Lobachevskii Journal of Mathematics Vol. 6, 2000, 3–17 \begin{matrix}c \\ ◯ \end{matrix}A.Addou, S.Lahrech

A.Addou, S.Lahrech

SUFFICIENT CONDITIONS FOR ELLIPTIC PROBLEM OF OPTIMAL CONTROL IN ℝ^{2}

(submitted by A.V. Lapin)

Abstract. It is known that when we look for sufficient conditions of local extremum for Gâteaux functionals (G.f) associated to Dirichlet problem of second order in ℝ^{2}, the (G.f) is not necesseraly Frechet differentiable. In this note, using a recent extension of Frechet Differentiability, (see [5]), we obtain that the (G.f) is differentiable with respect to the new notion. Thus we can give sufficient conditions for obtaining local minimum.

1. Preliminaries

1.1. Description of the optimization problem. Let A be an elliptic operator of second order :

Au ≡ ∑_{|l|≤1,|s|≤1}(−1)^{l}D^{l}(a_{ls}(x)D^{s}u), where a_{ls}(x) ∈ D(\overline{Q}).

Suppose that Q is sufficiently smooth and bounded in ℝ^{2}, let us consider the problem:

\{ \begin{matrix}Au = f, & (1.1) \\ u/_{∂Q} = 0 & (1.2) \end{matrix}

where Q is an open bounded domain in ℝ^{2}. For this problem, let us state Agmon’s-Douglis-Niremberg’s theorem

Theorem 1.1. [Agmon-Douglis-Niremberg] Let 1 < q < ∞, then we have ∀f ∈ L^{q}(Q), ∃ u ∈ W^{2,q}(Q) ⋂ W_{0}^{1,q}(Q) unique solution of problem (1.1),(1.2). Moreover ∀m ≥ 0 if f ∈ W^{m,q}(Q),then u ∈ W^{m+2,q}(Q) and ∥u∥_{W^{m+2,q}(Q)} ≤ c∥f∥_{W^{m,q}(Q)}.

Let f ∈ F ⊂ W_{0}^{1,2}(Q) be a control. Let us consider

J_{k}(f) = ∫_{Q} v_{k}(x, u, f)dx + c_{k}∥f∥_{W^{1,2}(Q)}^{2}, (k = 0, 1, 2, ..., s_{1})

and J_{k}(f) = ∫_{Q} v_{k}(x, u, f)dx , (k = s_{1}+1, s_{1}+2, ..., s_{1}+s_{2}), where u is a solution

of problem (1.1),(1.2) in W_{0}^{1,2}(Q) ⋂ W^{2,2}(Q). We consider three problems of minimizing the functional J_{0}(f) :

i)J_{0}(f) → min,  (1.3)
ii)J_{0}(f) → min, J(f) = 0 with J = (J_{s_{1}+1}, ......., J_{s_{1}+s_{2}}),  (1.4)
iii)J_{0}(f) → min, J(f) = 0, J_{k}(f) ≤ 0 , (k = 1, 2, ..., s_{1}).  (1.5)

We must choose a control f^{0} in order that a solution u^{0} of the problem (1.1),(1.2) with f = f^{0} satisfies the inequality type : J_{k}(f) ≤ 0 , (1 ≤ k ≤ s_{1}) and the equality type : J_{k}(f) = 0 , (s_{1} + 1 ≤ k ≤ s_{1} + s_{2}) and a functional J_{0}(f) takes a minimum value. This control f^{0} will be called optimal.

1.2. Taylor mapping and lower semi-Taylor mapping. Let ∥.∥_{W^{1,2}(Q)} be the usual norm in W_{0}^{1,2}(Q), F subset of W_{0}^{1,2}(Q), τ be a topology in F, Y a norm space with a norm ∥.∥_{Y}. According to [5] a map r : F ⟶ Y, (respectively r : F ⟶ ℝ), is called (τ, ∥.∥_{W^{1,2}(Q)}) infinitely small, (respectively (τ, ∥.∥_{W^{1,2}(Q)}) lower infinitely semi-small), of order p_{1} at f ∈ F if : ∀ ε > 0, ∃ O_{f} ∈ τ , ∀ h ∈ W_{0}^{1,2}(Q) we have

f + h ∈ O_{f} ⇒ ∥r(f + h)∥_{Y} ≤ ε∥h∥_{W^{1,2}(Q)}^{p_{1}},

(respectively ∀ ε > 0, ∃ O_{f} ∈ τ , ∀ h ∈ W_{0}^{1,2}(Q) we have

f + h ∈ O_{f} ⇒ ∥r(f + h)∥_{Y} ≥ −ε∥h∥_{W^{1,2}(Q)}^{p_{1}}),

where O_{f} is a neighbourhood of f in (F, τ).

A map J : F → Y,(respectively J : F → ℝ), is called a (τ, ∥.∥_{W^{1,2}(Q)}) Taylor mapping (respectively a (τ, ∥.∥_{W^{1,2}(Q)}) lower semi-Taylor mapping) of order p_{1} at f ∈ F if there exists such symmetrical linear maps J^{(k)}(f) : W_{0}^{1,2}(Q) → Y (not necesseraly continuous), (respectively J^{(k)}(f) : W_{0}^{1,2}(Q) → ℝ (k = 1, .., p_{1})) and a neighbourhood O_{f} ∈ τ: ∀ h ∈ W_{0}^{1,2}(Q) we have

J(f+h)−J(f) = J^{(1)}(f)h+2^{−1}J^{(2)}(f)(h, h)+...(p_{1})!^{−1}J^{(p_{1})}(f)(h, ...h)+r(f+h);

where r is (τ, ∥.∥_{W^{1,2}(Q)}) infinitely small (respectively (τ, ∥.∥_{W^{1,2}(Q)}) lower infinitely semi-small), of order p_{1} at f ∈ F. Let us remark that (J^{(1)}(f), .....J^{(p_{1})}(f)) are not unique. Let us solve the three problems (1.3),(1.4) and (1.5). For the problem (1.5) let us introduce the lagrangian function :

L(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}(f)+ < y^{∗}, J(f) >, (1.6)

L_{f}(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}^{(1)}(f)+ < y^{∗}, J^{(1)}(f) >, (1.7)

