����������� ������ 55 (2003), 107–119

UDK 517.977

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SUFFICIENT CONDITIONS FOR ELLIPTIC PROBLEM OF OPTIMAL CONTROL IN R^{n}, WHERE n > 2

S. Lahrech and A. Addou

Abstract. This paper is concerned with the local minimization problem for a variety of non Frechet-differentiable Gâteaux functional J(f) ≡ ∫_{Q} v(x, u, f) dx in the Sobolev space (W_{0}^{1,2}(Q), ∥·∥_{p}), where u is the solution of the Dirichlet problem for a linear uniformly elliptic operator with nonhomogenous term f and ∥ · ∥_{p} is the norm generated by the metric space L^{p}(Q), (p > 1). We use a recent extension of Frechet-differentiability (approach of Taylor mappings, see [5]), and we give various assumptions on v to guarantee a critical point to be a strict local minimum. Finally, we give an example of a control problem where classical Frechet differentiability cannot be used and their approach of Taylor mappings works.

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 a sufficiently smooth and bounded domain inR^{n}. Let us consider the problem

Au = f,  (1.1)
u/_{∂Q} = 0.  (1.2)

For this problem, let us state Agmon’s-Douglis-Niremberg’s theorem.

Theorem 1.1. If 1 < q < ∞, then we have that ∀f ∈ L^{q}(Q), there exists a unique solution u ∈ W^{2,q}(Q) ∩ W_{0}^{1,q}(Q) 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 and let u be the solution of problem (1.1), (1.2) in W_{0}^{1,2}(Q) ∩ W^{2,2}(Q) associated to f. 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 the sequence of functions v_{k}: Q×R×R → R is measurable on Q × R × R and has second derivative with respect to (u, f) on R × R for almost all x ∈ Q.

AMS Subject Classification: 49 K 20

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, where 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 the 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 the functional J_{0}(f) takes the minimum value. This control f^{0} will be called optimal.

1.2. Taylor mappings and lower semi-Taylor mappings

Let ∥ · ∥_{W^{1,2}(Q)} be the usual norm in W_{0}^{1,2}(Q), F a subset of W_{0}^{1,2}(Q), τ a topology in F, Y a normed space, and ∥ · ∥_{Y} the norm in Y. According to [5], a mapping r: F → Y (respectively, r: F → R) is said to be infinitesimally (τ, ∥ · ∥_{W^{1,2}(Q)})-small (respectively, infinitesimally lower (τ, ∥ · ∥_{W^{1,2}(Q)})-semismall) 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) ≥ −ε∥h∥_{W^{1,2}(Q)}^{p_{1}});

here and below, O_{f} is a neighborhood of f in (F, τ).

A mapping J: F → Y (respectively, J: F → R) is called a (τ, ∥ · ∥_{W^{1,2}(Q)})- Taylor (respectively, lower (τ, ∥ · ∥_{W^{1,2}(Q)})-semi-Taylor) mapping of order p_{1} at f ∈ F if there exist k linear symmetric (not necessarily continuous) mappings J^{(k)}(f): (W_{0}^{1,2}(Q))^{k} → Y (respectively, J^{(k)}(f): (W_{0}^{1,2}(Q))^{k} → R), k = 1,…, p_{1}, such that

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: F → Y (respectively, r: F → R) is an infinitesimally (τ, ∥·∥_{W^{1,2}(Q)})-small (respectively, infinitesimally lower (τ, ∥ · ∥_{W^{1,2}(Q)})-semismall) mapping of order p_{1} at f ∈ F.

We note that J^{(1)}(f),…, J^{(p_{1})}(f) are not in general single-valued. The set of tuples (J^{(1)}(f),…, J^{(p_{1})}(f)) is denoted by S_{n}(J, f).

Let us solve the problems (1.3), (1.4) and (1.5).

For the problem (1.5) let us introduce the Lagrange functions:

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}})^{∗}.

Similarly, for the problem (1.4), let us introduce the Lagrange functions:

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 give the following lemma where the proof can be traced back to [5].

Lemma 1.1. Let (Ω, Σ, μ) be a measure space with σ-finite measure, and let X be a complete linear metric space continuously imbedded in the metric space M(Ω) of equivalence classes of measurable almost everywhere finite functions x: Ω → R, with the metrizable topology τ(meas) of convergence in measure on each set of σ-finite measure.

