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Hindawi Publishing Corporation Advances in Difference Equations Volume 2007, Article ID 40160, 17 pages doi:10.1155/2007/40160 Research Article Necessary Conditions of Optimality for Second-Order Nonlinear Impulsive Differential Equations Y. Peng, X. Xiang, and W. Wei Received 2 February 2007; Accepted 5 July 2007 Recommended by Paul W. Eloe We discuss the existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive differential equations in infinite dimensional spaces. We apply a direct approach to derive the maximum principle for the problem at hand. An example is also presented to demonstrate the theory. Copyright © 2007 Y. Peng et al. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction It is well know n that Pontryagin maximum principle plays a central role in optimal control theory. In 1960, Pontryagin derived the maximum principle for optimal control problems in finite dimensional spaces (see [1]). Since then, the maximum principle for optimal control problems involving first-order nonlinear impulsive differential equations in finite (or infinite) dimensional spaces has been extensively studied (see [2–10]). How- ever, there are a few papers addressing the existence of optimal controls for the systems governed by the second-order nonlinear impulsive differential equations. By reducing wave equation to the customary vector form, Fattorini obtained the maximum principle for time optimal control problem of the semilinear wave e quations (see [6, Chapter 6]). Recently, Peng and Xiang [11, 12] applied the semigroup theory to establish the existence of optimal controls for a class of second-order nonlinear differential equations in infinite dimensional spaces. Let Y beareflexiveBanachspacefromwhichthecontrolsu take the values. We denote a class of nonempty closed and convex subsets of Y by P f (Y). Assume that the multifunc- tion ω : I = [0,T] → P f (Y) is measurable and ω(·) ⊂ E where E is a b ounded set of Y, the admissible control set U ad ={u ∈ L p ([0,T],Y) | u(t) ∈ ω(t)a.e}. U ad =∅(see [13, Page 142 Proposition 1.7 and Page 174 Lemma 3.2]). In this paper, we develop a direct 2AdvancesinDifference Equations technique to derive the maximum principle for a Lagrange problem of systems governed by a class of the second-order nonlinear impulsive differential equation in infinite di- mensional spaces. Consider the following second-order nonlinear impulsive differential equations: ¨ x(t) = A ˙ x(t)+ f  t,x(t), ˙ x(t)  + B(t)u(t), t ∈ (0,T] \ Θ, x(0) = x 0 ,Δ l x  t i  = J 0 i  x  t i  , t i ∈ Θ, i = 1,2, ,n, ˙ x(0) = x 1 ,Δ l ˙ x  t i  = J 1 i  ˙ x  t i  , t i ∈ Θ, i = 1,2, ,n, (1.1) where the A is the