L_{ff}(f, y^{∗}, λ, λ_{0}) = ∑_{k=0}^{s_{1}} λ_{k}J_{k}^{(2)}(f)+ < y^{∗}, J^{(2)}(f) >, (1.8)

where λ_{0} ∈ ℝ, y^{∗} ∈ (ℝ^{s_{2}})^{∗},λ ∈ (ℝ^{s_{1}})^{∗}. Also for the problem (1.4), let us introduce the lagrangian function : L(f, y^{∗}, λ_{0}) = λ_{0}J_{0}(f)+ < y^{∗}, J(f) >, (1.9) L_{f}(f, y^{∗}, λ_{0}) = λ_{0}J_{0}^{(1)}(f)+ < y^{∗}, J^{(1)}(f) >, (1.10) L_{ff}(f, y^{∗}, λ_{0}) = λ_{0}J_{0}^{(2)}(f)+ < y^{∗}, J^{(2)}(f) >, (1.11) where λ_{0} ∈ ℝ, y^{∗} ∈ (ℝ^{s_{2}})^{∗}.

Let us formulate the following lemma where the proof can be traced back to [5].

Lemma 1.1. Let (Ω, ∑, μ) be a measurable space with a finite measure σ and X be a complete metric linear space in M(Ω). By M(Ω) we denote the set of a measurable functions and almost everywhere finite x : Ω ⟶ ℝ with a metrizable topology τ(mes). Suppose that ∀ x ∈ X | x |∈ X and ρ(x, 0) = ρ(| x | , 0) ∀ x ∈ X. Then, ∀ x_{n} → 0 in X , ∃ (x_{n_{k}})−subsequence of (x_{n}) and z ∈ X : | x_{n_{k}}(s) |≤ z(s) almost everywhere in Ω.

2. Sufficient conditions of local minimum for Gâteaux functional of second order Dirichlet problem

Let Q be a bounded domain in ℝ^{2} and F subspace of W_{0}^{1,2}(Q). Let G be a functional defined by: G(f) = ∫_{Q} v(x, u(x), f(x))dx , where u(x) is a solution

of problem (1.1),(1.2), and v : Q × ℝ × ℝ → ℝ. By τ_{p} we denote the topology generated by L^{p}(Q).

Theorem 2.1. Suppose that with every requirements (1.1) (1.2) following conditions hold true: v_{uf}^{(2)}, v_{fu}^{(2)} are continuous in Q × ℝ × ℝ, p > 1. Let us suppose also that

\begin{matrix}| v(x, u, f) | + | v_{u}^{(1)}(x, u, f) | + | v_{f}^{(1)}(x, u, f) |≤ a(| u |^{q+2} + | f |^{q+2}) + b_{3}(x), \\ | v_{uu}^{(2)}(x, u, f) | +2 | v_{uf}^{(2)}(x, u, f) | + | v_{ff}^{(2)}(x, u, f) |≤ a(| u |^{q} + | f |^{q}) + b_{4}(x), \end{matrix}

where q ∈ [0, p[, b_{3}, b_{4} ∈ L^{α}(Q) (α > 1). Then G is a (τ_{p}, ∥.∥_{W^{1,2}(Q)}) Taylor mapping of second order at every f ∈ F. Moreover G^{(2)}(f) ∈ B((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ) and G^{(1)}(f) ∈ L((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ).

Proof Let us prove first that the functional G is finite . We have

|G(f)|  = |∫_{Q} v(x, u, f)dx| ≤ ∫_{Q} |v(x, u, f)|dx

≤ a(∫_{Q} |u(x)|^{q+2} dx + ∫_{Q} |f(x)|^{q+2} dx) + ∫_{Q} b_{3}(x)dx

≤ c_{1}(∥u(x)∥_{W^{1,2}(Q)}^{q+2} + ∥f(x)∥_{W^{1,2}(Q)}^{q+2}) + ∥b_{3}(x)∥_{L^{1}(Q)}

< ∞.

Thus the functional G is finite.

Let R : W_{0}^{1,2}(Q) → W_{0}^{1,2}(Q), h → (R(h))(x), where (R(h))(x) is a solution of problem

\{ \begin{matrix}Au = h \\ u/_{∂Q} = 0 \end{matrix}  \begin{matrix}(2.1) \\ (2.2) \end{matrix}

6 A.ADDOU, S.LAHRECH

Such a solution exists for all h ∈ W_{0}^{1,2}(Q). Let G^{(1)}(f) be defined by

G^{(1)}(f)h = lim_{λ→0} λ^{−1}(G(f + λh) − G(f))
= lim_{λ→0} λ^{−1} ∫_{Q} [v(x, u + λR(h), f + λh) − v(x, u, f)]dx
= lim_{λ→0} λ^{−1} ∫_{Q} [v(x, u + λR(h), f + λh) − v(x, u, f + λh) + v(x, u, f + λh)
−v(x, u, f)]dx
= lim_{λ→0} ∫_{Q} [∫_{0}^{1} v_{u}^{(1)}(x, u + θλR(h), f + λh)R(h)dθ + ∫_{0}^{1} v_{f}^{(1)}(x, u, f + ρλh)hdρ]dx
= lim_{λ→0} ∫_{Q} [∫_{0}^{1}[v_{u}^{(1)}(x, u + θλR(h), f + λh) − v_{u}^{(1)}(x, u, f)]R(h)dθ
+ ∫_{0}^{1} v_{u}^{(1)}(x, u, f)R(h)dθ + ∫_{0}^{1}[v_{f}^{(1)}(x, u, f + ρλh) − v_{f}^{(1)}(x, u, f)]hdρ
+h ∫_{0}^{1} v_{f}^{(1)}(x, u, f)dρ]dx
= ∫_{Q} v_{u}^{(1)}(x, u, f)R(h)dx + ∫_{Q} hv_{f}^{(1)}(x, u, f)dx.

