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Optimal control problem for a generalized sixth order Cahn-Hilliard type equation with nonlinear diffusion
Boundary Value Problems volume 2015, Article number: 58 (2015)
Abstract
In this paper, we study the initial-boundary-value problem for a generalized sixth order Cahn-Hilliard type equation, which describes the separation properties of oil-water mixtures when a substance enforcing the mixing of the phases is added. The optimal control under boundary condition is given and the existence of optimal solution is proved.
1 Introduction
We consider the equation
in \(\Omega\times(0, T)\), where \(\Omega=(0,1)\), \(\gamma>0\), \(k>0\), and \(\gamma_{2}>0\) with the initial and boundary conditions
The function \(f(u)\) stands for the derivative of a potential \(F(u)\) with \(F(u)\), \(a(u)\) approximated, respectively, by a sixth and a second order polynomial
where \(a_{2}>0\).
The free energy functional proposed by Gompper et al. [1–4] has the form
with the density given by
Here u is the scalar order parameter, which is proportional to the local difference between oil and water concentrations. The properties of the amphiphile and its concentration enter model (1.1) implicitly via (1.4) and (1.5). \(F(u)\) has three minima at \(u=-1\), \(u=1\), and \(u=0\), which describe the oil, water and disordered microemulsion phases. In [2–4], the coefficient \(a(u)\) is approximated by the quadratic function (1.5) with constants \(a_{0}\) of arbitrary sign and \(a_{2}\) positive.
Like in the classical Cahn-Hilliard the theory the order parameter u is a conserved quantity. Thus it satisfies the conservation law
with the mass flux j given by the constitutive equation
and μ representing the chemical potential
where \(\mathcal{D}\geq0\), the dissipation potential, has the form
and M is the mobility, k, \(\gamma_{2}\) are the viscosity coefficients corresponding to the rate of the order parameter and its spatial gradient.
The first variation \(\frac{\delta\psi}{\delta u}\) is defined by the condition that
most hold for all test functions \(\zeta\in C_{0}^{\infty}(\Omega)\). In the case of free energy this leads to the following expressions:
From the above discussions we know that
Combining (1.6)-(1.13) we get the following conserved evolution system:
where \(\Omega\subset\mathbb{R}^{3}\) is a bounded domain with the boundary ∂Ω, occupied by the ternary mixture, and \((0, T)\) is the time interval. We endow this system with the initial and boundary condition (1.2) and (1.3), in this paper we consider the one-dimensional case with \(M=1\).
Schimperna and Pawłow [5] studied (1.1) when \(\gamma_{2}=0\) with logarithmic potential
They investigated the behavior of the solutions to the sixth order system as the parameter γ tended to 0, the uniqueness and regularization properties of the solutions have been discussed.
Pawłow and Zaja̧czkowski [6] proved that the problem (1.1)-(1.5) with \(k=\gamma_{2}=0\) under consideration is well posed in the sense that it admits a unique global smooth solution which depends continuously on the initial datum.
In past decades, the optimal control of distributed parameter system had received much attention in the academic field. A wide spectrum of problems in applications can be solved by methods of optimal control, such as chemical engineering and vehicle dynamics. Modern optimal control theories and applied models are not only represented by ODEs, but also by PDEs. Kunisch and Volkwein solved open-loop and closed-loop optimal control problems for the Burgers equation [7], Armaou and Christofides studied the feedback control of Kuramto-Sivashing equation [8].
Recently, many authors studied the optimal control problem for the pseudo-parabolic equation, such as Tian et al. [9–11], Zhao and Liu [12].
In this paper, we consider the optimal control problem for the following equation:
When \(y=u-kD^{2}u+\gamma_{2}D^{4}u\), we take the distributed optimal control problem
For fixed \(T > 0\), we set \(\Omega= (0, 1)\) and \(Q =\Omega\times(0, T)\). Let \(Q_{0}\subset Q\) be an open set with positive measure.
Let \(V=H_{0}^{1}(0, 1)\), \(H=L^{2}(0, 1)\); \(V^{*}=H^{-1}(0, 1)\), and \(H^{*}=L^{2}(0, 1)\) are dual spaces, respectively, and we have
The extension operator \(B^{*}\in L(L^{2}(0, T; Q_{0}), L^{2}(0, T; V^{*}))\) is given by
The space \(W(0, T; V)\) is defined by
which is a Hilbert space endowed with the common inner product.
