ M 1 . By the Sobolev embedding theorem, we have

Step 2. Multiplying the first equation of (2.2) with − u n , x x and integrating by parts with respect to x in Ω , we have

That is,

Next, we estimate each term of the right hand side of (2.11). Owing to the Sobolev embedding theorem, we deduce that

where K 1 , K 2 > 0 are embedding constants.

Owing to f + B ∗ ω ∈ L 2 ( 0 , T ; V ∗ ) is a control item, there exists a constant M > 0 such that ∥ f + B ∗ ω ∥ L 2 ( 0 , T ; V ∗ ) ≤ M . Thus, by the Sobolev embedding theorem, we deduce

where K 3 > 0 is the embedding constant. It follows from (2.9), (2.11)–(2.14) and Young’s inequality that

Then, we have

where M 2 2 = K 2 2 | 2 a − b | 2 4 ε , M 3 2 = K 1 2 M 1 4 | a + b | 2 4 ε + K 3 2 M 2 ε .

It follows from the aforementioned inequality that

where t ∈ [ 0 , T ] and T < π 2 M 2 M 3 .

Step 3. We prove a uniform L 2 ( 0 , T ; V ) bound on a sequence { y n } . Taking the inner product of (2.2) with y n in Ω , we have

First, we estimate the first term of the right hand side of (2.16). It follows from (2.10), the Poincaré inequality and the Hölder inequality that

where C 4 = C | 2 a − b | 2 ( C + 1 ) .

Second, we estimate the second term of the right hand side of (2.16) in the following way:

It follows from (2.16), (2.17) and (2.18) that

By (2.19) and Young’s inequality, we have

That is,

where ε − C 4 > 0 . Integrating (2.20) with respect to t on [ 0 , T ] , we obtain

Moreover, we deduce

On the other hand, it follows from (2.20) that

Integrating (2.23) with respect to t on [ 0 , T ] , we obtain

Step 4. We prove a uniform L 2 ( 0 , T ; V ∗ ) bound on a sequence { y n , t } . By (2.2), (2.9), (2.10), (2.15) and the Sobolev embedding theorem, we have

It follows from (2.25) that

Integrating (2.26) with respect to t on [ 0 , T ] , we have

Observing the precious discussions, we have

1. the sequence { y n } is bounded in L 2 ( 0 , T ; H ) as well as L 2 ( 0 , T ; V ) ;

2. the sequence { y n , t } is bounded in L 2 ( 0 , T , V ∗ ) .

Therefore, by (i) and (ii), we obtain the boundedness of { y n } in W ( 0 , T ; V ) . Hence, by standard compactness arguments, we can know that there exists a weak limit y ( t , x ) to the subsequence of { y n } in W ( 0 , T ; V ) . Moreover, one concludes convergence of the subsequence of, again denoted by { y n } weak into W ( 0 , T ; V ) , weak-star in L ∞ ( 0 , T ; H ) and strong in L 2 ( 0 , T ; H ) to a function y ( t , x ) ∈ W ( 0 , T ; V ) . The uniqueness of weak solution can be easily derived by following ideas in [54].

This completes the proof of Theorem 2.1.□

Theorem 2.2

Let ϕ ∈ H , f + B ∗ ω ∈ L 2 ( V ∗ ) . Then there exist two constants L 1 > 0 and L 2 > 0 , such that

where

and

Proof

Multiplying the first equation of (2.1) by y and integrating by parts with respect to x in Ω , we have

By Poincaré’s inequality and Sobolev embedding theorem, we obtain

where K 4 > 0 is the embedding constant and λ 1 > 0 is the Poincaré coefficient.

By using the same argument as in the proof of Theorem 2.1, we have

where M 1 and M 4 are positive constants.

It follows from (2.28) and (2.29) that

where M 8 = | 2 a − b | 2 K 4 ( λ 1 + 1 ) M 1 .

Combining (2.27) with (2.30), we get

Integrating (2.31) with respect to t on [ 0 , T ] , we have

That is,

where ε > M 8 . By Hölder’s inequality and Young’s inequality, we have

From (2.32) and (2.33), we have

where C 0 = max { 1 ε − M 8 , 1 ( ε − M 8 ) 2 } .

On the other hand, in view of (2.31), we get

Integrating the aforementioned inequality with respect to t and observing (2.34) yields

where L 0 = 2 max { 1 , 2 C 0 , 2 C 0 M 8 } .

It follows from (2.1) and (2.29) that

It follows from (2.36) that

Integrating (2.37) with respect to t on [ 0 , T ] , we have

Taking into account (2.34) and (2.38), we have

where

and L 2 = 4 ( k M 1 + γ M 4 ) 2 T .

Thus, this completes the proof of Theorem 2.2.□