Therefore,

G^{(1)}(f) = ∫_{Q} v_{u}^{(1)}(x, u, f)R(h)dx + ∫_{Q} hv_{f}^{(1)}(x, u, f)dx.  (2.3)

Let G^{(2)}(f) be defined by :

G^{(2)}(f)(h_{1}, h_{2}) = lim_{λ→0} λ^{−1}[G^{(1)}(f + λh_{2}) − G^{(1)}(f)]h_{1}
= lim_{λ→0} λ^{−1}[∫_{Q} [v_{u}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{u}^{(1)}(x, u, f)]R(h_{1})dx

+ ∫_{Q} [v_{f}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{f}^{(1)}(x, u, f)]h_{1}dx]

= lim_{λ→0} λ^{−1}[∫_{Q} [v_{u}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{u}^{(1)}(x, u, f + λh_{2})

+v_{u}^{(1)}(x, u, f + λh_{2}) − v_{u}^{(1)}(x, u, f)]R(h_{1})dx SUFFICIENT CONDITIONS FOR ELLIPTIC PROBLEM 7

+ ∫_{Q} [v_{f}^{(1)}(x, u + λR(h_{2}), f + λh_{2}) − v_{f}^{(1)}(x, u, f + λh_{2})

+v_{f}^{(1)}(x, u, f + λh_{2}) − v_{f}^{(1)}(x, u, f)]h_{1}dx]

= lim_{λ→0} λ^{−1}[∫_{Q} [∫_{0}^{1} v_{uu}^{(2)}(x, u + θλR(h_{2}), f + λh_{2})λR(h_{2})dθ

+ ∫_{0}^{1} v_{fu}^{(2)}(x, u, f + ρλh_{2})λh_{2}dρ]R(h_{1})dx

+ ∫_{Q} [∫_{0}^{1} v_{uf}^{(2)}(x, u + θλR(h_{2}), f + λh_{2})λR(h_{2})dθ

+ ∫_{0}^{1} v_{ff}^{(2)}(x, u, f + ρλh_{2})λh_{2}dρ]h_{1}dx]

= ∫_{Q} v_{uu}^{(2)}(x, u, f)R(h_{1})R(h_{2})dx + ∫_{Q} v_{uf}^{(2)}(x, u, f)R(h_{1})h_{2}dx

+ ∫_{Q} v_{fu}^{(2)}(x, u, f)h_{1}R(h_{2})dx + ∫_{Q} v_{ff}^{(2)}(x, u, f)h_{1}h_{2}dx. (2.4)

The linearity and bilinearity of G^{(1)}(f) and G^{(2)}(f) is evident.

Let us prove now that they are bounded. Indeed,

|G^{(1)}(f)h| ≤ ∫_{Q} |v_{u}^{(1)}(x, u, f)||R(h)|dx + ∫_{Q} |v_{f}^{(1)}(x, u, f)||h|dx
≤ ∫_{Q} [a(|u(x)|^{q+2} + |f(x)|^{q+2}) + |b_{3}(x)|][|R(h)| + |h)|]dx
≤ [a[(∫_{Q} |u(x)|^{\frac{2α}{α+1} (q+2)}dx)^{\frac{α+1}{2α}} + (∫_{Q} |f(x)|^{\frac{2α}{α+1} (q+2)}dx)^{\frac{α+1}{2α}} ] + (∫_{Q} |b_{3}(x)|^{\frac{2α}{α+1}} dx)^{\frac{α+1}{2α}} ]

[[∫_{Q} (|R(h)|)^{\frac{2α}{α−1}} dx]^{\frac{α−1}{2α}} + [∫_{Q} (|h|)^{\frac{2α}{α−1}} dx]^{\frac{α−1}{2α}} ]

= [a(∥u(x)∥_{L^{\frac{2α}{α+1} (q+2)}(Q)}^{q+2} + ∥f(x)∥_{L^{\frac{2α}{α+1} (q+2)}(Q)}^{q+2}) + ∥b_{3}(x)∥_{L^{\frac{2α}{α+1}} (Q)}]

[∥R(h)∥_{L^{\frac{2α}{α−1}} (Q)} + ∥h∥_{L^{\frac{2α}{α−1}} (Q)}]

≤ c_{2}[∥R(h)∥_{W^{1,2}(Q)} + ∥h∥_{W^{1,2}(Q)}][a(∥u∥_{W^{1,2}(Q)}^{q+2} + ∥f∥_{W^{1,2}(Q)}^{q+2}) + ∥b_{3}(x)∥_{L^{α}(Q)}].

Thus, there exists c_{2} > 0 such that |G^{(1)}(f)h| ≤ c_{2}(∥R(h)∥_{W^{1,2}(Q)}+∥h∥_{W^{1,2}(Q)}).

Since R(h) depends continuously on h, |G^{(1)}(f)h| ≤ c_{3}∥h∥_{W^{1,2}(Q)}, where c_{3} > 0. Consequently , G^{(1)}(f) ∈ L((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ).

|G^{(2)}(f)(h_{1}, h_{2})| ≤ ∫_{Q} |v_{uu}^{(2)}(x, u, f)R(h_{1})R(h_{2}|dx + ∫_{Q} |v_{uf}^{(2)}(x, u, f)R(h_{1})h_{2}|dx

+ ∫_{Q} |v_{fu}^{(2)}(x, u, f)h_{1}R(h_{2})|dx + ∫_{Q} |v_{ff}^{(2)}(x, u, f)h_{1}h_{2}|dx

≤ ∫_{Q} |v_{uu}^{(2)}(x, u, f)R(h_{1})R(h_{2})|dx

+ ∫_{Q} 2|v_{uf}^{(2)}(x, u, f)|[|R(h_{1})||h_{2}| + |R(h_{2})||h_{1}|]dx

+ ∫_{Q} |v_{ff}^{(2)}(x, u, f)h_{1}h_{2}|dx

≤ ∫_{Q} [|v_{uu}^{(2)}(x, u, f)| + 2|v_{uf}^{(2)}(x, u, f)| + |v_{ff}^{(2)}(x, u, f)|]

[|R(h_{1})R(h_{2})| + |R(h_{1})h_{2}| + |h_{1}R(h_{2})| + |h_{1}h_{2}|]dx

≤ ∫_{Q} [a(|u|^{q} + |f|^{q}) + |b_{4}(x)|]