Suppose that X contains with each element x(s) the function |x(s)|, the metric in X is translation-invariant, and ρ(x, 0) = ρ(|x|, 0) for each x ∈ X. Then for each sequence x_{n} → 0 in X there exist a subsequence x_{n_{k}} and an element z ∈ X such that: |x_{n_{k}}(s)| ≤ z(s), k = 1, 2,… in the sense of the natural order on classes of functions.

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

Suppose that Q is a sufficiently smooth and bounded domain in R^{n}, where n > 2. Let F be a subset of W_{0}^{1,2}(Q). Let G be the functional defined on F by G(f) = ∫_{Q} v(x, u(x), f(x)) dx, where u(x) is the solution of problem (1.1), (1.2) in W_{0}^{1,2}(Q)∩W^{2,2}(Q) and the function v: Q×R×R → R is measurable on Q×R×R and has second derivative with respect to (u, f) on R × R for almost all x ∈ Q. Suppose also that v, v_{uf}^{(2)}, v_{fu}^{(2)} are continuous in Q × R × R.

Let τ_{p} be the topology generated by the metric space L^{p}(Q), where p > 1. In the rest of this section a = const.

Theorem 2.1. Suppose that the following conditions are added to the conditions of paragraph (1) and (2):

|v(x, u, f)| ≤ a(|u|^{ν} + |f|^{ν}) + b_{0}(x),
|v_{u}^{(1)}(x, u, f)| + |v_{f}^{(1)}(x, u, f)| ≤ a(|u|^{ν−1} + |f|^{ν−1}) + b_{1}(x),
|v_{uu}^{(2)}(x, u, f)| + 2|v_{uf}^{(2)}(x, u, f)| + |v_{ff}^{(2)}(x, u, f)| ≤ a(|u|^{\frac{2p}{n}} + |f|^{\frac{2p}{n}}) + b_{2}(x),

where ν = \frac{2n}{n−2}, b_{0}(x) ∈ L^{1}(Q), b_{1}(x) ∈ L^{\frac{2n}{n+2}} (Q), b_{2}(x) ∈ L^{\frac{n}{2}} (Q), and 1 < p ≤ \frac{2n}{n−2}. Then G is a (τ_{p}, ∥.∥_{W^{1,2}(Q)})-Taylor mapping of second order at each point f ∈ F. Moreover,1 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)|^{ν} dx + ∫_{Q} |f(x)|^{ν} dx) + ∫_{Q} b_{0}(x) dx
≤ a(∥u(x)∥_{L^{ν}(Q)}^{ν} + ∥f(x)∥_{L^{ν}(Q)}^{ν}) + ∥b_{0}(x)∥_{L^{1}(Q)}
≤ c_{1}(∥u(x)∥_{W^{1,2}(Q)}^{ν} + ∥f(x)∥_{W^{1,2}(Q)}^{ν}) + ∥b_{0}(x)∥_{L^{1}(Q)} < ∞.

Thus the functional G is finite.

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

Au = h,  (2.1)
u/_{∂Q} = 0.  (2.2)

Such a solution exists ∀h ∈ W_{0}^{1,2}(Q).

Let G^{(1)}(f) and G^{(2)}(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)h dρ] 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)]h dρ

+ 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

and

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

+ ∫_{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.

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

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.

112 S. Lahrech, A. Addou

The linearity and bilinearity of G^{(1)}(f) and G^{(2)}(f) are obvious. Let us prove now that they are bounded.

We have

|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)|^{ν−1} + |f(x)|^{ν−1}) + |b_{1}(x)|][|R(h)| + |h|] dx ≤ [a[∫_{Q}(|u(x)|^{ν−1})^{\frac{2n}{n+2}} dx]^{\frac{n+2}{2n}} + a[∫_{Q}(|f(x)|^{ν−1})^{\frac{2n}{n+2}} dx]^{\frac{n+2}{2n}}

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

= [a(∥u(x)∥L^{p}(Q)^{\frac{n+2}{2n}} + ∥f(x)∥L^{p}(Q)^{\frac{n+2}{2n}}) + ∥b_{1}(x)∥_{L^{\frac{2n}{n+2}} (Q)}]×

