AN APPROACH TO SECOND-ORDER NECESSARY CONDITIONS FOR MULTIPARAMETER DISCRETE INCLUSIONS

AN APPROACH TO SECOND-ORDER NECESSARY CONDITIONS FOR MULTIPARAMETER

DISCRETE INCLUSIONS

AURELIAN CERNEA

We study an optimization problem given by a multiparameter discrete inclusion that describes the Roesser model. An approach concerning second-order optima- lity conditions is proposed.

AMS 2000 Subject Classification: 49J30.

Key words: tangent cone, discrete inclusion, necessary optimality conditions.

1. INTRODUCTION Consider the problem

(1.1) minimize g(x_NN)

over the solutions of the multiparameter discrete inclusion

(1.2) x_ij ∈F_ij(x_ij−1, x_i−1j, x_i−1j−1), i= 0,1, . . . , N, j = 0,1, . . . , N, with endpoint constraints of the form

(1.3) x_NN ∈X_N,

where F_ij(·,·,·) : Rⁿ×Rⁿ×Rⁿ → P(Rⁿ), i= 0,1, . . . , N, j = 0,1, . . . , N are given set-valued maps,X_N ⊂Rⁿ and g(·) :Rⁿ →Rare given.

There are several papers devoted to first-order necessary optimality con- ditions for this problem ([6], [7], [8], [9] etc.). The aim of the present paper is to develop an approach concerning second-order necessary optimality conditions for problem (1.1)–(1.3). The general idea is to consider our problem as the problem of minimizing the terminal payoff on the intersection of the (known) target set with the (unknown) reachable set and to use a general result from nonsmooth analysis. This general (abstract) optimality condition was formu- lated for the first time by Zheng [10], but this result (Theorem 2.2 below) is, in fact, an obvious consequence of Theorems 6.3.1, 6.6.2, 4.7.4, Proposition 6.2.4 and Corollary 4.3.5 in [1].

REV. ROUMAINE MATH. PURES APPL.,52(2007),5, 529–538

In order to apply the general abstract optimality conditions (namely, Theorem 2.2 below) we must check a certain constraint qualification, so we are naturally led to study first and second order approximations of the reachable set along optimal solutions.

Let us mention that this idea has been already used in [3], [4], [5] and [10] to obtain second-order necessary optimality conditions for problems given by differential inclusions, hyperbolic differential inclusions and other classes of discrete inclusions.

The paper is organized as follows. In Section 2 we present the notation and definitions to be used in the sequel while in Section 3 we present our main results.

2. PRELIMINARIES

Since the reachable set that appears in optimization problems is, in gen- eral, neither a differentiable manifold nor a convex set, its infinitesimal prop- erties may be characterized only by tangent cones in a generalized sense, ex- tending the classical concepts of tangent cones in Differential Geometry and Convex Analysis, respectively. From the rather large number of tangent cones in the literature (e.g. [1]) we only use the following concepts.

LetX⊂Rⁿ and x∈cl(X) (the closure of X).

Definition 2.1. (a) The quasitangent (intermediate) cone to X at x is defined as

Q_xX ={v∈Rⁿ; ∀s_m→0+, ∃v_m→v: x+s_mv_m∈X};

(b) the second-order quasitangent setto X at x relative to v ∈Q_xX is defined as

Q²_(x,v)X={w∈Rⁿ; ∀s_m→0+, ∃w_m→w: x+s_mv+s²_mw_m ∈X}; (c)Clarke’s tangent cone to X at x is defined as

C_xX=

v∈Rⁿ; ∀(x_m, s_m)→(x,0+), x_m ∈X, ∃y_m ∈X : y_m−x_m s_m →v

.

For equivalent definitions and for several properties of these cones we refer to [1]. We recall that unlikeQ_xX, Clarke’s tangent cone C_xX is convex and one hasC_xX⊂Q_xX.

We denote byC⁺ the positive dual cone of C ⊂Rⁿ, namely, C⁺={q ∈Rⁿ; q, v ≥0, ∀v∈C}.

The negative dual cone ofC⊂Rⁿ isC⁻=−C⁺.