[|R(h_{1})R(h_{2})| + |R(h_{1})h_{2}| + |h_{1}R(h_{2})| + |h_{1}h_{2}|]dx

≤ [a(∥u(x)∥_{L^{αq}(Q)}^{q} + ∥f(x)∥_{L^{αq}(Q)}^{q}) + ∥b_{4}(x)∥_{L^{α}(Q)}]

[∥R(h_{1})∥_{L^{\frac{2α}{α−1}} (Q)}∥R(h_{2})∥_{L^{\frac{2α}{α−1}} (Q)} + ∥R(h_{1})∥_{L^{\frac{2α}{α−1}} (Q)}∥h_{2}∥_{L^{\frac{2α}{α−1}} (Q)}

+∥h_{1}∥_{L^{\frac{2α}{α−1}} (Q)}∥R(h_{2})∥_{L^{\frac{2α}{α−1}} (Q)} + ∥h_{1}∥_{L^{\frac{2α}{α−1}} (Q)}∥h_{2}∥_{L^{\frac{2α}{α−1}} (Q)}]

≤ c_{4}∥h_{1}∥_{W^{1,2}(Q)}∥h_{2}∥_{W^{1,2}(Q)}.

Consequently, G^{(2)}(f) ∈ B((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ). Let us prove now that G is (τ_{p}, ∥.∥_{W^{1,2}(Q)}) Taylor mapping, where τ_{p} is a topology generated by L^{p}(Q). Let f ∈ F, and let us prove that

r(h) ≡ G(f + h) − G(f) − G^{(1)}(f)h − 2^{−1}G^{(2)}(f)(h, h)

is (τ_{p}, ∥.∥_{W^{1,2}(Q)}) infinitely small of second order at zero.

Assume the contrary, then ∃(h̃_{m})_{m∈ℕ} ∈ F and ε > 0 such that h̃_{m} → 0 in L^{p}(Q) and r(h̃_{m}) ≥ ε∥h̃_{m}∥_{W^{1,2}(Q)}^{2}.

Using the Agmon’s-Douglis-Niremberg’s theorem, we obtain R(h̃_{m}) → 0 in L^{p}(Q), and using the lemma (1.1), we deduce that there exists Z̃(x) in L^{p}(Q) : |(R(h̃_{m}))(x)| ≤ z̃(x). let Z̃_{0}(x) = z̃(x) + |u(x)|, then |u(x)| + |(R(h̃_{m}))(x)| ≤ Z̃_{0}(x) and Z_{0} ∈ L^{p}(Q). Analogously, for f ∈ W^{1,2}(Q), we obtain |f(x)|+|h̃_{m}| ≤ Z̃_{1}, where Z_{1} ∈ L^{p}(Q) We have

r(h) = ∫_{Q} [v(x, u + R(h), f + h) − v(x, u, f) − v_{u}^{(1)}(x, u, f)R(h) − v_{f}^{(1)}(x, u, f)h

−2^{−1}[v_{uu}^{(2)}(x, u, f)R^{2}(h) + 2v_{uf}^{(2)}(x, u, f)R(h)h + v_{ff}^{(2)}(x, u, f)h^{2}]]dx.

And since

v(x, u + R(h), f + h) − v(x, u, f) = v(x, u + R(h), f + h) − v(x, u, f + h)
+v(x, u, f + h) − v(x, u, f)
= ∫_{0}^{1} v_{u}^{(1)}(x, u + θR(h), f + h)R(h)dθ
+ ∫_{0}^{1} v_{f}^{(1)}(x, u, f + λh)hdλ
= ∫_{0}^{1} R(h)[v_{u}^{(1)}(x, u + θR(h), f + h)
−v_{u}^{(1)}(x, u + θR(h), f) + v_{u}^{(1)}(x, u + θR(h), f)]dθ
+ ∫_{0}^{1} v_{f}^{(1)}(x, u, f + λh)hdλ
= ∫_{0}^{1} R(h)v_{u}^{(1)}(x, u + θR(h), f)dθ
+ ∫_{0}^{1} v_{f}^{(1)}(x, u, f + λh)hdλ
+ ∫_{0}^{1} ∫_{0}^{1} v_{fu}^{(2)}(x, u + θR(h), f + λh)hR(h)dλdθ,

then

r(h) = ∫_{Q} ∫_{0}^{1}[v_{u}^{(1)}(x, u + θR(h), f + h)R(h) − v_{u}^{(1)}(x, u, f)R(h) − 2^{−1}v_{uu}^{(2)}(x, u, f)R^{2}(h)]dθdx

+ ∫_{Q} ∫_{0}^{1}[v_{f}^{(1)}(x, u, f + λh)h − v_{f}^{(1)}(x, u, f)h − 2^{−1}v_{ff}^{(2)}(x, u, f)h^{2}]dλdx

− ∫_{Q} ∫_{0}^{1} v_{uf}^{(2)}(x, u, f)R(h)hdλdx + ∫_{Q} ∫_{0}^{1} ∫_{0}^{1} v_{uf}^{(2)}(x, u + θR(h), f + λh)hR(h)dλdθdx

10 A.ADDOU, S.LAHRECH

= ∫_{Q} ∫_{0}^{1}[v_{u}^{(1)}(x, u + θR(h), f) − v_{u}^{(1)}(x, u, f) − θv_{uu}^{(2)}(x, u, f)R(h)]R(h)dθdx

+ ∫_{Q} ∫_{0}^{1}[v_{f}^{(1)}(x, u, f + λh) − v_{f}^{(1)}(x, u, f) − λv_{ff}^{(2)}(x, u, f)h]hdλdx

− ∫_{Q} ∫_{0}^{1} v_{uf}^{(2)}(x, u, f)R(h)hdλdx + ∫_{Q} ∫_{0}^{1}∫_{0}^{1} v_{uf}^{(2)}(x, u + θR(h), f + λh)hR(h)dλdθdx.

