2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 135–147. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

OPTIMAL CONTROLS FOR A CLASS OF NONLINEAR EVOLUTION SYSTEMS

ABDELHAQ BENBRIK, MOHAMMED BERRAJAA, SAMIR LAHRECH

Abstract. We consider the abstract nonlinear evolution equation ż + Az = uBz +f. Viewing u as control, we seek to minimize J(u) = ∫_{0}^{T} L(z(t), u(t)) dt. Under suitable hypotheses, it is shown that there exists an optimal control \overline{u} and that it satisfies the appropriate optimality system. An example involving the p-Laplacian operator demonstrates the applicability of our results.

1. Introduction

In this paper, we investigate the optimal control problem governed by the abstract non linear evolution equation

ż + Az = uBz + f  (1.1)

These systems with linear operators A and B are called bilinear systems (see. [2, 3, 11]). They appear in many mathematical models from physical processes, for example, models involving the p Laplacian operator (see [4]). Our aim is to investigate the case where A is not linear.

We organize this work as follows: After formulating the problem, we address the question of existence and uniqueness of solutions to these systems. In section 3, we prove the existence theorem of optimal control and we give the necessary conditions of optimality. Finally, we present an example involving the p-Laplacian operator which illustrates the applications of the abstract framework and the results of the theory developed in the previous sections.

2. Setting of the problem

Throughout the paper, H denotes a separable Hilbert space and V a subspace of H having the structure of a reflexive Banach space which is continuously and densely embedded in H.

Identifying H with its dual H′, we have the Gelfand triplet V ↪ H ↪ V′ where V′ is the dual of V. We suppose that these embeddings are compact. Let ⟨., .⟩ be the duality pairing between V and V′ as well as the inner product on H. Let ∥ · ∥, | · | and ∥ · ∥_{V′} denote the norms on V, H and V′ respectively. Given a fixed real number T > 0 and 2 < p < +∞ we introduce the spaces:

2000 Mathematics Subject Classification. 49J20, 49K20.

Key words and phrases. Optimal control; monotone operator; compact embedding;

p-Laplacian; bilinear system.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

L^{p}(V) = L^{p}(0, T; V), L^{p}(H) = L^{p}(0, T; H), L^{p′}(V′) = L^{p′}(0, T; V′),

where (\frac{1}{p} + \frac{1}{p′} = 1) and W = \{w ∈ L^{p}(V) : ẇ ∈ L^{p′}(V′)\}. Here the derivative is understood in the sense of vector valued distributions.

It is well known that every w ∈ W is after eventual modification on a set of measure zero, continuous from [0, T] in H and the embedding W ↪ C([0, T]; H) is continuous [6, 7]. Furthermore, if V ↪ H compactly, then also W ↪ L^{p}(H) compactly.

We study the control problem

inf_{u} J(u)  (2.1)

subject to the state equation

ż + Az(t) = u(t)Bz(t) + f(t)
z(0) = z_{0},

where the cost functional is

J(u) = ∫_{0}^{T} L(z(t), u(t)) dt.

Our aim is to provide conditions under which the optimal solutions (2.1) exist. By an optimal solution we mean a control \overline{u} on which the infimum is attained.

For problem (2.1) we need the following hypotheses:

(H1) A: V → V′ is such that:

(i) ∥Aφ∥_{V′} ≤ α_{1}∥φ∥^{p−1} with α_{1} > 0. (ii) ⟨Aφ, φ⟩ ≥ α_{2}∥φ∥^{p} with α_{2} > 0. (iii) ⟨Aφ_{1} − Aφ_{2}, φ_{1} − φ_{2}⟩ ≥ α_{3}∥φ∥^{2} with β > 0. (iv) φ → A(φ) is continuously Frechet differentiable.

(H2) B: H → H is linear and continuous with |Bφ| ≤ b|φ|, b > 0.

(H3) u ∈ L^{r}(0, T) with r = p/(p − 2).

(H4) f ∈ L^{p′}(V′).

(H5) z_{0} ∈ H.

(H6) L: H × ℝ → ℝ is a integrand convex such that:

(i) L is coercive: lim_{∥u∥_{L^{r} (0,T)}→∞} ∫_{0}^{T} L(z(t), u(t)) dt = +∞ (ii) (x, u) → L(x, u) is continuously Frechet differentiable. (iii) for every x ∈ C([0, T], H) and every u ∈ L^{r}(0, T), J(u) is finite.

Remark 2.1. A(φ) ∈ L^{p′}(V′) if φ ∈ L^{p}(V) and then

∥Aφ∥_{L^{p′} (V′)} ≤ α_{1}∥φ∥_{L^{p}(V)}^{p−1}.

For u ∈ L^{r}(0, T) and φ ∈ L^{p}(V) we have uBφ ∈ L^{p′}(V′) and

∥uBφ∥_{L^{p′} (V′)} ≤ β_{1}∥u∥_{L^{r}(0,T)}∥φ∥_{L^{p}(V)},

where β_{1} > 0. Therefore, the choice of control space is compatible with the equation.