As was often remarked, the geometric interpretation of the classical (Fr´echet) derivative suggests the possibility of the introduction of general- ized differentiability concepts corresponding to each type of tangent cone (to the graph, to the epigraph or to the subgraph of the function) but, of course, not all these concepts are equally important. In what follows, for a mapping g(·) : X ⊂ Rⁿ → R which is not differentiable, we shall only use the first and second order uniform lower Dini derivative. We refer to [1] for the main properties of such derivatives:

D_↑g(x;v) = lim inf

(v,θ)→(v,0+)

g(x+θv)−g(x)

θ ,

D_↑²g(x, v;w) = lim inf

(w,θ)→(w,0+)

g(x+θv+θ²w)−g(x)−θD_↑g(x;v)

θ² .

Wheng(·) is of class C² one has

D_↑g(x, v) =g(x)^Tv, D²_↑g(x, v;w) =g(x)^Tz+1

2v^Tg(x)v.

The key tool in the proof of our main result is the abstract optimality condition, below.

Theorem 2.2 ([10]). Let g : Rⁿ → R be a locally Lipschitz function at z ∈ Rⁿ and S₁, S₂ nonempty subsets of Rⁿ containing z. If z solves the minimization problem

minimize g(x) over allx∈S₁∩S₂ and also satisfies the constraint qualification

(C_zS₁)⁻∩(C_zS₂)⁺={0}, then we have the first-order necessary condition

D_↑g(z;v) ≥0 ∀v∈Q_zS₁∩Q_zS₂.

Furthermore, if equality holds for some v₀, then we have the second-order necessary condition

D_↑²g(z, v₀;w)≥0 ∀w∈Q²_(z,v0)S₁∩Q²_(z,v0)S₂.

Corresponding to each type of tangent cone, sayτ_xX, one may introduce (e.g., [1]) a set-valued directional derivative of a multifunction G(·) : X ⊂ Rⁿ → P(Rⁿ) (in particular of a single-valued mapping) at a point (x, u) ∈ graph(G) as

τ_uG(x, ξ) ={ν ∈Rⁿ; (ξ, ν)∈τ_(x,u)graph(G)}, ν∈τ_xX.

This first-order derivative may be characterized, equivalently, by graphτ_uG(x,·) =τ_(x,u)(graphG(·)).

If the set-valued mapG(·) is Lipschitz, i.e., there existsL >0 such that G(x₁)⊂G(x₂) +Lx₁−x₂B ∀x₁, x₂ ∈X,

whereB denotes the closed unit ball inRⁿ, then the first order quasitangent derivative is given by (e.g., [1])

Q_uG(x;ξ) =

ν ∈Rⁿ; lim

θ→0+

1

θd(u+θν, G(x+θξ)) = 0

.

Similarly, one may define (e.g., [1]) second-order directional deivatives of a set-valued mapG(·). For example the second-order quasitangent derivative of G at (x, u) relative to (y, v) ∈ Q_(x,u)(graph(G(·)) is the set-valued map Q²_(u,v)G(x, y,·) defined by

graphQ²_(u,v)G(x, y;.) =Q²((x,u),(y,v))(graphG(·)).

We recall that a set-valued map A(·) : Rⁿ → P(Rⁿ) is called a closed (respectively, convex) process if graph(A(·)) is a closed (respectively, con- vex) cone.

For the basic properties of convex processes we refer to [1], but we shall only use here the above definition.

If G(·) : X ⊂ Rⁿ → P(Rⁿ) is a given set-valued map and (x, u) ∈ graph(G), as a closed convex process one may take the Clarke directional derivative A(·) =C_uG(x,·) of G(·) at (x, u).

The adjoint processA^∗ :Rⁿ→ P(Rⁿ) of the closed convex process Ais defined by

A^∗(p) ={q ∈Rⁿ;q, v ≤ p, v ∀(v, v)∈graphA(·)}. Denote by S_F the solution set of inclusion (1.2), i.e.,

S_F :=

x= (x₀, x₁, . . . , x_N); x_i= (x_i0, x_i1, . . . , x_iN), x_ij ∈Rⁿ, i, j = 0,1, . . . , N, x is a solution of (1.2), x_ij = 0 if i <0 or j <0

. and byR_F^N :={x_NN; x∈S_F}the reachable set of inclusion (1.2).

In what follows we consider a solution x = (x₀, x₁, . . . , x_N) ∈ S_F of problem (1.1)–(1.3) and shall assume the following hypotheses.