The plan of the paper is as follows. In Section 2, we prove the existence of the weak solution in a special space. The optimal control is discussed in Section 3, and the existence of an optimal solution is proved.
2 Existence of weak solution
Consider the following the sixth order Cahn-Hilliard type equation:
under the initial value
and boundary condition
where \(B^{*}\overline{\omega}\in L^{2}(0, T; V^{*})\) and the control item \(\overline{\omega}\in L^{2}(0, T; Q_{0})\).
Let \(y=u-kD^{2}u+\gamma_{2}D^{4}u\); the above problem is rewritten as
Now, we give the definition of the weak solution to the problem (2.2) in the space \(W(0, T; V)\).
Definition 2.1
A function \(y(x, t)\in W(0, T; V)\) is called a weak solution to problem (2.2), if
for all \(\phi\in V\), a.e. \(t\in[0, T]\) and \(y_{0}\in H\) are valid.
Theorem 2.1
The problem (2.2) admits a weak solution \(y(x, t)\in W(0, T; V)\) in the interval \([0, T]\), if \(B^{*}\overline{\omega}\in L^{2}(0, T; V^{*})\) and \(y_{0}\in H\).
Proof
Employ the standard Galerkin method.
The differential operator \(A=-\partial^{2}_{x}\) is a linear unbounded self-adjoint operator in H with \(D(A)\) dense in H, where H is a Hilbert space with a scalar product \((\cdot, \cdot)\) and norm \(\|\cdot\|\).
There exists an orthogonal basis \(\{\psi_{i}\}\) of H. Let \(\{\psi_{i}\}_{i=1}^{\infty}\) be the eigenfunctions of the operator \(A=-\partial^{2}_{x}\) with
For \(n\in\mathbb{N}\), we define the discrete ansatz space by
Set \(y_{n}(t)=y_{n}(x, t)=\sum_{i=1}^{n}y_{i}^{n}(t)\psi_{i}(x)\) and require \(y_{n}(0, \cdot)\mapsto y_{0}\) in H holds true.
To prove the existence of a unique weak solution to the problem (2.2), we are going to analyze the limiting behavior of sequences of smooth functions \(\{y_{n}\}\) and \(\{u_{n}\}\).
Performing the Galerkin procedure for the problem (2.2), we have
According to ODE theory, there is a unique solution to (2.3) in the interval \([0, t_{n}]\). We should show that the solution is uniformly bounded when \(t_{n}\rightarrow T\).
As a first step, multiplying the first equation of (2.3) by
and integrating with respect to x, we obtain
where
and
Applying a simple calculation, we have
where \(C_{1}>0\) and \(C_{0}\geq0\).
Since \(B^{*}\overline{\omega}\in L^{2}(0, T; V^{*})\) is a control item, we assume
Taking into account (2.4), (2.7), (2.8), (1.4), and integrating (2.4) with respect to time from 0 to t, we know
Choosing \(\varepsilon_{1}\), \(\varepsilon_{2}\), and ε sufficiently small, from the above inequality and the Poincaré inequality, we have
From (2.11), we know
By virtue of (2.9), (2.10), and (2.13), we obtain
By Sobolev’s imbedding theorem it follows from (2.14) that
As a second step, multiplying (1.1) by \(D^{2}u_{n}\) and integrating with respect to x, we obtain
From a simple calculation, we have
where
Thus it follows from (2.14), (2.18), and (2.19) that
where ε is sufficiently small.
By the Gronwall inequality, (2.21) implies
As a third step, multiplying (1.1) by \(D^{4} u_{n}\) and integrating with respect to x, we obtain
On account of (2.15) and (2.10), we know
On the other hand, by the Nirenberg inequality, we have
Hence, by the Hölder and Young inequalities, we obtain
Similarly,
the \(\varepsilon_{3}\) is sufficiently small.
Therefore, by the Gronwall inequality, we have
From a simple calculation, we have
From (2.14), (2.15), (2.26), and (2.27), we obtain
As a fourth step, from (2.2), (2.14), (2.15), and the Sobolev embedding theorem, we have
Then
Thus, we have:
-
(i)
For every \(t\in[0, T]\), the sequence \(\{y_{n}\}_{n\in\mathbb{N}}\) is bounded in \(L^{2}(0, T; H)\) as well as in \(L^{2}(0, T; V)\), which is independent of the dimension of the ansatz space n.
-
(ii)
For every \(t\in[0, T]\), the sequence \(\{y_{n, t}\}_{n\in\mathbb{N}}\) is bounded in \(L^{2}(0, T; V^{*})\), which is independent of the dimension of the ansatz space n.
Hence, we get \(\{y_{n, t}\}_{n\in\mathbb{N}}\subset W(0, T; V)\), and \(\{y_{n, t}\}_{n\in\mathbb{N}}\) weak in \(W(0, T; V)\), weak star in \(L^{\infty}(0, T; H)\) and strong in \(L^{2}(0, T; H)\) to a function \(y(x, t)\in W(0, T; V)\). Obviously, the uniqueness of the solution is easy to obtain [13]. We omit it here. □
To ensure that the norm of weak solution in the space \(W(0, T; V)\) can be controlled by the initial value and the control item, we need the following theorem.
Theorem 2.2
If \(B^{*}\overline{\omega}\in L^{2}(0, T; V^{*})\) and \(y_{0}\in H\), then there exist constants \(C_{3}>0\) and \(C_{4}>0\), such that
Proof
Similar to the proof of Theorem 2.1, we obtain
Multiplying the equation by y and integrating the equation with respect to x, we obtain
From the Hölder and Young inequalities, we have
From (2.30), we have
and
Note that
Integrating the above inequality with respect to t yields
By (2.36), (2.2), and (2.30), we deduce that
From (2.36) and (2.37), we have
The proof is completed. □
3 Optimal problem
In this section, we will study the distributed optimal control and the existence of the optimal solution is obtained based on Lions’ theory.
We study the following problem when \(\overline{\omega}\in L^{2}(0, T;Q_{0})\),
where \(y=u-kD^{2}u+\gamma_{2}D^{4}u\).
As we know that there exists a weak solution y to (2.2), due to \(u=(1-k\partial_{x}^{2}+\gamma_{2}\partial_{x}^{2})^{-1}y\), we know that there exists a weak solution u to (2.1). Let there be given an observation operator \(C\in L(W(0, T; V), S)\), in which S is a real Hilbert space and C is continuous.
We choose a performance index of tracking type
where \(z\in S\) is a desired state and \(\delta>0\) is fixed.
The optimal control problem as regards the further generalized sixth order Cahn-Hilliard equation is
where \((y, \overline{\omega})\) satisfies the problem (2.2).
Let \(X=W(0, T; V)\times L^{2}(0, T; Q_{0})\) and \(Y=L^{2}(0, T; V)\times H\).
We define an operator \(e=e(e_{1}, e_{2}): X\rightarrow Y\) by
where
and \(D^{2}\) is an operator from \(H^{1}(0, 1)\) to \(H^{-1}(0, 1)\).
Then (3.2) is rewritten as
Now, we have the following theorem.
Theorem 3.1
There exists an optimal control solution to the problem.
Proof
Let \((y, \overline{\omega})\in X\) satisfy the equation \(e=e(y, \overline{\omega})=0\). In view of (3.1), we have
From Theorem 2.2, we have
Hence
As the norm is weakly lowered semi-continuous [14], we find that \(\mathcal{J}\) is weakly lowered semi-continuous.
Since \(\mathcal{J}(y, \overline{\omega})\geq0\) for all \((y, \overline{\omega})\in X\) holds, there exists
which means that there exists a minimizing sequence \(\{(y_{n}, \overline{\omega^{n}})\}_{n\in\mathbb{N}}\) in X such that
From (3.3), there exists an element \((y^{*}, \overline{\omega}^{*})\in X\) such that
when \(n\rightarrow\infty\).
From (3.4), we have
Since \(W(0, T; V)\) is compactly embedded into \(L^{2}(0, T; L^{\infty})\) and continuously embedded into \(C(0, T; H)\), we derive that \(y_{n}\rightarrow y^{*}\) strongly in \(L^{2}(0, T; L^{\infty})\) and \(y_{n}\rightarrow y^{*}\) strongly in \(C(0, T; H)\), as \(n\rightarrow\infty\). Then we also derive that \(u_{n}\rightarrow u^{*}\), \(Du_{n}\rightarrow Du^{*}\), \(D^{2}u_{n}\rightarrow D^{2}u^{*}\), \(D^{3}u_{n}\rightarrow D^{3}u^{*}\), \(D^{4}u_{n}\rightarrow D^{4}u^{*}\) strongly in \(C(0, T; H)\), as \(n\rightarrow\infty\).