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

≤ c_{0}[∥u(x)∥W^{1,2}(Q)^{\frac{n+2}{2n}} + ∥f(x)∥W^{1,2}(Q)^{\frac{n+2}{2n}} + ∥b_{1}(x)∥_{L^{\frac{2n}{n+2}} (Q)}]×

× [∥R(h)∥_{W^{1,2}(Q)} + ∥h∥_{W^{1,2}(Q)}]. Thus ∃ c_{2} > 0 such that

|G^{(1)}(f)h| ≤ c_{2}(∥R(h)∥_{W^{1,2}(Q)} + ∥h∥_{W^{1,2}(Q)}).

On the other hand, R(h) depends continually on h, thus |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).

Let us prove now that G^{(2)}(f) is also bounded. We have

|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|^{\frac{2p}{n}} + |f|^{\frac{2p}{n}} ) + |b_{2}(x)|]

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

≤ [a[∫_{Q}(|u(x)|^{\frac{2p}{n}} )^{\frac{n}{2}} dx]^{\frac{2}{n}} + a[∫_{Q}(|f(x)|^{\frac{2p}{n}} )^{\frac{n}{2}} dx]^{\frac{2}{n}} + [∫_{Q} |b_{2}(x)|^{\frac{n}{2}} dx]^{\frac{2}{n}} ]

× [[∫_{Q}(|R(h_{1})||R(h_{2})|)^{\frac{n}{n−2}} dx]^{\frac{n−2}{n}} + [∫_{Q}(|R(h_{1})||h_{2}|)^{\frac{n}{n−2}} dx]^{\frac{n−2}{n}}

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

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

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

× [a(∥u(x)∥L^{p}(Q)^{\frac{2p}{n}} + ∥f(x)∥L^{p}(Q)^{\frac{2p}{n}}) + ∥b_{2}(x)∥_{L^{\frac{n}{2}} (Q)}]

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

+ ∥h_{1}∥_{W^{1,2}(Q)}∥R(h_{2})∥_{W^{1,2}(Q)} + ∥h_{1}∥_{W^{1,2}(Q)}∥h_{2}∥_{W^{1,2}(Q)}]

≤ c_{5}∥h_{1}∥_{W^{1,2}(Q)}∥h_{2}∥_{W^{1,2}(Q)}.

Thus G^{(2)}(f) ∈ B((W_{0}^{1,2}(Q), ∥ · ∥_{W^{1,2}(Q)}), R).

Let us prove now that G is a (τ_{p}, ∥.∥_{W^{1,2}(Q)})-Taylor mapping, where τ_{p} is the 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 infinitesimally (τ_{p}, ∥.∥_{W^{1,2}(Q)})-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 Lemma 1.1, we deduce that ∃Z̃(x) ∈ L^{p}(Q) such that |(R(h̃_{m}))(x)| ≤ z̃(x).

Let Z̃_{0}(x) = z̃(x) + |u(x)|, then |u(x)| + |(R(h̃_{m}))(x)| ≤ Z̃_{0}(x), where Z̃_{0}(x) ∈ L^{p}(Q). Analogously for f ∈ W^{1,2}(Q), we obtain |f(x)| + |h̃_{m}| ≤ Z̃_{1}, where Z̃_{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.

Indeed,

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)

114 S. Lahrech, A. Addou

= ∫_{0}^{1} v_{u}^{(1)}(x, u + θR(h), f + h)R(h) dθ + ∫_{0}^{1} v_{f}^{(1)}(x, u, f + λh)h dλ

= ∫_{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)h dλ
= ∫_{0}^{1} R(h)v_{u}^{(1)}(x, u + θR(h), f) dθ + ∫_{0}^{1} v_{f}^{(1)}(x, u, f + λh)h dλ
+ ∫_{0}^{1} ∫_{0}^{1} v_{fu}^{(2)}(x, u + θR(h), f + λh)hR(h) dλ dθ.

So,

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

= ∫_{Q} ∫_{0}^{1}[v_{u}^{(1)}(x, u + θR(h), f) − v_{u}^{(1)}(x, u, f) − 2^{−1}(v_{uu}^{(2)}(x, u, f)R(h)]R(h) dθ dx + ∫_{Q} ∫_{0}^{1}[v_{f}^{(1)}(x, u, f + λh) − v_{f}^{(1)}(x, u, f) − 2^{−1}v_{ff}^{(1)}(x, u, f)h]h dλ dx − ∫_{Q} ∫_{0}^{1} v_{uf}^{(2)}(x, u, f)R(h)h dλ dx + ∫_{Q} ∫_{0}^{1} ∫_{0}^{1} v_{uf}^{(2)}(x, u + θR(h), f + λh)hR(h) dλ dθ dx.

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), & R(h̃_{m}) ≠ 0, \\ 0, & 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), & h̃_{m} ≠ 0, \\ 0, & h̃_{m} = 0. \end{matrix}

Elliptic problem of optimal control 115

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). So,

|r(h̃_{m})| = |∫_{Q} ∫_{0}^{1} A_{m}(x, θ)R^{2}(h̃_{m}) dθ dx + ∫_{Q} ∫_{0}^{1} B_{m}(x, λ)h̃_{m}^{2} dλ dx
+ ∫_{Q} ∫_{0}^{1} ∫_{0}^{1} F_{m}(x, θ, λ)R(h̃_{m})h̃_{m} dλ dθ dx|.

Thus

|r(h̃_{m})| ≤ ∫_{0}^{1} ∫_{Q} |A_{m}(x, θ)R^{2}(h̃_{m})| dx dθ + ∫_{0}^{1} ∫_{Q} |B_{m}(x, λ)h̃_{m}^{2}| dλ dx

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

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

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

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

= ∫_{0}^{1}[∫_{Q} |A_{m}(x, θ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dθ ∥R(h̃_{m})∥_{L^{\frac{2n}{n−2}} (Q)}^{2}

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

+ ∫_{0}^{1} ∫_{0}^{1}[∫_{Q} |F_{m}(x, θ, λ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dθ dλ×

× [(∫_{Q} |R(h̃_{m})|^{\frac{2n}{n−2}} dx)^{\frac{1}{2}} (∫_{Q} |h̃_{m}|^{\frac{2n}{n−2}} dx)^{\frac{1}{2}} ]^{\frac{n−2}{n}}

≤ c_{6} ∫_{0}^{1}[∫_{Q} |A_{m}(x, θ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dθ ∥R(h̃_{m})∥_{W^{1,2}(Q)}^{2}

+ c_{7}∥h̃_{m}∥_{W^{1,2}(Q)}^{2} ∫_{0}^{1} ∫_{0}^{1}[∫_{Q} |F_{m}(x, θ, λ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dθ dλ

≤ c_{8}[∫_{0}^{1}[∫_{Q} |A_{m}(x, θ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dθ + ∫_{0}^{1}[∫_{Q} |B_{m}(x, λ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dλ

+ ∫_{0}^{1} ∫_{0}^{1}[∫_{Q} |F_{m}(x, θ, λ|^{\frac{n}{2}} dx]^{\frac{2}{n}} dθ dλ]|h̃_{m}∥_{W^{1,2}(Q)}^{2}  (2.3)

Let us remark that A_{m}(x, θ), B_{m}(x, λ), F_{m}(x, θ, λ) → 0 almost everywhere.

On the other hand, using the mean value theorem, we deduce that there exists a sequence k_{m}(x) such that 0 ≤ k_{m}(x) ≤ 1 and

|A_{m}(x, θ)|^{\frac{n}{2}} = [|v_{uu}^{(2)}(x, u(x) + k_{m}(x) θ[R(h̃_{m})](x), f) − v_{uu}^{(2)}(x, u(x), f)|θ]^{\frac{n}{2}}
≤ [|v_{uu}^{(2)}(x, u(x) + k_{m}(x) θ [R(h̃_{m})](x), f)| + |v_{uu}^{(2)}(x, u(x), f)|]^{\frac{n}{2}}
≤ [a(|u(x) + k_{m}(x)θ[R(h̃_{m})](x)|^{\frac{2p}{n}} + |f(x)|^{\frac{2p}{n}} ) + |b_{2}(x)|
+ a(|u(x)|^{\frac{2p}{n}} + |f(x)|^{\frac{2p}{n}} ) + |b_{2}(x)|]^{\frac{n}{2}}
≤ [a[(|u(x)| + |R(h̃_{m})(x)|)^{\frac{2p}{n}} + |f(x)|^{\frac{2p}{n}} ]
+ a[|u(x)|^{\frac{2p}{n}} + |f(x)|^{\frac{2p}{n}} ] + 2|b_{2}(x)|]^{\frac{n}{2}}
≤ [2a(|Z̃_{0}(x)|^{\frac{2p}{n}} + |f(x)|^{\frac{2p}{n}}) + 2|b_{2}(x)|]^{\frac{n}{2}} ∈ L^{1}(Q).

Analogously, for B_{m} we deduce that there exists S_{m}(x): 0 ≤ S_{m}(x) ≤ 1 and

|B_{m}(x, λ)|^{\frac{n}{2}} = |v_{ff}^{(2)}(x, u(x), f(x) + ξ_{m}λh̃_{m}) − v_{ff}^{(2)}(x, u(x), f(x))λ|^{\frac{n}{2}}
≤ [a(|u(x)|^{\frac{2p}{n}} + |f(x) + ξ_{m}λh̃_{m}|^{\frac{2p}{n}}) + |b_{2}(x)|
+ a(|u(x)|^{\frac{2p}{n}} + |f(x)|^{\frac{2p}{n}}) + |b_{2}(x)|]^{\frac{n}{2}}
≤ [2a(|u(x)|^{\frac{2p}{n}} + 2|Z̃_{1}(x)|^{\frac{2p}{n}}) + 2|b_{2}(x)|]^{\frac{n}{2}} ∈ L^{1}(Q).

Analogously, for F_{m} we obtain

|F_{m}(x, θ, λ)|^{\frac{n}{2}} = |v_{uf}^{(2)}(x, u(x) + θR(h̃_{m}), f + λh̃_{m}) − v_{uf}^{(2)}(x, u(x), f)|^{\frac{n}{2}}
≤ [a(|u(x) + θR(h̃_{m})|^{\frac{2p}{n}} + |f + λh̃_{m}|^{\frac{2p}{n}}
+ |u(x)|^{\frac{2p}{n}} + |f(x)|^{\frac{2p}{n}}) + 2|b_{2}(x)|]^{\frac{n}{2}}
≤ [2a(|Z̃_{0}(x)|^{\frac{2p}{n}} + |Z̃_{1}(x)|^{\frac{2p}{n}}) + 2|b_{2}(x)|]^{\frac{n}{2}} ∈ L^{1}(Q).

Let us remark that A_{m}(x, θ) → 0, B_{m}(x, λ) → 0, F_{m}(x, θ, λ) → 0 almost everywhere. Thus, using the dominated convergence theorem, we conclude that

∫_{0}^{1}[∫_{Q} |A_{m}(x, θ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dθ → 0,

∫_{0}^{1}[∫_{Q} |B_{m}(x, λ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dλ → 0,

∫_{0}^{1} ∫_{0}^{1}[∫_{Q} |F_{m}(x, θ, λ)|^{\frac{n}{2}} dx]^{\frac{2}{n}} dλ dθ → 0,

but this contradicts (2.3).

Theorem 2.2. Let the following condition be added to the conditions of Theorem 2.1:

|v_{u}^{(1)}(x, u, f)| + |v_{f}^{(1)}(x, u, f)| ≤ a(|u|^{\frac{2p}{n}} + |f|^{\frac{2p}{n}} ) + |b̂_{1}(x)|.

Then the functional G is a (τ_{p}, ∥.∥_{W^{1,2}(Q)})-Taylor mapping of first and second order at each point f ∈ F.

Proof. We must estimate r(h) ≡ G(f + h) − G(f) − G^{(1)}(f)h as in the proof of Theorem 2.1, representing r(h_{m}) in the form:

r(h_{m}) = ∫_{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), & R(h̃_{m}) ≠ 0, \\ 0, & 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), & h̃_{m} ≠ 0, \\ 0, & h̃_{m} = 0, \end{matrix}

while h_{m} is the same as in the proof of Theorem 2.1. These estimates are omitted.

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

Theorem 2.3. Suppose that in the problem (1.4), v_{k} satisfies the conditions of Theorems 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 mappings of first and second order at each point 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 (τ_{p}, ∥·∥_{W^{1,2}(Q)})-semi-Taylor mappings of first and second order at each point f ∈ F. Consequently, ∃J_{k}^{(1)}(f) and ∃J_{k}^{(2)}(f), (k = 0,…, s_{1} + s_{2}).