Hypothesis2.3. i)X_N ⊂Rⁿ is a closed set.

ii) The set-valued maps F_ij(·) have nonempty closed values ∀i, j ∈ {0, 1, . . . , N} and there exists L >0 such that F_ij(·) is Lipschitz with Lipschitz constant L,∀i, j∈ {0,1, . . . , N}.

Hypothesis 2.4. There exists a family A_ij(·) : R³ⁿ → P(Rⁿ), i, j = 0,1, . . . , N, of closed convex processes such that

A_ij(u, v, w) ⊂Q_xijF_ij((x_ij−1, x_i−1j, x_i−1j−1); (u, v, w))

∀u, v, w∈Rⁿ,∀i, j∈ {0,1, . . . , N}.

Let A₀ ⊂Q_x00F₀₀(0,0,0) be a closed convex cone. With problem (1.2) we associate the linearized problem

(2.1) w_ij∈A_ij(w_ij−1, w_i−1j, w_i−1j−1), w₀₀∈A₀, i, j= 0,1, . . . , N, i+j >0 with the boundary conditionsw_ij = 0 fori <0 or j <0.

Denote byS_Athe solution set of inclusion (2.1) and byR^N_A the reachable set of inclusion (2.1).

In the study of our optimization problem we need the duality result below.

Lemma 2.5 ([9]). Assume that Hypothesis 2.4 is verified and let r(·) : Rⁿ→ P(Rⁿ) be the set-valued map defined by

r(α) :={w_NN; w= (w₀, . . . , w_N)is a solution of (2.1), w₀₀=α}. Then, for all b∈Rⁿ,

r^∗(b) =

u¹₀₁+u²₁₀+u³₁₁; (u¹_ij, u²_ij, u³_ij)∈A^∗_ij(u¹_ij+1+u²_i+1j+u³_i+1j+1), i, j= 0,1, . . . , N, 0< i+j <2N, (u¹_NN, u²_NN, u³_NN)∈A^∗_NN(b)

.

Corollary 2.6. Assume that Hypothesis 2.4 is satisfied and A₀ ⊂ Q_x00F₀₀(0,0,0) is a closed convex cone. Then

(R^N_A)⁺⊂

q∈Rⁿ; ∃u¹_ij, u²_ij, u³_ij ∈Rⁿ such that(u¹_ij, u²_ij, u³_ij)∈

∈A^∗_ij(u¹_ij+1+u²_i+1j+u³_i+1j+1), i, j = 0,1, . . . , N, 0< i+j <2N, (u¹_NN, u²_NN, u³_NN)∈A^∗_NN(q), u¹₀₁+u²₁₀+u³₁₁∈A⁺₀

.

Proof. Obviously,R^N_A =r(A₀), where r(·) was defined in the statement of Lemma 2.5. Therefore, (R^N_A)⁺= (r(A₀))⁺ =r^∗−1(A⁺₀). If b∈(R^N_A)⁺ then it follows thatr^∗(b)∈A⁺₀ and the corollary now follows from Lemma 2.5.

Remark 2.7. Hypothesis 2.4 that appears in Lemma 2.5, is satisfied if we take

A_ij(u, v, w) :=C_xijF_ij((x_ij−1, x_i−1j, x_i−1j−1); (u, v, w))

∀u, v, w∈Rⁿ,∀i, j∈ {0,1, . . . , N}.

3. THE MAIN RESULTS

We prove first an approximation of the reachable setR^N_F at x_N.

Theorem3.1. Assume that Hypothesis2.3is satisfied and denote byR^N_Q the reachable set of the discrete inclusion

(3.1) w_ij∈Q_xijF_ij((x_ij−1, x_i−1j, x_i−1j−1); (w_ij−1, w_i−1j, w_i−1j−1)), i+j >0 w₀₀∈Q_x00F₀₀(0,0,0), w_ij = 0 for i <0 or j <0.

ThenR^N_Q ⊂Q_xNNR^N_F.

Proof. Let ¯w∈R^N_Q and s_k→0+. It follows that there exists a solution (w₀, w₁, . . . , w_N) to (3.1) such that ¯w=w_NN.

In particular, w₀₀ ∈ Q_x00F₀₀(0,0,0) and, therefore, there exists w₀₀^k → w₀₀ such thatx₀₀+s_kw₀₀^k ∈F₀₀(0,0,0) ∀k∈N.