As the sequence \(\{y_{n}\}_{n\in\mathbb{N}}\) converges weakly, \(\|y_{n}\|_{W(0, T; V)}\) is bounded. Also, we see that \(\|y_{n}\|_{L^{2}(0, T; L^{\infty})}\) is bounded based on the embedding theorem.
Since \(y_{n}\rightarrow y^{*}\) strongly in \(L^{2}(0, T; L^{\infty})\), we derive that \(\|y^{*}\|_{L^{2}(0, T; L^{\infty})}\), \(\|u^{*}\|_{L^{2}(0, T; L^{\infty})}\), \(\|D^{2}u^{*}\|_{L^{2}(0, T; L^{\infty})}\) and \(\|D^{4}u^{*}\|_{L^{2}(0, T; L^{\infty})}\) are bounded.
Notice that
As we know
Note that
For \(I^{1}_{1}\), we have
Also we have
Further, similar to (3.6), we have
From (3.5), we have
In view of the above discussion, we can conclude that
Since \(y^{*}\in W(0, T; V)\), we have \(y^{*}(0)\in H\). From \(y_{n}\rightharpoonup y^{*}\) in \(W(0, T; V)\), we can infer that \(y_{n}(0)\rightharpoonup y^{*}(0)\). Thus we obtain
which means that \(e_{2}(y^{*}, \overline{\omega}^{*})=0\), \(\forall n\in\mathbb{N}\).
Hence, we can derive that \(e(y^{*}, \overline{\omega}^{*})=0\), \(\forall n\in\mathbb{N}\).
In conclusion, there exists an optimal solution \((y^{*}, \overline{\omega}^{*})\) to the problem. We can infer that there exists an optimal solution \((y^{*}, \overline{\omega}^{*})\) to the viscous generalized Cahn-Hilliard equation due to \(u=(1-k\partial_{x}^{2}+\gamma_{2}\partial_{x}^{4})^{-1}y\). □
References
Gompper, G, Kraus, M: Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations. Phys. Rev. E 47, 4289-4300 (1993)
Gompper, G, Kraus, M: Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations. Phys. Rev. E 47, 4301-4312 (1993)
Gompper, G, Goos, J: Fluctuating interfaces in microemulsion and sponge phases. Phys. Rev. E 50, 1325-1335 (1994)
Gompper, G, Zschocke, S: Ginzburg-Landau theory of oil-water-surfactant mixtures. Phys. Rev. A 46, 4836-4851 (1992)
Schimperna, G, Pawłow, I: On a class of Cahn-Hilliard models with nonlinear diffusion. SIAM J. Math. Anal. 45, 31-63 (2013)
Pawłow, I, Zaja̧czkowski, W: A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures. Commun. Pure Appl. Anal. 10, 1823-1847 (2011)
Kunisch, K, Volkwein, S: Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102, 345-371 (1999)
Armaou, A, Christofides, PD: Feedback control of the Kuramoto-Sivashinsky equation. Physica D 137, 49-61 (2000)
Shen, C, Tian, L, Gao, A: Optimal control of the viscous Dullin-Gottwald-Holm equation. Nonlinear Anal., Real World Appl. 11, 480-491 (2010)
Shen, C, Gao, A, Tian, L: Optimal control of the viscous generalized Camassa-Holm equation. Nonlinear Anal., Real World Appl. 11, 1835-1846 (2010)
Tian, L, Shen, C, Ding, D: Optimal control of the viscous Camassa-Holm equation. Nonlinear Anal., Real World Appl. 10, 519-530 (2009)
Zhao, X, Liu, C: Optimal control problem for viscous Cahn-Hilliard equation. Nonlinear Anal. TMA 74, 6348-6357 (2011)
Hinze, M, Volkwein, S: Analysis of instantaneous control for the Burgers equation. Nonlinear Anal. TMA 50, 1-26 (2002)
Wouk, A: A Course of Applied Functional Analysis. Wiley-Interscience, New York (1979)
Acknowledgements
The authors would like to express their sincere thanks to the referee’s valuable suggestions for the revision and improvement of the manuscript. This work is supported by the National Science Foundation of China (No. J1310022).
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Shi, G., Liu, C. & Wang, Z. Optimal control problem for a generalized sixth order Cahn-Hilliard type equation with nonlinear diffusion. Bound Value Probl 2015, 58 (2015). https://doi.org/10.1186/s13661-015-0321-9
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DOI: https://doi.org/10.1186/s13661-015-0321-9