Applied Mathematics E-Notes, 1(2001), 97-103 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

Optimal Boundary Control of a Nonlinear Diffusion Equation ^{∗}^{†}

Jing-xue Yin^{‡}, Wen-mei Huang^{§}

Received 26 April 2001

Abstract

We discuss the optimal boundary control governed by a nonlinear diffusion equation, and establish the existence and stability of the optimal control.

In this paper, we are concerned with the optimal boundary control governed by the following nonlinear heat conduction equation

\frac{∂u}{∂t} + divJ⃗ + λu = 0, (x, t) ∈ Q_{T} = Θ × (0, T),  (1)

subject to the initial value condition

u(x, 0) = u_{0}(x), x ∈ Θ,  (2)

and the boundary value condition

J⃗ · n⃗ = −h(u − α), (x, t) ∈ ∂Θ × (0, T),  (3)

where J⃗ = −|∇u|^{p−2}∇u is the heat flux, p > 2, Θ ⊂ R^{N} is a bounded domain with smooth boundary, n⃗ denotes the outward normal to the boundary ∂Θ, λ is a positive constant, u_{0}(x) is a nonnegative bounded function and h is the heat transfer coefficient which we take as our control. The cost functional is chosen as

J(h) = \frac{1}{2} \{β ∫ ∫_{Q_{T}} (u − Z_{d})^{2}dxdt + γ ∫_{∂Θ×(0,T)} h^{2}dsdt\} , h ∈ U_{M},  (4)

where U_{M} is the admissible set, namely,

U_{M} = {h| 0 ≤ h ≤ M, h ∈ L^{∞}(∂Θ × (0, T)), h ≡ 0 on ∂Θ \ Γ} .

Here Γ is a partial boundary of Θ with mes Γ > 0, Z_{d} is the desired temperature distribution, the coefficient β and γ are per unit costs associated with failing to achieve the desired temperature distribution and with imposing a heat transfer coefficient different from zero. According to different requests, β and γ can take different values. Then the optimal control problem of the temperature system is

∗Mathematics Subject Classifications: 35K99

†Supported by the Ministry of Science and Technology of China

‡Department of Mathematics, Jilin University, Changchu, Jilin 130012, P. R. China

§Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, P. R. China

To find a h^{∗} ∈ U_{M}, s.t. J(h^{∗}) = inf_{h∈U_{M}} J(h).  (5)

Thus, the state equation (1) with the initial and boundary value condition (2), (3), together with the cost functional (4), and question (5) compose a mathematical model of the optimal boundary control of the heat transfer system. If u > α, the system releases heat, while if u < α, the system absorbs heat.

It was Lenhart and Wilson [1] who first studied the optimal control for such kind of system with p = 2, established the existence, uniqueness and stability of the optimal control, and proved that the optimal control can be formulated by h^{∗} = qu, where q is a nonnegative function independent of u. Later on, similar results were obtained by several authors, see for example [5], [6] and [7]. It is well known that for the classical heat conduction equation, i.e., the case where p = 2, the speed of propagation is infinite. However, for the case where p > 2, the state equation becomes the p-Laplace equation, whose solutions possess the property of finite speed of propagation of disturbances. Hence, it is more natural to consider the heat transfer system governed by the p-Laplace equation.

Due to the degeneracy of our equations, we are only interested in weak solutions to our problem (1)–(3) in the following sense: A nonnegative function u ∈ C(0, T; L^{2}(Θ)) ⋂ L^{p}(0, T; W^{1,p}(Θ)) is said to be a weak solution of the problem (1)–(3) if the following integral equality holds

∫_{Θ} u(x, τ)φ(x, τ)dx + ∫ ∫_{Q_{τ}} |∇u|^{p−2}∇u · ∇φdxdt
+ ∫_{∂Θ×(0,τ)} h(s, t)(u(s, t) − α)φ(s, t)dsdt + ∫ ∫_{Q_{τ}} λu(x, t)φ(x, t)dxdt
− ∫_{Θ} u_{0}(x)φ(x, 0)dx − ∫ ∫_{Q_{τ}} u(x, t)φ_{t}(x, t)dxdt
= 0,  (6)

where φ is an arbitrary test function in C^{1}(\overline{Q}_{T})), τ ∈ (0, T) and Q_{τ} = Θ × (0, τ). The main results of this paper are as follows. THEOREM 1. Assume that Z_{d} ∈ L^{2}(Q_{T}), u_{0} ∈ C^{1}(\overline{Θ}) and satisfies the following

compatibility condition

|∇u_{0}|^{p−2}∇u_{0} · n⃗ = −hu_{0}, x ∈ ∂Θ.