On the other hand,w₀₁∈Q_x01F₀₁((x₀₀,0,0); (w₀₀,0,0)) and by the defi- nition of the quasitangent derivative ofF₀₁ there exists ( ˜w₀₁^k ,w˜^k₀₀,w˜^k₋₁₁,w˜₋₁₀^k )

→(w₀₁, w₀₀,0,0) such that

x₀₁+s_kw˜^k₀₁∈F₀₁(x₀₀+s_kw˜^k₀₀, s_kw˜^k₋₁₁, s_kw˜^k₋₁₀) ∀k∈N.

By the Lipschitz property of the set-valued map F₀₁(·), we can write x₀₁+s_kw˜₀₁^k ∈F₀₁(x₀₀+s_kw^k₀₀,0,0)+s_kL(w˜^k₀₀−w^k₀₀+w˜^k₋₁₁+w˜^k₋₁₀)B

∀k∈N. Thus, there exists b¹_k∈B such that

x₀₁+s_k[ ˜w₀₁^k −Lb¹_k(w˜₀₀^k −w₀₀^k +w˜₋₁₁^k +w˜^k₋₁₀)]∈F₀₁(x₀₀+s_kw₀₀^k ,0,0)

∀k∈N, and if we definew₀₁^k := ˜w₀₁^k −Lb¹_k(w˜^k₀₀−w₀₀^k +w˜₋₁₁^k +w˜^k₋₁₀), then we have w₀₁^k →w₀₁ ask→ ∞and

x₀₁+s_kw^k₀₁∈F₀₁(x₀₀+s_kw₀₀^k ,0,0) ∀k∈N.

By repeating this construction forj= 2, . . . , N we find that there exists w_0j^k ∈Rⁿ such that w_0j^k →w_0j ask→ ∞ and

x_0j+s_kw^k_0j ∈F_0j(x_0j−1+s_kw_0j−1^k ,0,0) ∀k∈N, ∀j= 2, . . . , N.

Next, we repeat this construction for i ∈ {1,2, . . . , N} fixed and j ∈ {0,1, . . . , N} and find that there exists w^k_ij ∈ Rⁿ such that w^k_ij → w_ij as k→ ∞ and

x_ij+s_kw^k_ij∈F_ij(x_ij−1+s_kw^k_ij−1, x_i−1j+s_kw^k_i−1j, x_i−1j−1+s_kw_i−1j−1^k )∀k∈N. In particular, for s_k → 0+ there exists w^k_NN → w_NN such that x_NN + s_kw^k_NN ∈R^N_F, i.e., w=w_NN ∈Q_xNNR^N_F and the proof is complete.

Another first-order approximation of the reachable setR_F^N atx_N can be obtained in terms of the variational inclusion defined by the Clarke derivative of the set valued map.

Theorem3.2. Assume that Hypothesis2.3is satisfied and denote byR^N_C the reachable set of the discrete inclusion

(3.2) w_ij∈C_xijF_ij((x_ij−1, x_i−1j, x_i−1j−1); (w_ij−1, w_i−1j, w_i−1j−1)), i+j >0 w₀₀∈C_x00F₀₀(0,0,0), w_ij = 0 for i <0 or j <0.

ThenR^N_C ⊂C_xNNR^N_F.

The proof of Theorem 3.2 can be done using the same arguments em- ployed to prove Theorem 3.1.

In order to apply Theorem 2.2 for our problem (1.1)–(1.4) we need to know the second-order quasitangent set to the reachable setR_F^N at x_NN.

Theorem 3.3. Assume that Hypothesis 2.3 is satisfied, let y = (y₀, y₁, . . . , y_N) satisfy (3.1) and let R²_Q denote the reachable set of the discrete inclusion

(3.3)

v_ij ∈Q²_(x

ij,y_ij)F_ij((x_ij−1, x_i−1j, x_i−1j−1),(y_ij−1, y_i−1j, y_i−1j−1); (v_ij−1, v_i−1j, v_i−1j−1)), i+j >0

v₀₀∈Q²_(x

00,y₀₀)F₀₀(0,0,0), v_ij = 0 for i <0 or j <0.

ThenR²_Q ⊂Q²_(x

NN,y_NN)R^N_F.

Proof. Letv ∈R²_Q and t_k →0+. It follows that there exists a solution (v₀, v₁, . . . , v_N) to (3.3) such thatv=v_NN.