Let us suppose also that J(f̂) = 0, ∃y^{∗} ∈ (R^{s_{2}})^{∗}, ∃α > 0 L_{f}(f̂, y^{∗}, 1) = 0 and ∀h ∈ kerJ^{(1)}(f̂) L_{ff}(f̂, y^{∗}, 1)(h, h) ≥ 2α∥h∥_{W^{1,2}(Q)}^{2}, where L_{f}(f̂, y^{∗}, 1) and

L_{ff}(f̂, y^{∗}, 1) are given by formulas (1.10), (1.11). Then f̂ is a strict τ_{p}-local minimum point.

Proof. All conditions of Theorem 1.5 in [5] are satisfied, so f̂ is a strict τ_{p}-local minimum point.

Theorem 2.4. Suppose that in the problem (1.5), v_{k} satisfies the conditions of Theorems 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 mappings of first and second order at each point 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 (τ_{p}, ∥·∥_{W^{1,2}(Q)})-semi-Taylor mappings of first and second order at each point f ∈ F. Consequently, ∃J_{k}^{(1)}(f) and ∃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}})^{∗}, ∃y^{∗} ∈ (R^{s_{2}})^{∗}, ∃γ ≥ 0, ∃λ̂_{k} > 0, (k = 1, .., s_{1}): L_{f}(f̂, y^{∗}, λ̂, 1) = 0 and ∀h ∈ L L_{ff}(f̂, y^{∗}, λ̂, 1)(h, h) ≥ 2γ∥h∥_{W^{1,2}(Q)}^{2}, where

L_{f}(f̂, y^{∗}, λ̂, 1) and L_{ff}(f̂, y^{∗}, λ̂, 1) are defined by formulas 1.7 and 1.8. Then f̂ is a strict τ_{p}-local minimum point.

Proof. All conditions of Theorem 1.6 in [5] are satisfied, so f̂ is a strict τ_{p}-local minimum point.

Theorem 2.5. Suppose that in the problem 1.3, v_{k} satisfies the conditions of Theorems 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 mappings of first and second order at each point 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 (τ_{p}, ∥·∥_{W^{1,2}(Q)})-semi-Taylor mappings of first and second order at each point 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 a strict τ_{p}-local minimum point.

Proof. All conditions of Theorem 1.4 in [5] are satisfied, so f̂ is a strict τ_{p}-local minimum point.

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

Indeed, suppose we have n = 3 and \frac{3}{2} < p < 1. 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}} , so α_{m} → +∞ and ∀x ∈ Q ∀u ∈ R ∀m ∈ N

|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). By the countable additivity of Lebesgue measure, ∃c > 0 ∃Q′ ⊂ Q: μ(Q′) > 0 and ρ(Q′, ∂Q) > 0 and ∀x ∈ Q′ |f̃(x)| ≤ c.

In this case 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} defined by:

h̃_{m}(x) = \{ \begin{matrix}d_{m} − f̃(x), & when x ∈ Q_{m}, \\ 0, & when 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}} .

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

On the other hand, we have

|G(f̃+ h̃_{m}) − G(f̃)| =

= |∫_{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}} → +∞.

Therefore, |G(f̃+ h̃_{m}) − G(f̃)| → +∞. Thus G is not Frechet differentiable at each point f ∈ W_{0}^{1,2}(Q).

REFERENCES

[1] V.M. Alekseev, V.M. Tikhomirov, S.V. Fomin, Optimal Control, Nauka, Moskva 1979 (in Russian).

[2] F.H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc, 205 (1975), 247–262.

[3] F.H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res. 1, 2 (1976), 165–174 [4] G. Köthe, Topological Vector Spaces, I, Springer, Berlin-Heidelberg-New York, 1983.

[5] M.F. Sukhinin, Lower semi-Taylor mappings and sufficient conditions for an extremum, Math. USSR Sbornik 73, 1 (1992), 257–271.

[6] A. Taylor, D. Lay, Introduction to Functional Analysis, 2nd ed., Krieger publ. co., Malabar, 1980.

(received 02.06.2002)

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