Let A_{m}, B_{m} be two functions defined by

A_{m}(x, θ) = \{ \begin{matrix}\frac{v_{u}^{(1)}(x, u + θR(h̃_{m}), f) − v_{u}^{(1)}(x, u, f)}{R(h̃_{m})} − θv_{uu}^{(2)}(x, u, f) & if R(h̃_{m}) ≠ 0 \\ 0 & if R(h̃_{m}) = 0 \end{matrix}

B_{m}(x, λ) = \{ \begin{matrix}\frac{v_{f}^{(1)}(x, u, f + λh̃_{m}) − v_{f}^{(1)}(x, u, f)}{h̃_{m}} − λv_{ff}^{(2)}(x, u, f) & if h̃_{m} ≠ 0 \\ 0 & if h̃_{m} = 0 \end{matrix}

Let F_{m} be defined by

F_{m}(x, θ, λ) = v_{uf}^{(2)}(x, u(x) + θR(h̃_{m}), f + λh̃_{m}) − v_{uf}^{(2)}(x, u(x), f), then

|r(h̃_{m})| = |∫_{0}^{1} ∫_{Q} A_{m}(x, θ)R^{2}(h̃_{m})dxdθ + ∫_{0}^{1} ∫_{Q} B_{m}(x, λ)h̃_{m}^{2}dxdλ

+ ∫_{0}^{1} ∫_{0}^{1} ∫_{Q} F_{m}(x, θ, λ)R(h̃_{m})h̃_{m}dxdλdθ|

so we obtain

|r(h̃_{m})| ≤ ∫_{0}^{1}(∫_{Q} |A_{m}(x, θ)|^{α}dx)^{\frac{1}{α}} dθ[∫_{Q} |R(h̃_{m})|^{\frac{2α}{α−1}} dx]^{\frac{α−1}{α}}

+ ∫_{0}^{1}(∫_{Q} |B_{m}(x, λ)|^{α}dx)^{\frac{1}{α}} dλ[∫_{Q} |h̃_{m}|^{\frac{2α}{α−1}} dx]^{\frac{α−1}{α}}

+ ∫_{0}^{1} ∫_{0}^{1}(∫_{Q} |F_{m}(x, θ, λ)|^{α}dx)^{\frac{1}{α}} dλdθ[(∫_{Q} |R(h̃_{m})|^{\frac{2α}{α−1}} dx)^{\frac{1}{2}} ]^{\frac{α−1}{α}} [(∫_{Q} |h̃_{m}|^{\frac{2α}{α−1}} dx)^{\frac{1}{2}} ]^{\frac{α−1}{α}}

≤ ∫_{0}^{1}(∫_{Q} |A_{m}(x, θ)|^{α}dx)^{\frac{1}{α}} dθ∥R(h̃_{m})∥_{L^{\frac{2α}{α−1}} (Q)}^{2} + ∫_{0}^{1}(∫_{Q} |B_{m}(x, λ)|^{α}dx)^{\frac{1}{α}} dλ∥h̃_{m})∥_{L^{\frac{2α}{α−1}} (Q)}^{2}

+ ∫_{0}^{1} ∫_{0}^{1}(∫_{Q} |F_{m}(x, θ, λ)|^{α}dx)^{\frac{1}{α}} dλdθ∥R(h̃_{m})∥_{L^{\frac{2α}{α−1}} (Q)}∥h̃_{m})∥_{L^{\frac{2α}{α−1}} (Q)}

≤ c_{5} ∫_{0}^{1}(∫_{Q} |A_{m}(x, θ)|^{α}dx)^{\frac{1}{α}} dθ∥R(h̃_{m})∥_{W^{1,2}(Q)}^{2}+ c_{5} ∫_{0}^{1}(∫_{Q} |B_{m}(x, λ)|^{α}dx)^{\frac{1}{α}} dλ∥h̃_{m})∥_{W^{1,2}(Q)}^{2}

+c_{5} ∫_{0}^{1} ∫_{0}^{1}(∫_{Q} |F_{m}(x, θ, λ)|^{α}dx)^{\frac{1}{α}} dλdθ∥R(h̃_{m})∥_{W^{1,2}(Q)}∥h̃_{m})∥_{W^{1,2}(Q)}

≤ c_{6}∥h̃_{m})∥_{W^{1,2}(Q)}^{2}[∫_{0}^{1}(∫_{Q} |A_{m}(x, θ)|^{α}dx)^{\frac{1}{α}} dθ + ∫_{0}^{1}(∫_{Q} |B_{m}(x, λ)|^{α}dx)^{\frac{1}{α}} dλ

+ ∫_{0}^{1} ∫_{0}^{1}(∫_{Q} |F_{m}(x, θ, λ)|^{α}dx)^{\frac{1}{α}} dλdθ]. (2.8)

Let us remark that A_{m}(x, θ), B_{m}(x, λ), F_{m}(x, θ, λ) → 0, as m → +∞ almost everywhere. We can suppose that αq ≤ p. So using the mean value theorem we obtain

|A_{m}(x, θ)|^{α} ≤ [|v_{uu}^{(2)}(x, u(x) + k_{m}(x)θR(h̃_{m})(x), f)| + |v_{uu}^{(2)}(x, u(x), f)|]^{α}

≤ [a[(|u(x)| + |R(h̃_{m})|)^{q} + |f(x)|^{q}] + |b_{4}(x)|

+a[|u(x)|^{q} + |f(x)|^{q}] + |b_{4}(x)|]^{α}

≤ [a[|Z̃_{0}(x)|^{q} + |u(x)|^{q} + 2|f(x)|^{q}] + 2|b_{4}(x)|]^{α} ∈ L^{1}(Q).

Analogously,

|B_{m}(x, λ)|^{α} ≤ [|v_{ff}^{(2)}(x, u(x), λS_{m}(x)h̃_{m}(x) + f)| + |v_{ff}^{(2)}(x, u(x), f)|]^{α}

≤ [a[(|u(x)|^{q} + (|h̃_{m}| + |f(x)|)^{q}] + |b_{4}(x)|

+a[|u(x)|^{q} + |f(x)|^{q}] + |b_{4}(x)|]^{α}

≤ [a[2|u(x)|^{q} + |Z̃_{1}(x)|^{q}] + a|f(x)|^{q} + 2|b_{4}(x)|]^{α} ∈ L^{1}(Q).

Analogously, we have

|F_{m}(x, θ, λ)|^{α} ≤ [|v_{uf}^{(2)}(x, u(x) + θR(h̃_{m}), f + λh̃_{m})| + |v_{uf}^{(2)}(x, u(x), f)|]^{α}

≤ [a[[|u(x)| + |R(h̃_{m})|]^{q} + [|h̃_{m}| + |f(x)|]^{q}] + |b_{4}(x)|

+a[|u(x)|^{q} + |f(x)|^{q}] + |b_{4}(x)|]^{α}

≤ [a[|Z̃_{0}(x)|^{q} + |Z̃_{1}(x)|^{q}] + a|u(x)|^{q} + a|f(x)|^{q} + 2|b_{4}(x)|]^{α} ∈L^{1}(Q).

Thus,using the dominated convergence theorem, we obtain

∫_{0}^{1}(∫_{Q} |A_{m}(x, θ)|^{α}dx)^{\frac{1}{α}} dθ → 0, as m → +∞,

∫_{0}^{1}(∫_{Q} |B_{m}(x, λ)|^{α}dx)^{\frac{1}{α}} dλ → 0, as m → +∞,

and

∫_{0}^{1} ∫_{0}^{1}(∫_{Q} |F_{m}(x, θ, λ)|^{α}dx)^{\frac{1}{α}} dλdθ → 0, as m → +∞,

which contradicts the assumption (2.8). Thus we achieve the proof.

Theorem 2.2. Suppose that in addition to the conditions of theorem (2.1) we have

|v_{u}^{(1)}(x, u, f)| + |v_{f}^{(1)}(x, u, f)| ≤ a(|u|^{q} + |f|^{q}) + |b̂_{2}(x)|,

where b̂_{2}(x) ∈ L^{1}(Q). Then G is (τ_{p}, ∥.∥_{W^{1,2}(Q)}) Taylor mapping of first order at every f ∈ F and G^{(1)}(f) ∈ L((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ).

Proof. Let f ∈ F, and let us prove that the map r(h) ≡ G(f + h) − G(f) − G^{(1)}(f)h is (τ_{p}, ∥.∥_{W^{1,2}(Q)}) infinitely small of first order at zero.

Assume the contrary, then ∃ h̃_{m} ∈ L^{p}(Q) such that h̃_{m} → 0 in L^{p}(Q) but |r(h̃_{m})| ≥ ε∥h̃_{m}∥_{W^{1,2}(Q)}. Thus R(h̃_{m}) → 0 in L^{p}(Q) , as m → +∞.

Using the lemma (1.1),we deduce that there exists Z̃_{0}(x) in L^{p}(Q) and there exists Z̃_{1}(x) in L^{p}(Q): forall m in ℕ |u(x)| + |R(h̃_{m})| ≤ Z̃_{0}(x) almost everywhere in Q. We have also for all m in ℕ |f(x)| + |h̃_{m}| ≤ Z̃_{1}(x) almost everywhere in Q. So,

|r(h̃_{m})| = |∫_{Q} [v(x, u(x) + [R(h̃_{m})](x), f(x) + h̃_{m}(x)) − v(x, u(x), f(x))

−v_{u}^{(1)}(x, u(x), f(x))R(h̃_{m}) − v_{f}^{(1)}(x, u(x), f(x))h̃_{m}]dx|

≤ |∫_{Q} [v(x, u(x) + R(h̃_{m}), f(x) + h̃_{m}) − v(x, u(x), f(x) + h̃_{m})
−v_{u}^{(1)}(x, u(x), f(x))R(h̃_{m}) + v(x, u(x), f(x) + h̃_{m})
−v(x, u(x), f(x)) − v_{f}^{(1)}(x, u(x), f(x))h̃_{m}]dx|

= |∫_{Q} A_{m}(x)R(h̃_{m})dx| + |∫_{Q} B_{m}(x)h̃_{m}dx|,

where,

A_{m}(x) = \{ \begin{matrix}\frac{v(x, u + R(h̃_{m}), f + h̃_{m}) − v(x, u, f + h̃_{m})}{R(h̃_{m})} − v_{u}^{(1)}(x, u, f) & if R(h̃_{m}) ≠ 0 \\ 0 & if R(h̃_{m}) = 0 \end{matrix}

B_{m}(x) = \{ \begin{matrix}\frac{v(x, u, f + h̃_{m}) − v(x, u, f)}{h̃_{m}} − v_{f}^{(1)}(x, u, f) & if h̃_{m} ≠ 0 \\ 0 & if h̃_{m} = 0 \end{matrix} .

We have

|r(h̃_{m})| ≤ (∫_{Q} |A_{m}(x)|^{\frac{2α}{α+1}} dx)^{\frac{α+1}{2α}} ∥R(h̃_{m})∥_{L^{\frac{2α}{α−1}} (Q)} + (∫_{Q} |B_{m}(x)|^{\frac{2α}{α+1}} dx)^{\frac{α+1}{2α}} ∥h̃_{m}∥_{L^{\frac{2α}{α−1}} (Q)}

≤ c_{4}[(∫_{Q} |A_{m}(x)|^{\frac{2α}{α+1}} dx)^{\frac{α+1}{2α}} ∥R(h̃_{m})∥_{W^{1,2}(Q)} + (∫_{Q} |B_{m}(x)|^{\frac{2α}{α+1}} dx)^{\frac{α+1}{2α}} ∥h̃_{m}∥_{W^{1,2}(Q)}]

≤ c_{5}[(∫_{Q} |A_{m}(x)|^{α}dx)^{\frac{1}{α}} + (∫_{Q} |B_{m}(x)|^{α}dx)^{\frac{1}{α}} ]∥h̃_{m}∥_{W^{1,2}(Q)}.  (2.9)

Using the mean value theorem, we obtain

|A_{m}(x)|^{α} ≤ [a[(|u(x)| + |[R(h̃_{m})](x)|)^{q} + (|f(x)| + |h̃_{m}|)^{q}] + |b̂_{2}(x)|

+a[|u(x)|^{q} + |f(x)|^{q}] + |b̂_{2}(x)|]^{α}

≤ [a(|Z̃_{0}(x)|^{q} + |Z̃_{1}(x)|^{q} + |u(x)|^{q} + |f(x)|^{q}) + 2|b̂_{2}(x)|]^{α} ∈ L^{1}(Q).

Analogously,

|B_{m}(x)|^{α} ≤ [a[|u(x)|^{q} + (|f(x)| + |h̃_{m}|)^{q}] + |b̂_{2}(x)| + a[|u(x)|^{q} + |f(x)|^{q}] + |b̂_{2}(x)|]^{α}

≤ [a[2|u(x)|^{q} + |Z̃_{1}(x)| + |f(x)|^{q}] + 2|b̂_{2}(x)|]^{α} ∈ L^{1}(Q).

On the other hand A_{m}(x) → 0 and B_{m}(x) → 0 almost everywhere in Q. Thus using the dominated convergence theorem, we obtain

(∫_{Q} |A_{m}(x)|^{α}dx)^{\frac{1}{α}} → 0, as m → +∞,

(∫_{Q} |B_{m}(x)|^{α}dx)^{\frac{1}{α}} → 0, as m → +∞,

which contradicts the assumption(2.9). Thus we achieve the proof.

Remark 2.1. If v_{k} satisfies the conditions of theorem (2.1) and (2.2) then the functionals

J_{k}(f) ≡ ∫_{Q} v_{k}(x, u, f)dx (k = 0, 1, .., s_{1} + s_{2})

are (τ_{p}, ∥.∥_{W^{1,2}(Q)}) Taylor mapping of first and second order at every f ∈ F and J_{k}^{(1)}(f) ∈ L((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ), J_{k}^{(2)}(f) ∈ B((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ) ∀k = 0, 1, .., s_{1} + s_{2}.

Now let us give the sufficient conditions of optimality for the problem (1.3),(1.4) and (1.5).

Theorem 2.3. Suppose that for the problem (1.5), v_{k} satisfies the conditions of theorem (2.1) and (2.2) then the functionals J_{k}(f) ≡ ∫_{Q} v_{k}(x, u, f)dx , (k =

s_{1} + 1, ..., s_{1} + s_{2}) are (τ_{p}, ∥.∥_{W^{1,2}(Q)}) Taylor mapping of first and second order

at every f ∈ F, and J_{k}(f) = ∫_{Q} v_{k}(x, u, f)dx + c_{k}∥f∥_{W^{1,2}(Q)}^{2} , (k = 0, .., s_{1})

are lower semi-Taylor mapping of first and second order at every f ∈ F. Consequently, ∃J_{k}^{(1)}(f) , ∃J_{k}^{(2)}(f) ,(k = 0, .., s_{1} + s_{2}).

Let us suppose also that f̂ ∈ F,J(f̂) = 0, J_{k}(f̂) = 0,(k = 0, .., s_{1}). Let us put L = {h ∈ W_{0}^{1,2}(Q)/J_{k}^{(1)}(f̂)h = 0, k = 1, .., s_{1}, J^{(1)}(f̂)h = 0}. Suppose that ∃λ̂ ∈ (ℝ^{s_{1}})^{∗}, ∃ŷ^{∗} ∈ (ℝ^{s_{2}})^{∗}, ∃γ ≥ 0, ∃λ̂_{k} > 0, (k = 1, .., s_{1}) : L_{f}(f̂, ŷ^{∗}, λ̂, 1) = 0 and ∀h ∈ L L_{ff}(f̂, ŷ^{∗}, λ̂, 1)(h, h) ≥ 2γ∥h∥_{W^{1,2}(Q)}^{2}, where L_{f}(f̂, ŷ^{∗}, λ̂, 1) and L_{ff}(f̂, ŷ^{∗}, λ̂, 1) are defined by formula (1.7) and (1.8). Then f̂ is τ_{p} strict local minimum.

Proof. We know that W_{0}^{1,2}(Q) is a Hilbert space, thus J^{(1)}(f̂) is an open map (W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}) ⟶ ℝ^{s_{2}}. On the other hand, J^{(1)}(f̂) ∈ L((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ^{s_{2}}) and J_{k}^{(1)}(f̂) ∈ L((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ). And using the theorem (1.6)in [5], we conclude that f̂ is τ_{p} strict local minimum.

Thus we achieve the proof.

Theorem 2.4. Suppose that in the problem (1.3), v_{k} satisfies the conditions of theorem (2.2) and (2.3) then the functionals

J_{k}(f) ≡ ∫_{Q} v_{k}(x, u, f)dx , (k = s_{1} + 1, .., s_{1} + s_{2})

are (τ_{p}, ∥.∥_{W^{1,2}(Q)}) Taylor mapping of first and second order at every f ∈ F

and J_{k}(f) = ∫_{Q} v_{k}(x, u, f)dx+c_{k}∥f∥_{W^{1,2}(Q)}^{2} ,(k = 0, .., s_{1}) are lower semi-Taylor

mapping of first and second order at every f ∈ F. Consequently, ∃J_{k}^{(1)}(f) and ∃J_{k}^{(2)}(f) ,(k = 0, .., s_{1} + s_{2}) Let us suppose also that J_{0}^{(1)}(f̂) = 0 and ∃α > 0, ∀h ∈ W_{0}^{1,2}(Q) J_{0}^{(2)}(f̂)(h, h) ≥ 2α∥h∥_{W^{1,2}(Q)}^{2}. Then f̂ is τ_{p} strict local minimum .

Proof. Let r(f̂+ h) = J_{0}(f̂+ h) − J_{0}(f̂) − 2^{−1}J_{0}^{(2)}(f̂)(h, h).

So ∃V_{f} ∈ τ_{p}, ∀h ∈ W_{0}^{1,2}(Q) : f̂+ h ∈ V_{f} r(f̂+ h) ≥ −2^{−1}α∥h∥_{W^{1,2}(Q)}^{2}.

In this case

J_{0}(f̂+ h) − J_{0}(f̂) = 2^{−1}J_{0}^{(2)}(f̂)(h, h) + r(f̂+ h)
≥ α∥h∥_{W^{1,2}(Q)}^{2} − 2^{−1}α∥h∥_{W^{1,2}(Q)}^{2}
> 2^{−1}α∥h∥_{W^{1,2}(Q)}^{2}
> 0.

Thus we achieve the proof.

Theorem 2.5. Suppose that in the problem (1.4), v_{k} satisfies the conditions of theorem (2.1) and (2.2) then the functionals

J_{k}(f) ≡ ∫_{Q} v_{k}(x, u, f)dx, (k = s_{1} + 1, .., s_{1} + s_{2})

are (τ_{p}, ∥.∥_{W^{1,2}(Q)}) Taylor mapping of first and second order at every f ∈ F

and J_{k}(f) = ∫_{Q} v_{k}(x, u, f)dx + c_{k}∥f∥_{W^{1,2}(Q)}^{2} are lower semi Taylor mapping of

first and second order at every f ∈ F, (k = 0, .., s_{1}). Consequently, ∃J_{k}^{(1)}(f) and ∃J_{k}^{(2)}(f),(k = 0, .., s_{1} + s_{2}) Let us suppose also that J(f̂) = 0, ∃ ŷ^{∗} ∈ (ℝ^{s_{2}})^{∗}, ∃α > 0 L_{f}(f̂, ŷ^{∗}, 1) = 0 and ∀h ∈ kerJ^{(1)}(f̂) L_{ff}(f̂, ŷ^{∗}, 1)(h, h) ≥ 2α∥h∥_{W^{1,2}(Q)}^{2}, where L_{f}(f̂, ŷ^{∗}, 1) and L_{ff}(f̂, ŷ^{∗}, 1) are given by formula (1.10),(1.11). Then f̂ is τ_{p} strict local minimum.

Proof We have L_{ff}(f̂, ŷ^{∗}, 1) ∈ B((W_{0}^{1,2}(Q), ∥.∥_{W^{1,2}(Q)}), ℝ) and using directly the theorem (1.5) in [5]), we conclude that f̂ is τ_{p} strict local minimum.

Remark 2.2. Let us remark that in the theorem (2.1) and (2.2),the increasing conditions verified by v are not sufficient to certify the Frechet differentiability of functional G : (W_{0}^{1,2}(Q), ∥.∥_{L^{p}(Q)}) → ℝ^{s_{2}}.

In fact, suppose we have \frac{3}{2} < q < p < 2. Let us define v : Q × ℝ × ℝ → ℝ

by: v(x, u, f) = a[|u|^{\frac{5}{2}} + |f|^{\frac{5}{2}} ] + |b_{0}(x)|, where b_{0}(x) ∈ C(\overline{Q}), a ∈ ℝ, a > 0. Let

d_{m} → +∞ , and put α_{m} = |d_{m}|^{\frac{1}{2}} ,then α_{m} → +∞ and ∀x ∈ Q ∀u ∈ ℝ ∀m ∈ ℕ We have

|v(x, u, d_{m})| ≥ a|d_{m}|^{\frac{5}{2}}
= a|d_{m}|^{\frac{1}{2}} |d_{m}|^{2}
≥ a|d_{m}|^{\frac{1}{2}} |d_{m}|^{p}
= aα_{m}|d_{m}|^{p}.

Let f̃ ∈ W_{0}^{1,2}(Q). Since the Lesbegue’s measure μ is σ additive, then ∃c > 0, ∃Q′ ⊂ Q : μ(Q′) > 0, ρ(∂Q′, Q) > 0 and ∀x ∈ Q′ |f̃(x)| ≤ c. Put D ≡ max{|v(x, u, f)|/|u| ≤ c, |f| ≤ c, x ∈ \overline{Q}\} < ∞. Let us choose Q_{m} ⊂ Q′ such that μ(Q_{m}) = |d_{m}|^{−p}α_{m}^{− \frac{1}{2}} . Let h̃_{m}(x) defined by

h̃_{m}(x) = \{ \begin{matrix}d_{m} − f̃(x) & if x ∈ Q_{m} \\ 0 & if x ∈ Q \ Q_{m} \end{matrix}

We have

∥h̃_{m}(x)∥_{L^{p}(Q)} ≤ ∥d_{m}∥_{L^{p}(Q_{m})} + |f̃(x)∥_{L^{p}(Q_{m})}
≤ d_{m}(μ(Q_{m}))^{\frac{1}{p}} + c(μ(Q_{m}))^{\frac{1}{p}}
≤ α_{m}^{− \frac{1}{2p}} + c(μ(Q_{m}))^{\frac{1}{p}} .

SUFFICIENT CONDITIONS FOR ELLIPTIC PROBLEM 17

Consequently ∥h̃_{m}(x)∥_{L^{p}(Q)} → 0, i.e h̃_{m}(x) → 0 in L^{p}(Q). We have also

|G(f̃(x) + h̃_{m}(x)) − G(f̃(x))| = | ∫_{Q_{m}} [v(x, [R(f̃+ h̃_{m})](x), f̃(x) + h̃_{m}(x))

−v(x, [R(f̃)](x), f̃(x))]dx|

≥ | ∫_{Q_{m}} v(x, [R(f̃+ h̃_{m})](x), d_{m})dx|

−| ∫_{Q_{m}} v(x, [R(f̃)](x), f̃(x))dx|

≥ aα_{m}|d_{m}|^{p} − Dμ(Q_{m})
= aαm^{\frac{1}{2}} → +∞.

Therfore, |G(f̃(x) + h̃_{m}(x)) − G(f̃(x))| → +∞.

Thus G is not Frechet differentiable at every f ∈ W_{0}^{1,2}(Q).

References

[1] V.M. ALEKSEEV,V.M. TIKHOMIROV,C.V. FOMIN, optimal control. - M.nauka,1979.(russian)

[2] F.H. CLARKE, generalized gradients and applications trans, Amer. math. soc. 1975. V. 205. -P. 247-262.

[3] F.H. CLARKE, a new approach to lagrange multipliers, Math. oper. Res. 1976. V. 1,No. 2.-P165-174.

[4] G. KOTHE, topological vector spaces I. springer-verlag berlin . heidelberg.new York. 1983.

[5] M.F. SUKHININ, Lower semi-Taylor mapping and sufficient conditions for an extremum. ,Math.USSR Sbornik.-1992-V.73, No.1.-P.257-271.

[6] A. TAYLOR,D. LAY, introduction to functional analysis, second edition, krieger publishing company,malabar,florida. 1980.

Université Mohamed I, Faculté des Sciences, Département de mathématiques et Informatique, Oujda, Maroc. E-mail address: lahrech@sciences.univ-oujda.ac.ma, addou@sciences.univ-oujda.ac.ma

Received August, 18, 1999