EJDE/CONF/14 OPTIMAL CONTROLS OF NONLINEAR EVOLUTION SYSTEMS 137

3. Results on the evolution problem

We consider the evolution problem

\begin{matrix}ż(t) + Az(t) = u(t)Bz(t) + f \\ z(0) = z_{0} \end{matrix}  (3.1)

We recall that by a solution to the above problem, we mean a function z ∈ W that satisfies (3.1).

Theorem 3.1. Under hypothesis (H1)(i), (H1)(ii), (H1)(iii), (H2), (H3), (H4) and (H5), equation (3.1) admits a unique solution z such that z ∈ L^{∞}(H) and z ∈ W.

Proof. Uniqueness: if z_{1} and z_{2} are solutions of (3.1), then z = z_{1} − z_{2} satisfies

\begin{matrix}ż + Az_{1} − Az_{2} = uBz \\ z(0) = 0 \end{matrix}  (3.2)

and for t ∈ [0, T],

\frac{1}{2}|z(t)|^{2} ≤ b ∫_{0}^{t} |u(τ)||z(τ)|^{2} dτ .

Using the Gronwall lemma, we obtain z_{1} = z_{2}.

The existence follows from a standard application of the Galerkin method [6] and the a priori estimates given in Lemma 3.2. We remark that by Theorem 3.1, z ∈ C([0, T]; H). □

Lemma 3.2. Under the hypothesis of Theorem 3.1, if z is a solution of (3.1) then

∥z∥_{L^{p}(V)} ≤ K_{1}[|z_{0}|^{2} + ∥u∥_{L^{r}(0,T)}^{r} + ∥f∥_{L^{p′} (V′)}^{p′}]^{1/p}  (3.3)
∥z∥_{L^{∞}(H)} ≤ K_{2}[|z_{0}|^{2} + ∥u∥_{L^{r}(0,T)}^{r} + ∥f∥_{L^{p′} (V′)}^{p′}]^{1/2}  (3.4)
∥ż∥_{L^{p′} (V′)} ≤ K_{3}[∥z∥_{L^{p}(V)}^{p−1} + ∥u∥_{L^{r}(0,T)}^{p−1/p−2} + ∥f∥_{L^{p′} (V′)}]  (3.5)

Proof. Let z be a solution of (3.1), then

∫_{0}^{T} ⟨ż(t), z(t)⟩ dt + ∫_{0}^{T} ⟨Az(t), z(t)⟩ dt = ∫_{0}^{T} ⟨u(t)Bz(t), z(t)⟩ dt + ∫_{0}^{T} ⟨f(t), z(t)⟩ dt.

Using (H1)(ii), (H2) and the continuity of the embedding V ↪ H, we have

\frac{1}{2}|z(T)|^{2} − \frac{1}{2}|z_{0}|^{2} + α_{2} ∫_{0}^{T} ∥z(τ)∥_{V}^{p} dτ

≤ K′_{1} ∫_{0}^{T} |u(τ)|∥z(τ)∥^{2} dτ + ∫_{0}^{T} ∥f(τ)∥_{V′}∥z(τ)∥ dτ .

By the Young inequality [10], for \frac{1}{r} + \frac{2}{p} = 1, we have

K′_{1} ∫_{0}^{T} |u(τ)|∥z(τ)∥^{2} dτ ≤ \frac{α_{2}}{4} ∥z∥_{L^{p}(V)}^{p} + K′_{2}∥u∥_{L^{r}(0,T)}^{r},

∫_{0}^{T} ∥f(τ)∥_{V} ∥z(τ)∥ dτ ≤ \frac{α_{2}}{4} ∥z∥_{L^{p}(V)}^{p} + K′_{3}∥f∥_{L^{p′} (V′)}^{p′}.

Hence

\frac{α_{2}}{2} ∥z∥_{L^{p}(V)}^{p} ≤ \frac{1}{2}|z_{0}|^{2} + K′_{2}∥u∥_{L^{r}(0,T)}^{r} + K′_{3}∥f∥_{L^{p′} (V′)}^{p′},

from which, we deduce then (3.3).

Multiplying (3.1) by z and integrating on [0, t] we obtain

\frac{1}{2}|z(t)|^{2} − \frac{1}{2}|z_{0}|^{2} + \frac{α_{2}}{2} ∫_{0}^{t} ∥z(τ)∥^{p} dτ ≤ K′′_{2}∥u∥_{L^{r}(0,T)}^{r} + K′′∥f∥_{L^{p′} (V′)}^{p′}

and then

|z(t)|_{H} ≤ K_{2}[|z_{0}|^{2} + ∥u∥_{L^{r}(0,T)}^{r} + ∥f∥_{L^{p′} (V′)}^{p′}]^{1/2}

which implies 3.4.