In particular,v₀₀∈Q²_(x

00,y₀₀)F₀₀(0,0,0) and therefore there existsv^k₀₀→ v₀₀ such that x₀₀+t_ky₀₀+t²_kv^k₀₀∈F₀₀(0,0,0).

On the other hand, v₀₁ ∈ Q²_(x

01,y₀₁)F₀₁((x₀₀,0,0),(y₀₀,0,0); (v₀₀,0,0)) and by the definition of the second-order quasitangent derivative ofF₀₁ there exists (˜v^k₀₁,˜v^k₀₀,v˜₋₁₁^k ,˜v^k₋₁₀)→(v₀₁, v₀₀,0,0) such that

(x₀₀,0,0, x₀₁) +t_k(y₀₀,0,0, y₀₁) +t²_k(˜v₀₀^k ,˜v^k₋₁₁,v˜₋₁₀^k ,v˜^k₀₁)∈graphF₀₁(·)

∀k∈N.

Using the Lipschitz property of the set-valued mapF₀₁(·), we can write x₀₁+t_ky₀₁+t²_kv˜₀₁^k ∈F₀₁(x₀₀+t_ky₀₀+t²_kv˜₀₀^k , t²_kv˜₋₁₁^k , t²_k˜v^k₋₁₀)⊂

⊂F₀₁(x₀₀+t_ky₀₀+t²_kv₀₀^k ,0,0) +Lt²_k(v₀₀^k −v˜^k₀₀+˜v^k₋₁₁+v˜₋₁₀^k )B.

Thus, there exists b¹_k∈B such that

x₀₁+t_ky₀₁+t²_k[˜v^k₀₁−Lb¹_k(v^k₀₀−v˜₀₀^k +˜v^k₋₁₁+˜v₋₁₀^k )]∈

∈F₀₁(x₀₀+t_ky₀₀+t²_kv₀₀^k ,0,0)

∀k∈Nand if we define

v^k₀₁:= ˜v₀₁^k −Lb¹_k(v^k₀₀−v˜₀₀^k +v˜₋₁₁^k +˜v^k₋₁₀),

then we have v₀₁^k → v₁ and

x₀₁+t_ky₀₁+t²_kv^k₀₁∈F₀₁(x₀₀+t_ky₀₀+t²_kv₀₀^k ,0,0) ∀k∈N.

By repeating this construction forj= 2, . . . , N, we find that there exists v_0j^k ∈Rⁿ such that v^k_0j →v_0j and

x_0j+t_ky_0j+t²_kv_0j^k ∈F_0j(x_0j+t_ky_0j+t²_kv_0j−1^k ,0,0)

∀k∈N,∀j= 2, . . . , N.

Next, we repeat this construction for i∈ {1,2, . . . , N} fixed andj ∈ {0, 1, . . . , N}and we find that there existsv^k_ij ∈Rⁿsuch thatv_ij^k →v_ij ask→ ∞ and

x_ij +t_ky_ij +t²_kv^k_ij ∈F_ij(x_ij−1+t_ky_ij−1+t²_kv^k_ij−1, x_i−1j+t_ky_i−1j+t²_kv_i−1j^k , x_i−1j−1+t_ky_i−1j−1+t²_kv_i−1j−1^k ) ∀k∈N.

In particular, fort_k→0+x_NN +t_ky_NN +t²_kv_NN^k ∈R_F^N, i.e.,v=v_NN ∈ Q²_(x

NN,y_NN)R^N_F and the proof is complete.

We are now able to prove our main result.

Theorem3.4. Assume that Hypothesis 2.3 is satisfied. Let g(·) :Rⁿ→ R be a locally Lipschitz function, A₀ ⊂Q_x00F₀₀(0,0,0) a closed convex cone, x = (x₀, x₁, . . . , x_N) ∈ S_F an optimal solution to problem (1.1)–(1.3) and assume that the constraint qualification below is satisfied:

(3.4)

−q∈Rⁿ; ∃u¹_ij, u²_ij, u³_ij ∈Rⁿ such that(u¹_ij, u²_ij, u³_ij)∈

∈(C_xijF_ij((x_ij−1, x_i−1j, x_i−1j−1); (·,·,·)))^∗(u¹_ij+1+u²_i+1j+u³_i+1j+1), i, j= 0,1, . . . , N, 0< i+j <2N,

(u¹_NN, u²_NN, u³_NN)∈(C_xNNF_NN((x_NN−1, x_N−1N, x_N−1N−1); (·,·,·)))^∗(q), u¹₀₁+u²₁₀+u³₁₁∈A⁺₀

∩(C_xNNX_N)⁺={0}.