Then there exists an optimal control h^{∗} ∈ U_{M} which minimizes the cost functional J(h) defined by (4).

As for the stability of the optimal control h, we have

THEOREM 2. Suppose u = u(h) and u_{ε} = u(h + εl) are solutions of problem (1)–(3), corresponding to h ∈ U_{M}, h + εl ∈ U_{M} respectively. Then

∥u_{ε} − u∥_{L^{2}(Q_{T})} = O(ε), ε → 0.

We need several lemmas which will be used in the proof of our main results.

LEMMA 1. For any u, v ∈ L^{p}(0, T; W^{1,p}(Θ)), the following inequality holds:

∫ ∫_{Q_{T}} (|∇u|^{p−2}∇u − |∇v|^{p−2}∇v) · ∇(u − v)dxdt ≥ 0.

Indeed, the above inequality follows easily from the convexity of Φ(X) = |X|^{p}. LEMMA 2 (Feng [3]). Let B_{0}, B and B_{1} be reflexive Banach spaces which satisfy

B_{0} ↪^{C} B ↪ B_{1}, where ↪ denotes imbedding and ↪^{C} denotes compact imbedding. Then we have

L^{r_{0}}(0, T; B_{0}) ∩ {ϕ|ϕ_{t} ∈ L^{r_{1}}(0, T; B_{1})} ↪^{C} L^{r_{0}}(0, T; B);

L^{∞}(0, T; B_{0}) ∩ {ϕ|ϕ_{t} ∈ L^{r_{2}}(0, T; B_{1})} ↪^{C} C(0, T; B);

L^{r_{0}}(0, T; B) ∩ {ϕ|ϕ_{t} ∈ L^{r_{0}}(0, T; B)} ↪^{C} C(0, T; B).

Here 1 ≤ r_{0}, r_{1} ≤ ∞, and 1 < r_{2} ≤ ∞.

LEMMA 3. Under the assumption in Theorem 1, there exists a unique solution u_{h} of the problem (1)–(3) for any h ∈ U_{M}.

PROOF. For the existence of solution u_{h}, we refer to [2]. The uniqueness can also be proved in a rather standard way as follows. Let u_{1}, u_{2} be solutions of the problem (1)–(3). From the definition of a solution of (1)-(3), we have

∫ ∫_{Q_{τ}} (|∇u_{1}|^{p−2}∇u_{1} − |∇u_{2}|^{p−2}∇u_{2})∇φdxdt − ∫ ∫_{Q_{τ}} (u_{1} − u_{2})φ_{t}dxdt
+ ∫_{∂Θ×(0,τ)} h(u_{1} − u_{2})φdsdt + ∫ ∫_{Q_{τ}} λ(u_{1} − u_{2})φdxdt
= ∫_{Θ}(u_{2}(x, τ) − u_{1}(x, τ))φ(x, τ)dx,

for all τ ∈ (0, T). Choosing φ = u_{1} − u_{2}, we obtain

∫ ∫_{Q_{τ}} (|∇u_{1}|^{p−2}∇u_{1} − |∇u_{2}|^{p−2}∇u_{2}) · ∇(u_{1} − u_{2})dxdt
+ ∫_{∂Θ×(0,τ)} h(u_{1} − u_{2})^{2}dsdt + ∫ ∫_{Q_{τ}} λ(u_{1} − u_{2})^{2}dxdt
= ∫ ∫_{Q_{τ}} (u_{1} − u_{2})(u_{1} − u_{2})_{t}dxdt − ∫_{Θ}(u_{1}(x, τ) − u_{2}(x, τ))^{2}dx,

for all τ ∈ (0, T). Noticing that

∫ ∫_{Q_{τ}} (|∇u_{1}|^{p−2}∇u_{1} − |∇u_{2}|^{p−2}∇u_{2}) · ∇(u_{1} − u_{2}) ≥ 0,

we get

∫_{∂Θ×(0,τ)} h(u_{1} − u_{2})^{2}dsdt + ∫ ∫_{Q_{τ}} λ(u_{1} − u_{2})^{2}dxdt ≤ −\frac{1}{2} ∫_{Θ}(u_{1} − u_{2})^{2}(x, τ)dx ≤ 0,

which implies that u_{1}(x, t) = u_{2}(x, t), a.e. (x, t) ∈ Q_{T}.

We are now in a position to present and prove our main results.

First consider Theorem 1. Without loss of generality, we assume that α = 0. Let {h_{n}} be a sequence in U_{M} for which

lim_{n→∞} J(h_{n}) = inf_{h∈U_{M}} J(h).

By Lemma 3, for each n, we can define u_{n} = u(h_{n}) as the solution of the problem (1)–(3) with h = h_{n}, namely, u_{n} satisfies the following integral equality

∫_{Θ} u_{n}(x, τ)φ(x, τ)dx + ∫ ∫_{Q_{τ}} |∇u_{n}|^{p−2}∇u_{n} · ∇φdxdt
+ ∫_{∂Θ×(0,τ)} h_{n}(s, t)u_{n}(s, t)φ(s, t)dsdt + ∫ ∫_{Q_{τ}} λu_{n}(x, t)φ(x, t)dxdt
− ∫_{Θ} u_{0}(x)φ(x, 0)dx − ∫ ∫_{Q_{τ}} u_{n}(x, t)φ_{t}(x, t)dxdt
= 0  (7)

Using the regularity results in [4] for the p-Laplace equation, we see that u_{nt} ∈ L^{2}(Q_{T}) and satisfies the following estimate

∫ ∫_{Q_{τ}} |u_{nt}(x, t)|^{2}dxdt ≤ C, τ ∈ (0, T),  (8)

where C is a positive constant independent of n. By virtue of this and the definition of weak solutions, after an approximation process, we may always choose u_{n}, or ψu_{n} for some smooth function ψ, as a test function in (7). First, take u_{n} as the test function in (7) and obtain

\frac{1}{2} ∫_{Θ}\{u_{n}^{2}(x, τ) − u_{0}^{2}(x)\}dx + ∫ ∫_{Q_{τ}} |∇u_{n}|^{p}dxdt

+ ∫_{0}^{τ} ∫_{∂Θ} h_{n}u_{n}^{2}dxdt + ∫ ∫_{Q_{τ}} λu_{n}^{2}dxdt

= 0.  (9)

Noticing that the last three terms in (8) are nonnegative, we have

∫_{Θ} u_{n}^{2}(x, τ)dx ≤ ∫_{Θ} u_{0}^{2}(x)dx, τ ∈ (0, T),  (10)

∫ ∫_{Q_{τ}} |∇u_{n}(x, t)|^{p}dxdt ≤ \frac{1}{2} ∫_{Θ} u_{0}^{2}(x)dx, τ ∈ (0, T).  (11)

From Lemma 2, there exists a subsequence of {u_{n}}, denoted also by {u_{n}}, u^{∗} ∈ C(0, T; L^{2}(Θ)) ∩ L^{p}(0, T; W^{1,p}(Θ)) and w ∈ L^{p/(p−1)}(Q_{T}), which satisfy u_{n} → u^{∗} a.e. Q_{T} , u_{nt} ⇀ u_{t}^{∗} in L^{2}(Q_{T}), ∇u_{n} ⇀ ∇u^{∗} in L^{p}(Q_{T}), and |∇u_{n}|^{p−2}∇u_{n} ⇀ w in L^{p/(p−1)}(Q_{T}). We claim that w = |∇u^{∗}|^{p−2}∇u^{∗}. Indeed, in view of

∫ ∫_{Q_{T}} u^{∗}φ_{t}dxdt − ∫ ∫_{Q_{T}} w_{i}φ_{x_{i}}dxdt − ∫ ∫_{Q_{T}} λu^{∗}φdxdt = 0,  (12)

for φ ∈ C_{0}^{∞}(Q_{T}), we need only to show the following

∫ ∫_{Q_{T}} |∇u^{∗}|^{p−2}∇u^{∗} · ∇φdxdt = ∫ ∫_{Q_{T}} w_{i}φ_{x_{i}}dxdt, φ ∈ C_{0}^{∞}(Q_{T}).  (13)

Actually, for any v ∈ L^{p}(0, T; W^{1,p}(Θ)) ∩ C(0, T; L^{2}(Θ)), ψ ∈ C_{0}^{∞}(Q_{T}), 0 ≤ ψ ≤ 1, suppψ ⊂ Θ, we have

∫ ∫_{Q_{T}} ψ(|∇u_{n}|^{p−2}∇u_{n} − |∇v|^{p−2}∇v) · ∇(u_{n} − v)dxdt ≥ 0.  (14)

Choosing φ = ψu_{n} in (7), we obtain

∫ ∫_{Q_{T}} λψu_{n}^{2}dxdt + ∫ ∫_{Q_{T}} ψ|∇u_{n}|^{p}dxdt
= \frac{1}{2} ∫ ∫_{Q_{T}} ψ_{t}u_{n}^{2}dxdt − ∫ ∫_{Q_{T}} u_{n}|∇u_{n}|^{p−2}∇u_{n}∇ψdxdt.

It follows from (14) that

\frac{1}{2} ∫ ∫_{Q_{T}} ψ_{t}u_{n}^{2}dxdt − ∫ ∫_{Q_{T}} u_{n}|∇u_{n}|^{p−2}∇u_{n}∇ψdxdt

− ∫ ∫_{Q_{T}} ψ|∇u_{n}|^{p−2}∇u_{n} · ∇vdxdt − ∫ ∫_{Q_{T}} λψu_{n}^{2}dxdt

− ∫ ∫_{Q_{T}} ψ|∇v|^{p−2}∇v · ∇(u_{n} − v)dxdt

≥ 0  (15)

Letting n → ∞ in (15), we get

\frac{1}{2} ∫ ∫_{Q_{T}} ψ_{t}u^{∗}^{2}dxdt − ∫ ∫_{Q_{T}} u^{∗}w_{i}ψ_{x_{i}}dxdt − ∫ ∫_{Q_{T}} ψw_{i}v_{x_{i}}dxdt

− ∫ ∫_{Q_{T}} λψu^{∗}^{2}dxdt − ∫ ∫_{Q_{T}} ψ|∇v|^{p−2}∇v · ∇(u^{∗} − v)dxdt

≥ 0  (16)

Take ψu^{∗} as a test function in (12) to obtain

\frac{1}{2} ∫ ∫_{Q_{T}} u^{∗}^{2}ψ_{t}dxdt − ∫ ∫_{Q_{T}} w_{i}ψ_{x_{i}}u^{∗}dxdt

− ∫ ∫_{Q_{T}} w_{i}ψu_{x_{i}}^{∗}dxdt − ∫ ∫_{Q_{T}} λu^{∗}^{2}ψdxdt

= 0  (17)

Using (16), we have

∫ ∫_{Q_{T}} ψ(w_{i} − |∇v|^{p−2}v_{x_{i}})(u_{x_{i}}^{∗} − v_{x_{i}})dxdt ≥ 0.  (18)

Choosing v = u^{∗} − θφ in (18), where θ ≥ 0, φ ∈ C_{0}^{∞}(Q_{T}), we get

∫ ∫_{Q_{T}} ψ(w_{i} − |∇(u^{∗} − θφ)|^{p−2}(u^{∗} − θφ)_{x_{i}})φ_{x_{i}}dxdt ≥ 0.

Letting θ → 0, we have

∫ ∫_{Q_{T}} ψ(w_{i} − |∇u^{∗}|^{p−2}u_{x_{i}}^{∗})φ_{x_{i}}dxdt ≥ 0, φ ∈ C_{0}^{∞}(Q_{T}).

Obviously, if we let θ ≤ 0, we can get another inequality which has reverse direction. Therefore, we can choose a function ψ, with suppφ ⊂ suppψ, and ψ = 1 on suppφ, such that (13) is true, which implies w = |∇u^{∗}|^{p−2}∇u^{∗}.

By

h_{n} ⇀^{∗} h^{∗} in L^{∞}((0, T) × ∂Θ),

and the continuity of the mapping from H^{1}(Θ) to L^{2}(∂Θ), we have

u_{n} → u^{∗} in L^{2}((0, T); L^{2}(∂Θ)),

and let n → ∞ in (7), we see that u^{∗} is a weak solution of the problem (1)–(3) with h^{∗} as the heat transfer coefficient.

At last, by the lower semicontinuity of the cost functional and using the weak convergencies derived above, we see that h^{∗} is an optimal control. The proof of Theorem 1 is complete.

We now turn to the proof of Theorem 1. From the definition of a solution to our problem (1)–(3), we see that u_{ε} and u satisfy the integral equality (7). Choosing φ = u_{ε} − u in (7), we have

\frac{1}{2} ∫_{Θ}(u_{ε}(x, T) − u(x, T))^{2}dx + ∫ ∫_{Q_{T}} |∇u_{ε}|^{p−2}∇u_{ε}∇(u_{ε} − u)dxdt

− ∫ ∫_{Q_{T}} |∇u|^{p−2}∇u · ∇(u_{ε} − u)dxdt + ∫ ∫_{Q_{T}} λ(u_{ε} − u)^{2}dxdt

+ ∫_{0}^{T} ∫_{∂Θ} h(u_{ε} − u)^{2}dsdt + ∫_{0}^{T} ∫_{∂Θ} εlu_{ε}(u_{ε} − u)dsdt

= 0.

Using

∫ ∫_{Q_{T}} (|∇u_{ε}|^{p−2}∇u_{ε} − |∇u|^{p−2}∇u) · ∇(u_{ε} − u)dxdt ≥ 0,

we get

∫ ∫_{Q_{T}} λ(u_{ε} − u)^{2}dxdt ≤ εl ∫_{0}^{T} ∫_{∂Θ} u_{ε}(u − u_{ε})dsdt ≤ Cε.

The proof is complete.

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