Multiplying 3.2 by ξ ∈ L^{p}(V), we have

| ∫_{0}^{T} ⟨ż(t), ξ(t)⟩ dt|
≤ | ∫_{0}^{T} ⟨Az(t), ξ(t)⟩ dt| + | ∫_{0}^{T} ⟨u(t)Bz(t), ξ(t)⟩| + | ∫_{0}^{T} ⟨f(t), ξ(t)⟩|.

Hence

| ∫_{0}^{T} ⟨ż(t), ξ(t)⟩ dt| ≤ [α_{1}∥z∥_{L^{p}(V)}^{p−1} + β_{1}∥u∥_{L^{r}(0,T)}∥z∥_{L^{p}(V)} + ∥f∥_{L^{p′} (V′)}]∥ξ∥_{L^{p}(V)},

which by Young inequality implies (3.5). □

4. Optimal controls

The aim of this section is to prove the existence of optimal controls for problem (2.1). The differentiability of the mapping u ↦ z permits the characterization of the optimal control \overline{u} by necessary conditions corresponding to J′(\overline{u}) = 0.

Existence theorem for the control problem.

Theorem 4.1. If (H1), (H2), (H3), (H4), (H5) and (H6) hold, then (2.1) admits an optimal solution.

Proof. Let (u_{n})_{n} be a minimizing sequence for (2.1), i.e. the pairs (z_{n}, u_{n}) are admissible for (2.1) and

lim_{n} J(u_{n}) = \overline{J}.

From (H6) we have ∥u_{n}∥_{L^{r}(0,T)} ≤ M.

And from Lemma 3.2, we know that (z_{n})_{n} belongs to a bounded subset of L^{∞}(H) ∩ W. By passing to a subsequence if necessary, we may assume that

u_{n} ⇀ \overline{u} w − L^{r}(0, T)
z_{n} ⇀ \overline{z} w ∗ −L^{∞}(H)
z_{n} ⇀ \overline{z} w − L^{p}(V)
Az_{n} ⇀ χ w − L^{p′}(V′)
u_{n}Bz_{n} ⇀ Ψ w − L^{p′}(V′)
ż_{n} ⇀ Λ w − L^{p′}(V′)

1. Using the convergence σ(D(0, T; V); D′(0, T; V′)) we obtain Λ = ˙ \overline{z}.

2. V ↪ H compactly implies that

z_{n} → \overline{z} s − L^{p}(H)
z_{n}(t) → \overline{z}(t) s − H for all t ∈ [0, T].

EJDE/CONF/14 OPTIMAL CONTROLS OF NONLINEAR EVOLUTION SYSTEMS 139

For φ ∈ L^{p}(V), we have ∫_{0}^{T} ⟨u_{n}(t)Bz_{n}(t), φ(t)⟩ dt

= ∫_{0}^{T} ⟨u_{n}(t)B(z_{n}(t) − \overline{z}(t); φ(t)⟩ dt + ∫_{0}^{T} ⟨u_{n}(t)B\overline{z}(t), φ(t)⟩ dt.

Note that

∫_{0}^{T} ⟨u_{n}(t)B(z_{n}(t) − \overline{z}(t); φ(t)⟩ dt ≤ K_{1}∥u_{n}∥_{L^{r}(0,T)}∥z_{n} − \overline{z}∥_{L^{p}(H)}∥φ∥_{L^{p}(H)} and

∫_{0}^{T} u_{n}(t)⟨B\overline{z}(t), φ(t)⟩ dt → ∫_{0}^{T} \overline{u}(t)⟨B\overline{z}(t), φ(t)⟩ dt

because ⟨B\overline{z}, φ⟩ ∈ L^{r′}(0, T). We deduce that Ψ = \overline{u}B\overline{z}.

3. For y ∈ L^{p}(V), we set X_{m} = ∫_{0}^{T} ⟨Az_{m}(t) − Ay(t); z_{m}(t) − y(t)⟩ dt .

We have X_{m} = ∫_{0}^{T} ⟨Az_{m}(t); z_{m}(t)⟩ dt − ∫_{0}^{T} ⟨Az_{m}(t); y(t)⟩ dt − ∫_{0}^{T} ⟨Ay(t); z_{m}(t) − y(t)⟩ dt and

∫_{0}^{T} ⟨Az_{m}(t), z_{m}(t)⟩ dt

= \frac{1}{2}|z_{m,0}|^{2} − \frac{1}{2}|z_{m}(T)|^{2} + ∫_{0}^{T} ⟨u_{m}Bz_{m}(t), z_{m}(t)⟩ dt + ∫_{0}^{T} ⟨f(t), z_{m}(t)⟩ dt.