Then we have the first-order necessary condition

(3.5) D_↑g(x_NN;y_NN)≥0 ∀y_NN ∈R^N_Q ∩Q_xNNX_N.

Furthermore, if equality holds for some y_NN, then we have the second-order necessary condition

(3.6) D_↑²g(x_NN, y_NN;w_NN)≥0 ∀w_NN ∈R²_Q∩Q²_(x

NN,y_NN)X_N. Proof. According to Theorem 3.2, R_C^N ⊂C_xNNR^N_F. Hence

(3.7) (C_xNNR_F^N)⁺⊂(R^N_C)⁺.

It follows from Lemma 2.6, applied with A_ij(u, v, w) := C_xijF_ij((x_ij−1, x_i−1j, x_i−1j−1); (u, v, w)),∀u, v, w∈Rⁿ,∀i, j∈ {0,1, . . . , N}, that

(3.8)

(R^N_C)⁺⊂

q; ∃u¹_ij, u²_ij, u³_ij ∈Rⁿ such that (u¹_ij, u²_ij, u³_ij)∈

∈(C_xijF_ij((x_ij−1, x_i−1j, x_i−1j−1); (·,·,·)))^∗(u¹_ij+1+u²_i+1j +u³_i+1j+1), i, j= 0,1, . . . , N, 0< i+j <2N,

(u¹_NN, u²_NN, u³_NN)∈(C_xNNF_NN((x_NN−1, x_N−1N, x_N−1N−1); (·,·,·)))^∗(q), u¹₀₁+u²₁₀+u³₁₁∈A⁺₀

.

Therefore, from (3.4), (3.7) and (3.8) we deduce that (3.9) (C_xNNR^N_F)⁻∩(C_xNNX_N)⁺={0}.

From Theorem 3.1 we have

(3.10) R^N_Q ⊂Q_xNNR^N_F

while from Theorem 3.3 we have

(3.11) R²_Q⊂Q²_(xNN,y

NN)R^N_F.

If x = (x₀, x₁, . . . , x_N) ∈ S_F is an optimal solution to problem (1.1)–

(1.3), then we have

g(x_NN) = min{g(z); z∈R_F^N∩X_N}. So, we apply Theorem 2.2 withS₁=R^N_F andS₂=X_N.

Condition (3.9) assures that the constraint qualification in Theorem 2.2 is satisfied. Hence from (3.10) and Theorem 2.2 we obtain (3.5) while from (3.11) and Theorem 2.2 we obtain (3.6).

REFERENCES

[1] J.P. Aubin and H. Frankowska,Set-valued Analysis. Birkh¨auser, Basel, 1990.

[2] A. Cernea, On some second-order necessary conditions for differential inclusion prob- lems. Lect. Notes Nonlinear Anal.2(1998), 113–121.

[3] A. Cernea,Some second-order necessary conditions for nonconvex hyperbolic differential inclusion problems. J. Math. Anal. Appl.253(2001), 616–639.

[4] A. Cernea,Second-order necessary conditions for differential-difference inclusion prob- lems. Nonlinear Anal.62(2005), 963–974.

[5] A. Cernea, Second-order necessary conditions for discrete inclusions with end point constraints. Discuss. Math. Differ. Incl. Control Optim.25(2005), 47–58.

[6] A. Cernea, Minimum principle for a class of discrete inclusions. Math. Reports8(58) (2006),4, 391–399.

[7] A. Cernea,Minimum principle and controllability for multiparameter discrete inclusions via derived cones. Discrete Dyn. Nat. Soc.2006(2006), 1–12.

[8] E.H. Mahmudov, Necessary conditions for optimal control problems described by dif- ferential inclusions with distributed parameters. Soviet. Math. Dokl.38(1989), 467–471.

[9] H.D. Tuan and Y. Ishizuka,On controllability and maximum principle for discrete in- clusions. Optimization 34(1995), 293–316.

[10] H. Zheng,Second-order necessary conditions for differential inclusion problems. Appl.

Math. Optim.30(1994), 1–14.

Received 18 September 2006 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania