Optimal Control of a Viscous Shallow Water Equation

y)m(y)dy, m = u − u xx , and p is chosen to be the Green’s function for the Helmholtz operator 1 − ∂2 x on the line. In a recent study of soliton equations, it is found that (1) for ε = 0 and any b ̸ = − 1 is included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations [2]. When ε = 0, (1) becomes the b-family of equations:


Introduction
Holm and Staley [1] studied the following family of evolutionary 1+1 PDEs: which describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids.Here  =  *  = ∫ ∞ −∞ ( − )(),  =  −   , and  is chosen to be the Green's function for the Helmholtz operator 1 −  2   on the line.In a recent study of soliton equations, it is found that (1) for  = 0 and any  ̸ = − 1 is included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations [2].
For  = 3 in (2), it becomes the Degasperis-Procesi (DP) equation: which can be used as a model for nonlinear shallow water dynamics, and its asymptotic accuracy is the same as (3).Degasperis et al. [5] presented that (4) has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to (3) peakons [4,10,19].Dullin et al. [20] showed that (4) can be obtained from the shallow water elevation equation by an appropriate Kodama transformation.The numerical stability of solitons and peakons, the multisoliton solutions Then, they studied the global solutions and blow-up phenomena to the following generalized equation: In this paper, we study the optimal control problem for the following equation: where ,  are positive constants.The optimal control is an important component of modern control theories and has a wider application in modern engineering.Two methods are introduced to study the control problems in PDE: one is using a low model method and then changing to an ODE model [33]; the other is using a quasi-optimal control method [34].No matter which one is chosen, it is necessary to prove the existence of optimal solution according to the basic theory [35].The control problems of nonlinear PDE have been studied extensively.Kunisch and Volkwein solved open-loop and closed-loop optimal control problems for the Burgers equation [36] and discussed the instantaneous control of the equation [37].Vedantham [38] developed a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the Burgers equation.Øksendal [39] proved a sufficient maximum principle for the optimal control systems described by a quasi-linear stochastic heat equation.In [40], Ghattas and Bark studied the optimal control of two-and three-dimensional incompressible Navier-Stokes flows.Lagnese and Leugering [41] considered the problem of boundary optimal control of a wave equation with boundary dissipation.In [42], Yong established a unified existence theory of optimal controls for general semilinear evolutionary distributed parameter systems under the framework of mild (or weak) solutions for evolution equations.Yong and Zheng [43] subject to where Ω = (0, 1).Clearly, our control target is to match the given desired state  by adjusting the body force  in a control volume  0 ⊆  = (0, ) × Ω in the  2 -sense.

Existence of Unique Weak Solution
In this section, we prove the existence of a weak solution for the following equation: with the boundary conditions and the initial value where  =  −   , Ω = (0, 1),  *  ∈  2 ( * ), and a control  ∈  2 ( 0 ).Now, we give the definition of the weak solution in the space (0, ; ).
By using the standard Galerkin method and some a priori estimates, one can obtain the following theorem, which ensures the existence of a unique weak solution to the viscous shallow water equation.
Proof.Let {  } ∞ =1 be an orthonormal basis in the space  consisting of eigenfunctions of the operator  = − 2  .For  ∈ N, we define the discrete ansatz space by By analyzing the limiting behavior of the sequences of smooth functions {  } and {  }, we can prove the existence of a weak solution to (12).
Performing the Galerkin procedure for the problem ( 12)-( 14), we obtain with the boundary conditions and the initial value Clearly, ( 16) is an ordinary differential equation, and, according to standard ODE's theory, there is a unique solution to (16) in the interval [0,   ).What we should do is to show that the solution is uniformly bounded as   → .We will prove the existence of weak solution in the following steps.
Step 1. Taking the inner product of ( 16) with   in Ω, we have Let us estimate the first term of the right hand side of ( Combining inequalities ( 22) and ( 23) with ( 19) and using the Young inequality, we have where From the above discussion, we know that ‖  ‖ ≤ Step 2. We prove a uniform  2 (0, ; ) bound on a sequence {  }.Taking the inner product of ( 16) with   in Ω, we have Combining inequalities ( 29) and ( 30) with ( 28), we have From (32), we also have Integrating the pervious inequality with respect to  <  on [0, ], we obtain Step 3. We prove a uniform From the pervious discussion one has the following.
The proof of the uniqueness of the solution is similar to Theorem 1 in [37], so we omit it here.This completes the proof of the theorem.
Our next result describes that the norm of weak solution can be controlled by initial value and control item.Theorem 3. Let  0 ∈  and  *  ∈  2 ( * ); then there exists a positive constant , such that Proof.Taking the inner product of (12) with  in Ω, by using the same argument as in the proof of Theorem 2, we have where  1 and  5 are positive constants as in Theorem 2. Taking the inner product of (12) with  in Ω, we have Let us estimate the first term of the right hand side of ( 42 where  9 = max{ 8 /( −  7 ),  8 3 /( −  7 ) 2 }.Combining inequality (45) with  >  7 , we also have Integrating the pervious inequality with respect to  yields where  =  9 +  11 .This completes the proof of the theorem.

Existence of Optimal Solution to the
Problem of (7) In this section, we consider the optimal control problem of (7) and prove the existence of optimal solution based on Lions' theory (see [50]).Allowing a control  ∈  2 ( 0 ), we study the following control system: where  =  −   and Ω = (0, 1).Let  ∈ ((0, ; ), ) be an observation operator, and let  be a real Hilbert space of Advances in Mathematical Physics observations.Then, we choose performance index of tracking type where  ∈  is a desired state and  > 0 is fixed.Optimal control problem of ( 7) is min  (, ) , where (, ) satisfies (56).