But ∫_{0}^{T} (⟨u_{m}(t)Bz_{m}(t), z_{m}(t)⟩ − ⟨\overline{u}(t)B\overline{z}(t), \overline{z}(t)⟩) dt

= ∫_{0}^{T} ⟨u_{m}(t)Bz_{m}(t), z_{m}(t) − \overline{z}(t)⟩ dt + ∫_{0}^{T} ⟨u_{m}(t)Bz_{m}(t) − \overline{u}(t)B\overline{z}(t), \overline{z}(t)⟩ dt The first integral in the right-hand side approaches zero because z_{m} → \overline{z} (s − L^{p}(H)). The second integral approaches zero because u_{m}Bz_{m} ⇀ \overline{u}B\overline{z} (w − L^{p′}(V′)). We deduce that

lim sup_{m} ∫_{0}^{T} ⟨Az_{m}(t), z_{m}(t)⟩ dt

≤ \frac{1}{2}|z_{0}|^{2} − \frac{1}{2}|\overline{z}(T)|^{2} + ∫_{0}^{T} ⟨\overline{u}(t)B\overline{z}(t), \overline{z}(t)⟩ dt + ∫_{0}^{T} ⟨f(t), \overline{z}(t)⟩ dt .

Since \overline{z} satisfies

˙\overline{z} + χ = \overline{u}B\overline{z} + f
\overline{z}(0) = z_{0}

it follows that

\frac{1}{2}|z_{0}|^{2} − \frac{1}{2}|\overline{z}(T)|^{2} + ∫_{0}^{T} ⟨\overline{u}(t)B\overline{z}(t), \overline{z}(t)⟩ dt + ∫_{0}^{T} ⟨f(t), \overline{z}(t)⟩ dt = ∫_{0}^{T} ⟨χ(t), \overline{z}(t)⟩ dt.

Hence

0 ≤ lim sup_{m} X_{m} ≤ ∫_{0}^{T} ⟨χ(t) − Ay(t), \overline{z}(t) − y(t)⟩ dt for all y ∈ L^{p}(V)

Using the continuity of the operator A we obtain χ = A\overline{z}. We deduce that (\overline{z}, \overline{u}) is admissible for (2.1). From (H6) we have

∫_{0}^{T} L(\overline{z}(t), \overline{u}(t)) dt ≤ lim inf_{m} ∫_{0}^{T} L(z_{m}(t), u_{m}(t)) dt = \overline{J}

Hence \overline{u} is an optimal control. □

Optimality conditions. Before proceeding with investigation of the mapping Θ: u ↦ z, where z is defined by (3.1), we introduce a technical lemma generalizing the Gronwall inequality.

Lemma 4.2. Let T > 0 and c ≥ 0. Assume that λ and m are integrable in [0, T] with positive values. Let φ: [0, T] → ℝ^{+} be such that:

(a) λφ and λφ^{2} are integrable on [0, T].

(b) \frac{1}{2}φ^{2}(t) ≤ \frac{1}{2}c^{2} + ∫_{0}^{t} λ(s)φ(s) ds + ∫_{0}^{t} m(s)φ^{2}(s) ds for t ≥ 0.

Then

φ(t) ≤ [c + ∫_{0}^{t} λ(s) ds] exp ( ∫_{0}^{t} m(s) ds).

Proof. Set

Ψ(t) = [c^{2} + 2 ∫_{0}^{t} λ(s)φ(s) ds + 2 ∫_{0}^{t} m(s)φ^{2}(s) ds]^{1/2}.

We have that φ(t) ≤ Ψ(t) and Ψ̇ ≤ λ(t) + m(t)Ψ(t). Then

\frac{d}{dt}[Ψ(t) exp ( − ∫_{0}^{t} m(s) ds) − ∫_{0}^{t} λ(s) exp ( − ∫_{0}^{s} m(τ) dτ)] ≤ 0.

Hence

Ψ(t) ≤ exp ( ∫_{0}^{t} m(τ) dτ)[c + ∫_{0}^{t} λ(τ) exp ( − ∫_{0}^{τ} m(s) ds)dτ],

which completes the proof. □

Lemma 4.3. Suppose the hypothesis (H1), (H2), (H3), (H4) and (H5) hold, then the mapping Θ: L^{r}(0, T) → L^{∞}(H) ∩ L^{2}(V), u ↦ z is locally Lipschitz.

Proof. Let \overline{u} and h be in L^{r}(0, T) with ∥h∥_{L^{r}(0,T)} ≤ 1. Set \overline{z} = Θ(\overline{u}), z_{h} = Θ(\overline{u}+h) and z = z_{h} − \overline{z}. Then z satisfies

ż + Az_{h} − A\overline{z} = \overline{u}Bz + hBz_{h}
z(0) = 0

Multiplying by z and integrating on [0, t] we have

\frac{1}{2}|z(t)|^{2} + β ∫_{0}^{t} ∥z(τ)∥_{V}^{2} dτ ≤ b ∫_{0}^{t} |\overline{u}(τ)||z(τ)|^{2} dτ + b ∫_{0}^{t} |h(τ)||z_{h}(τ)||z(τ)| dτ .

Invoking the Lemma 4.2, we have

|z(t)|_{H} ≤ exp (b ∫_{0}^{t} |\overline{u}(τ)| dτ)[b ∫_{0}^{t} |h(τ)||z_{h}(τ)| dτ]

but

∫_{0}^{t} |h(τ)||z_{h}(τ)| dτ ≤ K∥h∥_{L^{r}(0,T)}∥z_{h}∥_{L^{∞}(H)}

and

∥z_{h}∥_{L^{∞}(H)}∥ ≤ K_{1} [|z_{0}|^{2} + ∥\overline{u} + h∥_{L^{r}(0,T)}^{r} + ∥f∥_{L^{p′(V ′)}}^{p′}]^{1/2} ≤ K′

where K′ is a positive constant depending on z_{0}, \overline{u} and f (because ∥h∥ ≤ 1). We obtain

∥z∥_{L^{∞}(H)} ≤ K′_{1}∥h∥_{L^{r}(0,T)},
∥z∥_{L^{2}(V)} ≤ K′_{2}∥h∥_{L^{r}(0,T)}
□

Theorem 4.4. Suppose that:

(i) The hypothesis of Lemma 4.3 are satisfied with f = 0. (ii) For φ and Ψ in C([0, T); H) with ∥Ψ∥_{C([0,T];H)} ≤ 1 we have

∥A′(φ(t) + Ψ(t)) − A′(φ(t))∥_{L(H)} ≤ γ(t)|Ψ(t)|_{H}

where γ ∈ L^{1}(0, T). Then the mapping Θ: L^{r}(0, T) → L^{∞}(H) ∩ L^{2}(V) is Fréchet differentiable and the derivative Θ′ ; \overline{u}.h is a solution of

\begin{matrix}ẏ(t) + A′_{\overline{z}(t)}y(t) = \overline{u}(t)By(t) + h(t)B\overline{z}(t) \\ y(0) = 0 \end{matrix}  (4.1)

where \overline{z} = Θ(\overline{u}).

Proof. 1. Since A is strongly monotone. For λ > 0, φ and Ψ in V, we have

⟨\frac{1}{λ}(A(φ + λΨ) − A(φ)), Ψ⟩ ≥ β∥Ψ∥_{V}^{2} .

Hence ⟨A′_{φ}Ψ, Ψ⟩ ≥ β∥Ψ∥_{V}^{2}

2. For \overline{u} ∈ L^{r}(0, T), the mapping h ↦ y defined by (4.1) is linear. Multiplying (4.1) by y and integrating on [0, t] we obtain

\frac{1}{2}|y(t)|^{2} + β ∫_{0}^{t} ∥y(τ)∥_{V}^{2} dτ ≤ b ∫_{0}^{t} |\overline{u}(τ)||y(τ)|^{2} dτ + |h(τ)||\overline{z}(τ)||y(τ)| dτ

By Lemma 4.3,

|y(t)| ≤ b ∫_{0}^{t} |h(τ)||\overline{z}(τ)| dτ exp [b ∫_{0}^{t} |\overline{u}(τ)|],

but

∥\overline{z}∥_{L^{∞}(H)} ≤ K_{1}[|z_{0}|^{2} + ∥\overline{u}∥_{L^{r}(0,T)}^{r}]^{1/2}.

Then

∥y∥_{L^{∞}(H)} ≤ K′_{1}∥h∥_{L^{r}(0,T)},
∥y∥_{L^{2}(V)} ≤ K′_{2}∥h∥_{L^{r}(0,T)},

where K′_{i} are positive constants depending on z_{0} and \overline{u}. Hence the mapping h ↦ y is continuous.

3. Set \overline{z} = Θ(\overline{u}), z_{h} = Θ(\overline{u} + h), z = z_{h} − \overline{z} and w = z − y where y is a solution of (4.1). We have

\begin{matrix}ẇ(t) + A′_{\overline{z}(t)}w(t) = \overline{u}(t)Bw(t) + h(t)Bz(t) + g(t) \\ w(0) = 0 \end{matrix}  (4.2)

where

g(t) = A′_{\overline{z}(t)}z(t) − (Az_{h}(t) − A\overline{z}(t))
= ∫_{0}^{1}[A′\overline{z}(t) − A′(\overline{z}(t) + sz(t))]z(t) ds.

Then

|g(t)|_{H} ≤ ∫_{0}^{1} γ(t)|sz(t)||z(t)| ds = \frac{γ(t)}{2}|z(t)|^{2}

On the other hand, multiplying (4.2) by w and integrating on [0, t] we obtain

\frac{1}{2}|w(t)|^{2} + β ∫_{0}^{t} ∥w(τ)∥_{V}^{2} dτ

≤ b ∫_{0}^{t} |\overline{u}(τ)||w(τ)|^{2} dτ + b ∫_{0}^{t} |h(τ)||z(τ)||w(τ)| dτ + \frac{1}{2} ∫_{0}^{t} γ(τ)|z(τ)|^{2}|w(τ)| dτ

Then Lemma 4.3 gives

|w(t)| ≤ exp (b ∫_{0}^{t} |\overline{u}(τ)| dτ)[b ∫_{0}^{t} |h(τ)||z(τ)| dτ + \frac{1}{2} ∫_{0}^{t} γ(τ)|z(τ)|^{2} dτ],

but ∥z∥_{L^{∞}(H)} ≤ K∥h∥_{L^{r}(0,T)} then

∥w∥_{L^{∞}(H)} ≤ K_{1}∥h∥_{L^{r}(0,T)}^{2},
∥w∥_{L^{2}(V)} ≤ K_{2}∥h∥_{L^{r}(0,T)}^{2}.

It follows that Θ is fréchet differentiable from L^{r}(0, T) on L^{∞}(H)∩L^{2}(V) and Θ′ ; \overline{u}.h is a solution of (4.1). □

Theorem 4.5. Assume the hypotheses of Theorem 4.4 and (H6) hold. Then an optimal control \overline{u}, its corresponding state \overline{z}, and its adjoint state p are necessarily tied by the optimality system:

(1) ˙ \overline{z} + A\overline{z} = \overline{u}B\overline{z} \overline{z}(0) = z_{0}

(2) −ṗ + A′∗ ; \overline{z}p = \overline{u}B^{∗}p + ∂_{1}L(\overline{z}(t), \overline{u}(t)) p(T) = 0

(3) ⟨B\overline{z}(t), p(t)⟩ + ∂_{2}L(\overline{z}(t), \overline{u}(t)) = 0 a.e. in [0, T]

Proof. Since L is Fréchet differentiable, we deduce that the functional

J(u) = ∫_{0}^{T} L(z(t), u(t)) dt

is Fréchet differentiable on L^{r}(0, T). Since \overline{u} is a minimum point for J,

J ′ ; \overline{u}.h = 0, ∀h ∈ L^{r}(0, T)

but

J ′ ; \overline{u}.h = ∫_{0}^{T} ⟨∂_{1}L(\overline{z}(t), \overline{u}(t), y(t)⟩ dt + ∫_{0}^{T} ⟨∂_{2}L(\overline{z}(t), \overline{u}(t), h(t)⟩ dt

where y = Θ′ ; \overline{u}.h. We define p by (2), then

J′(u).h = ∫_{0}^{T} ⟨−ṗ(t) + A′_{\overline{z}(t)}^{∗}p(t) − \overline{u}(t)B^{∗}p(t), y(t)⟩ dt + ∫_{0}^{T} h(t)∂_{2}L(\overline{z}(t), \overline{u}(t)) dt
= ∫_{0}^{T} ⟨p(t), ẏ(t) + A′_{\overline{z}(t)}y(t) − \overline{u}(t)By(t)⟩ dt + ∫_{0}^{T} h(t)∂_{2}L(\overline{z}(t), \overline{u}(t)) dt
= ∫_{0}^{T} [⟨p(t), B\overline{z}(t)⟩ + ∂_{2}L(\overline{z}(t), \overline{u}(t))]h(t) dt

Hence part (3) of the theorem is consequence of the above equality. □

5. Example

In this section, we present an example which illustrates the application of the results of the theory developed in the previous sections. Let Ω be a bounded domain in ℝ^{N} with smooth boundary Γ = ∂Ω. We consider the control problem (2.1) with

J(u) = ∫_{Q} |z(x, t)|^{4} dx dt + ∫_{0}^{T} |u(t)|_{ℝ^{N}}^{2} dt

Where z satisfies the nonlinear evolution equation

\frac{∂z}{∂t} − div(|∇z|^{2}∇z) = ∑_{i=1}^{N} u_{i}(t)\frac{∂z}{∂x_{i}} in Q = Ω×]0, T [

z = 0 in Σ = Γ×]0, T [
z(x, 0) = z_{0}(x)

(5.1)

Setting V = W_{0}^{1,4}(Ω), H = L^{2}(Ω) and V′ = W^{−1,4/3}(Ω) we have V ↪ H ↪ V′ continuously and densely. Furthermore V ↪ H compactly.

The equation (5.1) can be written in the form

ż(t) + Az(t) = u(t)Bz(t)
z(0) = z_{0}

where

(1) A: V → V′, φ ↦ − div(|∇φ|^{2}∇φ) which satisfies (H1) (see [6]).

(2) B = (B_{1}, …, B_{N}) with B_{i}: V → H, φ ↦ B_{i}φ = φ_{x_{i}}. Hence ∥B_{i}φ∥_{H} ≤

b_{i}∥φ∥_{V}, b_{i} > 0 and ∥Bφ∥_{H^{N}} ≤ b∥φ∥_{V}.
(3) u = (u_{1}, …, u_{N}) ∈ U = L^{2}(0, T; ℝ^{N}) . Here

u(t)Bz(t) = ∑_{i=1}^{N} u_{i}(t)B_{i}z(t) = ∑_{i=1}^{N} u_{i}(t)\frac{∂z(t)}{∂x_{i}}.

The cost function becomes

J(u) = ∥u∥_{L^{2}(0,T;ℝ^{N})}^{2} + ∥z∥_{L^{4}(0,T;Q)}^{4}

Since ∫_{Ω} u(t)Bz(x, t)z(x, t) dx = 0, the a priori estimates given by Lemma 3.2 become

∥z∥_{L^{4}(V)} ≤ K_{1}|z_{0}|^{1/2},
∥z∥_{L^{∞}(H)} ≤ K_{2}|z_{0}|,
∥ż∥_{L^{4/3}(V′)} ≤ K_{3}[∥z∥_{L^{4}(V)}^{3/2} + ∥u∥_{L^{2}(0,T;ℝ^{N})}^{3/2}]

Corollary 5.1. For z_{0} in L^{2}(Ω) and u in L^{2}(0, T; ℝ^{N}), the equation (3.1) with f = 0 admits a unique solution which satisfies

z ∈ L^{∞}(0, T; L^{2}(Ω)) ∩ L^{4}(0, T; W_{0}^{1,4}(Ω)),
ż ∈ L^{4/3}(0, T; W^{−1,4/3}(Ω))

Proposition 5.2. The mapping Θ: U → C([0, T]; H), u ↦ z, with z the solution of (3.1) with f = 0. is differentiable in the sense of Fréchet, and Θ′ ; \overline{u}.h satisfies

\begin{matrix}ẏ + A′_{\overline{z}(t)}y(t) = \overline{u}(t)By(t) + h(t)B\overline{z}(t) \\ y(0) = 0, \end{matrix}  (5.2)

where z = (Θ(u))(t) and

A′_{φ}.h = − ∑_{i=1}^{N} [|∇φ|^{2}h_{x_{i}} + 2⟨∇φ, ∇h⟩_{1}φ_{x_{i}}]_{x_{i}}

with ⟨∇φ, ∇h⟩_{1} = ∑_{i=1}^{N} φ_{x_{i}}h_{x_{i}} (φ and h ∈ V).

Proof. 1. For φ ∈ V, the mapping, A′_{φ}: V → V′,

h ↦ A′_{φ}h = − ∑_{i=1}^{N} [|∇φ|^{2}h_{x_{i}} + 2⟨∇φ, ∇h⟩_{1}φ_{x_{i}}]_{x_{i}}

is linear and for v ∈ V we have

⟨A′_{φ}h, v⟩_{V′,V} = ∑_{i=1}^{N} ∫_{Ω} f_{i}v_{x_{i}} dx

with f_{i} = |∇φ|^{2}h_{x_{i}} + 2⟨∇φ, ∇h⟩_{1}φ_{x_{i}}. Furthermore,

∥f_{i}∥_{L^{4/3}(Ω)}^{4/3} ≤ K_{1}[ ∫_{Ω} |∇φ|^{8/3}|h_{x_{i}}|^{4/3} dx + ∫_{Ω} |⟨∇φ, ∇h⟩_{1}|^{4/3}|φ_{x_{i}}|^{4/3} dx]
≤ 2K_{1}[( ∫_{Ω} |∇φ|^{4})^{1/4}( ∫_{Ω} |∇h|^{4})^{1/3}].

Then

∥f_{i}∥_{L^{4/3}(Ω)} ≤ K∥φ∥_{W_{0}^{1,4}(Ω)}^{2}∥h∥_{W_{0}^{1,4}(Ω)}.

Using the norm in V′ it follows that A′_{φ} ∈ L(V, V′).

For φ and h in V, we have

A(φ + h) − A(φ) − A′_{φ}(h) = F (φ, h)

where

F (φ, h) = − ∑_{i=1}^{N}[|∇h|^{2}h_{x_{i}} + |∇h|^{2}φ_{x_{i}} + 2⟨∇φ, ∇h⟩_{1}h_{x_{i}}]_{x_{i}} .

For v ∈ V,

⟨F, v⟩_{V′,V} = ∑_{i=1}^{N} ∫_{Ω} f_{i}v_{x_{i}} dx,

where

f_{i} = |∇h|^{2}φ_{x_{i}} + |∇h|^{2}h_{x_{i}} + 2⟨∇φ, ∇h⟩_{1}h_{x_{i}}.

Then

∥f_{i}∥_{L^{4/3}(Ω)}^{4/3} ≤ K′[ ∫_{Ω} |∇φ|^{4/3}|∇h|^{8/3} dx + ∫_{Ω} |∇φ|^{4/3}|∇h|^{8/3} dx + ∫_{Ω} |∇h|^{4} dx]
≤ K′′[∥φ∥_{W_{0}^{1,4}(Ω)}^{4/3}∥h∥_{W_{0}^{1,4}(Ω)}^{8/3} + ∥h∥_{W_{0}^{1,4}(Ω)}^{4}].

We deduce that

∥A(φ + h) − A(φ) − A′_{φ}(h)∥_{V′} ≤ K[∥φ∥_{V} ∥h∥_{V}^{2} + ∥h∥_{V}^{3} ∥]

Hence A is differentiable in the sense of Frechet.

2. The equation (5.2) admits a unique solution satisfying

y ∈ L^{2}(V) ∩ L^{∞}(H), ẏ ∈ L^{2}(V′).

The existence follows from a standard application of the Galerkin method and the a priori estimates obtained for (4.1). We remark that y ∈ C([0, T]; H).

3. The mapping A′: V → L(V, V′), φ ↦ A′_{φ} is locally Lipschitz. Let φ and ψ be in V with ψ in neighbourhood of 0. For h in V, we have

A′_{φ+ψ}h = − ∑_{i=1}^{N} [|∇(φ + ψ)|^{2}h_{x_{i}} + 2⟨∇(φ + ψ), ∇h⟩_{1}((φ + ψ))_{x_{i}}]_{x_{i}}

and (A′_{φ+ψ} − A′_{φ})h = F, where

F = − ∑_{i=1}^{N} [|∇ψ|^{2}h_{x_{i}} + 2⟨∇φ, ∇ψ⟩_{1}h_{x_{i}} + 2⟨∇φ, ∇h⟩_{1}ψ_{x_{i}}
+ 2⟨∇ψ, ∇h⟩_{1}φ_{x_{i}} + 2⟨∇ψ, ∇h⟩_{1}ψ_{x_{i}}]_{x_{i}}.

Then for v ∈ V,

⟨F, v⟩_{V′,V} = ∑_{i=1}^{N} ∫_{Ω} f_{i}v_{x_{i}} dx,

where

f_{i} = |∇ψ|^{2}h_{x_{i}} +2⟨∇φ, ∇ψ⟩_{1}h_{x_{i}} +2⟨∇ψ, ∇h⟩_{1}φ_{x_{i}} +2⟨∇ψ, ∇h⟩_{1}ψ_{x_{i}} +2⟨∇φ, ∇h⟩_{1}ψ_{x_{i}}.

Hence

∥f_{i}∥_{L^{p′} (Ω)} ≤ K[∥ψ∥_{V}^{2} + ∥ψ∥_{V} ∥φ∥_{V} ]∥h∥_{V} .

Since ψ is in neighbourhood of 0,

∥(A′_{φ+ψ} − A′_{φ})h∥_{V′} ≤ K′∥φ∥_{V} ∥ψ∥_{V} ∥h∥_{V} .

Hence

∥A′_{φ+ψ} − A′_{φ}∥_{L(V,V′)} ≤ K′′∥ψ∥_{V} .

It follows by theorem 4.4. that Θ is Frechet differentiable and its derivative Θ′_{u}.h satisfies (5.2) □

Now the functional J can be written as J(u) = ∫_{0}^{T} L(z(t), u(t)) dt with L satisfying (H6).

The differentiability of Θ and the norm ensures the differentiability of J and the expression of derivative is

dJ(u).h = 4 ∫_{Q} |z(x, t)|^{2}z(x, t)y(x, t) dx dt + 2 ∫_{0}^{T} ⟨u(t), h(t)⟩_{ℝ^{N}} dt

where y = Θ′_{u}h.

From Theorems 3.1, 4.4 and 4.5, we get the following result.

Corollary 5.3. An optimal control \overline{u}, its corresponding state \overline{z}, and its adjoint state p are necessarily tied by the optimality system: For 1 ≤ i ≤ N and t ∈ [0, T],

\overline{u}_{i}(t) = −2 ∫_{Ω} p(x, t)\frac{∂z}{∂x_{i}} (x, t) dx

\frac{∂\overline{z}}{∂t} − div(|∇\overline{z}|^{2}∇\overline{z} = ∑_{i=1}^{N} \overline{u}_{i}(t)\frac{∂\overline{z}}{∂x_{i}} in Q

\overline{z}(x, t) = 0 in Σ
\overline{z}(x, 0) = z_{0}(x) in Ω

−\frac{∂\overline{p}}{∂t} + A′_{\overline{z}(t)}\overline{p} = − ∑_{i=1}^{N} \overline{u}_{i}(t)\frac{∂\overline{p}}{∂x_{i}} + |\overline{z}(x, t)|^{2}\overline{z}(x, t) in Q

\overline{p}(x, t) = 0 in Σ

\overline{p}(x, T) = 0 in Ω

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Abdelhaq Benbrik Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc

E-mail address: benbrik@sciences.univ-oujda.ac.ma

Mohammed Berrajaa Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc

E-mail address: berrajaa@sciences.univ-oujda.ac.ma

Samir Lahrech Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc

E-mail address: lahrech@sciences.univ-oujda.ac.ma