infinitesimal generator of a C 0 -semigroup in a B anach space X, Θ = { t i ∈ I | 0 = t 0 <t 1 < ··· <t n <t n+1 = T}, J 0 i , J 1 i (i = 1,2, ,n) are nonlinear maps, and Δ l x(t i ) = x(t i +0)− x(t i ), Δ l ˙ x(t i ) = ˙ x(t i +0)− ˙ x(t i ). We denote the jump in the state x, ˙ x at time t i , respectively, with J 0 i , J 1 i determining the size of the jump at time t i . As a first step, we use the semigroup {S(t), t ≥ 0} generated by A to construct the semigroup generated by the operator matr ix A (see Lemma 2.2). Then, the existence and uniqueness of PC l -mild solution for (1.1) are proved. Next, we consider a Lagrange prob- lem of system governed by (1.1) and prove the existence of optimal controls. In order to derive the optimality conditions for the system (1.1), we consider the associated adjoint equation and convert it to a first-order backward impulsive integro-differential equa- tion with unbounded impulsive conditions. We note that the resulting integro-differential equation cannot be turned into the original problem by simple transformation s = T − t (see (4.9)). Subsequently, we introduce a suitable mild solution for adjoint equation and give a generalized backward Gronwall inequality to find a priori estimate on the solution of adjoint equation. Finally, we make use of Yosida approximation to derive the optimal- ity conditions. The paper is organized as follows. In Section 2, we give associated notations and pre- liminaries. In Section 3, the mild solution of second-order nonlinear impulsive differen- tial equations is introduced and the existence result is also presented. In addition, the existence of optimal controls for a Lagrange problem (P)isgiven.InSection 4, we dis- cuss corresponding the adjoint equation and directly derive the necessary conditions by the calculus of variations and the Yosida approximation. At last, an example is given for demonstration. 2. Preliminaries In this section, we give some basic notations and preliminaries. We present some ba- sic notations and terminologies. Let £(X) be the class of (not necessary bounded) linear operators in Banach space X.£ b (X) stands for the family of bounded linear operators in X.ForA ∈ £(X), let ρ(A) denote the resolvent set and R(λ,A) the resolvent corre- sponding to λ ∈ ρ(A). Define PC l (I,X)(PC r (I,X)) ={x : I → X | x is continuous at t ∈ I \ Θ,x is continuous from left (r ight) and has right- (left-) hand limits at t i ∈ Θ}. PC 1 l (I, X) ={x ∈ PC l (I,X) | ˙ x ∈ PC l (I,X)}, PC 1 r (I,X) ={x ∈ PC r (I,X) | ˙ x ∈ PC r (I,X)}.Set x PC = max  sup t∈I   x(t +0)   ,sup t∈I   x(t − 0)    , x PC 1 =x PC +  ˙ x  PC . (2.1) Y. Pe n g et al. 3 It can be seen that endowed with the norm · PC (· PC 1 )PC l (I,X)(PC 1 l (I,X)) and PC r (I,X)(PC 1 r (I,X)) are Banach spaces. In order to construct the C 0 -semigroup generated by A, we need the following lemma ([14, Theorem 5.2.2]). Lemma 2.1. Let A be a densely defined linear operator in X with ρ(A) =∅.ThentheCauchy problem ˙ x(t) = Ax(t), t>0, x(0) = x 0 (2.2) has a unique classical solution for each x 0 ∈ D(A) if, and only if, A is the infinitesimal gen- erator of a C 0 -semigroup {S(t), t ≥ 0} in X. In the following lemma we construct the C 0 -semigroup generated by A. Lemma 2.2 [12, Lemma 1]. Suppose A is the infinitesimal generator of a C 0 -semigroup {S(t),t ≥ 0} on X. Then A = ( 0 I 0 A ) is the infinitesimal generator of a C 0 -semigroup {S(t),t ≥ 0} on X × X,givenby S(t) =  I  t 0 S(τ)dτ 0 S(t)  . (2.3) Proof. Obviously, A isadenselydefinedlinearoperatorinX × X with ρ(A) =∅accord- ing to assumption. Consider the following initial value problem: ¨ x(t) = A ˙ x(t), t ∈ (0,T], x(0) = x 0 , ˙ x(0) = x 1 ∈ D(A). (2.4) It is to see that the classical solution of (2.4)canbegivenby x(t) = x 0 +  t 0 S(τ)x 1 dτ, ˙ x(t) = S(t)x 1 . (2.5) Setting v 0 (t) = x(t), v 1 (t) = ˙ x(t), v(t) = ( v 0 (t) v 1 (t) ), v 0 = ( x 0 x 1 ) ∈ D(A) = X × D(A), (2.4) can be rewritten as ˙ v(t) = Av(t), t ∈ (0,T], v(0) = v 0 ∈ D(A), (2.6) and (2.6) has a unique classical solution v given by v(t) =  I  t 0 S(τ)dτ 0 S(t)  v 0 . (2.7) Using Lemma 2.1, A generates a C 0 -semigroup {S(t),t ≥ 0}.  In order to study the existence of optimal control and necessary conditions of optimal- ity, we also need some important lemmas. For reader’s convenience, we state the following results. 4AdvancesinDifference Equations Lemma 2.3 [7 , Lemma 3.2]. Suppose A is the infinitesimal generator of a compact sem igroup {S(t),t ≥ 0} in X.ThentheoperatorQ : L p ([0,T],X) → C([0,T],X) with p>1 given by (Qf)(t) =  t 0 S(t − τ) f (τ)dτ (2.8) is strongly continuous. Lemma 2.4 [15, Lemma 1.1]. Let ϕ ∈ C([0,T],X) satisfy the following inequality:   ϕ(t)   ≤ a + b  t 0   ϕ(s)   ds+ c  t 0   ϕ s   B ds ∀t ∈ [0,t], (2.9) where a,b,c ≥ 0 are c onstants, and ϕ s  B = sup 0≤τ≤s ϕ(τ). Then   ϕ(t)   ≤ ae (b+c)t . (2.10) 3. Existence of optimal controls In this section, we not only present the existence of PC l -mild solution of the controlled system (1.1) but also give the existence of optimal controls of systems governed by (1.1). We consider the following controlled system: ¨ x(t) = A ˙ x(t)+ f  t,x(t), ˙ x(t)  + B(t)u(t), t ∈ (0,T] \ Θ, Δ l x  t i  = J 0 i  x  t i  , Δ l ˙ x  t i  = J 1 i  ˙ x  t i  , t i ∈ Θ, x(0) = x 0 , ˙ x(0) = x 1 , u ∈ U ad , (3.1) and natural ly introduce its mild solution. Definit ion 3.1. A function x ∈ PC 1 l (I,X)issaidtobeaPC l -mild solution of the system (3.1)ifx satisfies the following integral equation: x(t) = x 0 +  t 0 S(s)x 1 ds+  t 0  t τ S(s − τ)  f  τ,x(τ), ˙ x(τ)  + B(τ)u(τ)  dsdτ +  0<t i <t  J 0 i  x  t i  +  t t i S  s − t i  J 1 i  ˙ x  t i  ds  . (3.2) For the forthcoming analysis, we need the following assumptions: [B]: B ∈ L ∞ (I,£(Y,X)); [F]: (1) f : I × X × X → X is measurable in t ∈ I and locally Lipschitz continuous w ith respect to last two variables, that is, for all x 1 ,x 2 , y 1 , y 2 ∈ X, satisfying x 1 ,x 2 ,y 1 , y 2 ≤ρ,wehave   f  t,x 1 , y 1  − f  t,x 2 , y 2    ≤ L(ρ)    x 1 − x 2   +   y 1 − y 2    ; (3.3) (2) there exists a constant a>0suchthat   f (t,x, y)   ≤ a  1+x + y  ∀x, y ∈ X; (3.4) Y. Pe n g et al. 5 [J]: (1) J 0 i (J 1 i ):X → X (i = 1,2, ,n) map bounded set of X to bounded set of X; (2) There exist constants e 0 i ,e 1 i ≥ 0 such that maps J 0 i ,J 1 i : X → X satisfy   J 0 i (x) − J 0 i (y)   ≤ e 0 i x − y,   J 1 i (x) − J 1 i (y)   ≤ e 1 i x − y∀x, y ∈ X (i = 1,2, ,n). (3.5) Similar to the proof of existence of mild solution for the first-order impulsive evolution equation (see [16]), one can verify the basic existence result. Here, we have to deal with space PC 1 l (I,X)instead. Theorem 3.2. Suppose that A is the infinitesimal generator of a C 0 -semigroup. Under as- sumptions [B], [F], and [J](1), the syste m (3.1)hasauniquePC l -mild solution for every u ∈ U ad . Proof. Consider the map H given by (Hx)(t) = x 0 +  t 0 S(s)x 1 ds+  t 0  t τ S(s − τ)  f  τ,x(τ), ˙ x(τ)  + B(τ)u(τ)  dsdτ (3.6) on B  x 0 ,x 1 ,1  =  x ∈ C 1  0,T 1  ,X  |   ˙ x(t) − x 1   +   x(t) − x 0   ≤ 1, 0 ≤ t ≤ T 1  , (3.7) where T 1 would be chosen. Using assumptions and properties of semigroup, we can show that H is a contraction map and obtain local existence of mild solution for the following differential equation without impulse: ¨ x(t) = A ˙ x(t)+ f  t,x(t), ˙ x(t)  + B(t)u(t), t ∈ (0,T], x(0) = x 0 , ˙ x(0) = x 1 , u ∈ U ad . (3.8) The global existence comes from a priori estimate of mild solution in space C 1 (I,X) which can be proved by Gronwall lemma. Step by step, the existence of PC l -mild solution of (3.1)canbederived.  Let x u denote the PC l -mild solution of system (3.1) corresponding to the control u ∈ U ad , then we consider the Lagrange problem (P): find u 0 ∈ U ad such that J  u 0  ≤ J(u), ∀u ∈ U ad , (3.9) where J(u) =  T 0 l  t,x u (t), ˙ x u (t),u(t)  dt. (3.10) Suppose that [L]: (1) the functional l : I × X × X × Y → R ∪ {∞} is Borel measurable; (2) l(t, ·,·,·) is sequentially lower semicontinuous on X × Y for almost all t ∈ I; (3) l(t,x, y, ·)isconvexonY for each (x, y) ∈ X × X and almost all t ∈ I; 6AdvancesinDifference Equations (4) there exist constants b ≥ 0, c>0andϕ ∈ L 1 (I,R)suchthat l(t,x, y,u) ≥ ϕ(t)+b   x + y  + cu p Y ∀x, y ∈ X, u ∈ Y. (3.11) Now we can give the following result on existence of the optimal controls for problem (P). Theorem 3.3. Suppose that A is the infinitesimal generator of a compact semigroup. Under assumptions [F], [L], and [J](2), the problem (P) has a solution. Proof. If inf {J(u) | u ∈ U ad }=+∞, there is nothing to prove. We assume that inf {J(u) | u ∈ U ad }=m<+∞. By assumption [L], we have m>−∞. By definition of infimum, there exists a sequence {u n }⊂U ad such that J(u n ) → m. Since {u n } is bounded in L p (I,Y), there exists a subsequence, relabeled as {u n },andu 0 ∈ L p (I,Y)suchthat u n w −→ u 0 in L p (I,Y). (3.12) Since U ad is closed and convex, from the Mazur lemma, we have u 0 ∈ U ad . Suppose x n is the PC l -mild solution of (3.1) corresponding to u n (n = 0,1,2, ). Then x n satisfies the fol lowing integral equation x n (t) = x 0 +  t 0 S(s)x 1 ds+  t 0  t τ S(s − τ)  f  τ,x n (τ), ˙ x n (τ)  + B(τ)u n (τ)  dsdτ +  0<t i <t J 0 i  x n  t i  +  0<t i <t  t t i S  s − t i  J 1 i  ˙ x n  t i  ds. (3.13) Using the boundedness of {u n } and Theorem 3.2, there exists a number ρ>0suchthat x n  PC 1 l (I,X) ≤ ρ. Define η n (t) =  t 0  t τ S(s − τ)B(τ)u n (τ)dsdτ −  t 0  t τ S(s − τ)B(τ)u 0 (τ)dsdτ. (3.14) According to Lemma 2.3,wehave η n −→ 0inC(I,X)asu n w −→ u 0 . (3.15) By assumptions [F], [J](2), Theorem 3.2, and Gronwall lemma with impulse (see [17, Lemma 1.7.1]), there exists a constant M>0suchthat   x n (t) − x 0 (t)   +   ˙ x n (t) − ˙ x 0 (t)   ≤ M   η n   C 1 (I,X) , (3.16) that is, x n −→ x 0 in PC 1 l (I,X)asn −→ ∞ . (3.17) Y. Pe n g et al. 7 Since PC 1 l (I,X)  L 1 (I,X), using the assumption [L] and Balder’s theorem (see [18]), we can obtain m = lim n→∞  T 0 l  t,x n (t),u n (t)  dt ≥  T 0 l  t,x 0 (t),u 0 (t)  dt = J  u 0  ≥ m. (3.18) This means that J attains its minimum at u 0 ∈ U ad .  4. Necessary conditions of optimality In this section, we present necessar y conditions of optimality for Lagrange problem (P). Let (x 0 ,u 0 ) be an optimal pair. [F ∗ ] f satisfies the assumptions [F], f is continuously Frechet differentiable at x 0 and ˙ x 0 , respectively, f 0 x ∈ L 1 (I,£(X)), f 0 ˙ x ∈ L ∞ (I,£(X)), f 0 x (t i ± 0) = f 0 x (t i ), f 0 ˙ x (t i ± 0) = f 0 ˙ x (t i ) for t i ∈ Θ,where f 0 x (t) = f x (t,x 0 (t), ˙ x 0 (t)), f 0 ˙ x (t) = f ˙ x (t,x 0 (t), ˙ x 0 (t)). [L ∗ ] l is continuously Frechet differentiable on x, ˙ x and u, respectively, l 0 x (·) ∈ L 1 (I, X ∗ ), l 0 ˙ x (·) ∈ W 1,1 (I,X ∗ ), l 0 u (·) ∈ L 1 (I,Y ∗ ), l 0 ˙ x (T) ∈ X ∗ , l 0 ˙ x (t i ± 0) = l 0 ˙ x (t i )fort i ∈ Θ,where l 0 x (·) = l x (·,x 0 (·), ˙ x 0 (·),u 0 (·)), l 0 ˙ x (·) = l ˙ x (·,x 0 (·), ˙ x 0 (·),u 0 (·)), l 0 u (·) = l u (·,x 0 (·), ˙ x 0 (·), u 0 (·)). [J ∗ ] J 0 i (J 1 i ) is continuously Frechet differentiable on x 0 ( ˙ x 0 ), and J 10∗ i ˙ x (t i )D(A ∗ ) ⊆ D( A ∗ ), where J 00 ix (t i ) = J 0 ix (x 0 (t i )), J 10 i ˙ x (t i ) = J 1 i ˙ x ( ˙ x 0 (t i )) (i = 1,2, ,n). In order to derive a pr iori estimate on solution of adjoint equation, we need the fol- lowing generalized backward Gronwall lemma. Lemma 4.1. Let ϕ ∈ C(I,X ∗ ) satisfy the following inequality:   ϕ(t)   X ∗ ≤ a +b  T t   ϕ(s)   X ∗ ds+ c  T t   ϕ s   B 0 ds ∀t ∈ I, (4.1) where a,b,c ≥ 0 are c onstants, and ϕ s  B 0 = sup s≤τ≤T ϕ(τ) X ∗ . Then   ϕ(t)   X ∗ ≤ aexp  (b + c)(T − t)  . (4.2) Proof. Setting ϕ(T − t) = ψ(t)fort ∈ I, ψ t  B = sup 0≤τ≤t ϕ(τ) X ∗ ,wehave   ψ(t)   X ∗ ≤ a + b  t 0   ψ(s)   X ∗ ds+ c  t 0   ψ s   B ds. (4.3) Using Lemma 2.4,weobtain   ψ(t)   X ∗ ≤ aexp  (b + c)t  ; (4.4) further,   ϕ(t)   X ∗ ≤ aexp  (b + c)(T − t)  . (4.5) The proof is completed.  8AdvancesinDifference Equations Let X be a reflexive Banach space, let A ∗ be the adjoint operator of A,andlet{S ∗ (t),t ≥ 0} be the a djoint semigroup of {S(t),t ≥ 0}.ItisaC 0 -semigroup and its generator is just A ∗ (see [14, Theorem 2.4.4]). We consider the following adjoint equation: ϕ  (t) =−  A ∗ ϕ(t)   −  f 0∗ ˙ x (t)ϕ(t)   + f 0∗ x (t)ϕ(t)+l 0 x (t) − l 0  ˙ x (t), t ∈ [0,T) \ Θ, ϕ(T) = 0, Δ r ϕ  t i  = J 10∗ i ˙ x  t i  ϕ  t i  , t i ∈ Θ, ϕ  (T) =−l 0 ˙ x (T), Δ r ϕ   t i  = G i  ϕ  t i  ,ϕ   t i  , t i ∈ Θ, (4.6) where G i  ϕ  t i  ,ϕ   t i  =  J 00∗ ix  t i  A ∗ + f 0∗ ˙ x  t i  −  A ∗ + f 0∗ ˙ x  t i  J 10∗ i ˙ x  t i  ϕ  t i  +J 00∗ ix  t i  ϕ   t i  + J 00∗ ix  t i  l 0 ˙ x  t i  . (4.7) A function ϕ ∈ PC 1 r (I,X ∗ )  PC r (I,D(A ∗ )) is said to be a PC r -mild solution of (4.6) if ϕ is given by ϕ(t) =  T t S ∗ (τ − t)   T τ  f 0∗ x (s)ϕ(s) − l 0 x (s)+l 0  ˙ x (s)  ds+ f 0∗ ˙ x (τ)ϕ(τ)+l 0 ˙ x (T)  dτ +  t i >t S ∗  t i − t  J 10∗ i ˙ x  t i  ϕ  t i  +  t i >t  t i t S ∗ (τ − t)G i  ϕ  t i  ,ϕ   t i  dτ. (4.8) Lemma 4.2. Assume that X is a reflexive Banach space. Under the assumptions [F ∗ ], [L ∗ ], [J ∗ ], the evolution (4.6)hasauniquePC r -mild solution ϕ ∈ PC 1 r (I,X ∗ ). Proof. Consider the following equation: ϕ  (t)+  A ∗ + f 0∗ ˙ x (t)  ϕ(t)+  T t  f 0∗ x (s)ϕ(s)+l 0 x (s) − l 0  ˙ x (s)  ds =  t i >t G i  ϕ  t i  ,ϕ   t i  − l 0 ˙ x (T), t ∈ I \ Θ, ϕ(T) = 0, Δ r ϕ  t i  = J 10∗ i ˙ x  t i  ϕ  t i  , t i ∈ Θ. (4.9) Equation (4.9) is a linear impulsive integro-differential equation. Setting t = T − s, ψ(s) = ϕ(T − s), (4.9)canberewrittenas ψ  (s) =  A ∗ + f 0∗ ˙ x (T − s)  ψ(s)+F(s)+  s i <s g i  ψ  s i  ,ψ   s i  , s ∈ [0,T) \ Λ, ψ(0) = 0, Δ l ψ  s i  = J 10∗ i ˙ x  t i  ψ  s i  , s i ∈ Λ =  s i = T − t i | t i ∈ Θ  , (4.10) Y. Pe n g et al. 9 where g i  ψ  s i  ,ψ   s i  =  A ∗ + f 0∗ ˙ x  t i  J 10∗ i ˙ x  t i  − J 00∗ ix  t i  A ∗ + f 0∗ ˙ x  t i  ψ  s i  + J 00∗ ix  t i  ψ   t i  − J 00∗ ix  t i  l 0 ˙ x  t i  , F(s) =  T T −s  f 0∗ x (θ)ψ(T − θ)+l 0 x (θ) − l 0  ˙ x (θ)  dθ +l 0 ˙ x (T). (4.11) Obviously, if ϕ is the classical solution of (4.9), then it must be the PC r -mild solution of (4.6). Now we show that (4.9) has a unique classical solution ϕ ∈ PC 1 (I,X ∗ )  PC(I, D( A ∗ )). For s ∈ [0,s n ], prove that the following equation: ψ  (s) = A ∗ ψ(s)+ f 0∗ ˙ x (T − s)ψ(s)+F(s), ψ(0) = 0, (4.12) has a unique classical solution ψ ∈ C 1 ([0,s n ],X ∗ )  C([0,s n ],D(A ∗ )) given by ψ(s) =  s 0 S ∗ (s − τ)  f 0∗ ˙ x (T − τ)ψ(τ)+F(τ)  dτ. (4.13) By following the same procedure as in [16, Theorem 4.A], one can verify that (4.12) has a unique mild solution ψ ∈ C([0,s n ],X ∗ ) given by expression (4.13). By the definition of F,itiseasytoseethatF ∈ L 1 ([0,s n ],X ∗ )  C((0,s n ),X ∗ ). Using (4.13) and the basic properties of C 0 -semigroup, we obtain ψ(s) ∈ D(A ∗ )fors ∈ [0, s n ] and ψ  (s) = f 0∗ ˙ x (T − s)ψ(s)+F(s)+A ∗  s 0 S ∗ (s − τ)  f 0∗ ˙ x (T − τ)ψ(τ)+F(τ)  dτ. (4.14) This implies ψ ∈ C 1 ((0,s n ),X ∗ )andψ  (s n −) = ψ  (s n ). Using [14, Theorem 5.2.13], (4.12) has a unique classical solution ψ ∈ C 1 ((0,s n ),X ∗ )  C([0,s n ],D(X ∗ )) given by the expres- sion (4.13). In addition, the expressions (4.13)and(4.12)implyψ(0) = 0, ψ  (0) = l 0 ˙ x (T), and ψ(s n − 0), ψ  (s n − 0) exist. Furthermore, ψ ∈ C 1 ([0,s n ],X ∗ )  C([0,s n ],D(A ∗ )). By assumption [J ∗ ], we have ψ 0 n = ψ  s n  + J 10∗ n ˙ x  t n  ψ  s n  ∈ D( A ∗ ), ψ 1 n = ψ   s n  + g n  ψ  s n  ,ψ   s n  ∈ X ∗ . (4.15) For s ∈ (s n ,s n−1 ], consider the following equation: ψ  (s) =  A ∗ + f 0∗ ˙ x (T − s)  ψ(s)+  T−s n T−s  f 0∗ x (θ)ψ(T − θ)+l 0 x (θ) − l 0  ˙ x (θ)  dθ +ψ 1 n , ψ  s n +  = ψ 0 n , (4.16) 10 Advances in Difference Equations that is, study the following equation: ψ  (s) =  A ∗ + f 0∗ ˙ x (T − s)  ψ(s)+F(s)+g n  ψ  s n  ,ψ   s n  , ψ  s n +  = ψ 0 n . (4.17) By following the same procedure as on time interval [0,s n ], it has a unique classical solu- tion given by ψ(s) = S ∗  s − s n  ψ 0 n +  s s n S ∗ (s − τ)  f 0∗ ˙ x (T − τ)ψ(τ)+F(τ)+g n  ψ  s n  ,ψ   s n  dτ. (4.18) In general, for s ∈ (s i ,s i−1 ](i = 0, 1, ,n), consider the following equation: ψ  (s) =  A ∗ + f 0∗ ˙ x (T − s)  ψ(s)+F(s)+g i  ψ  s i  ,ψ   s i  , ψ  s i  = ψ  s i  + J 10∗ i ˙ x  t i  ψ  s i  ∈ D  A ∗  . (4.19) It has a unique classical solution given by ψ(s) = S ∗  s − s i  ψ 0 i +  s s i S ∗ (s − τ)  f 0∗ ˙ x (T − τ)ψ(τ)+F(τ)+g i  ψ  s i  ,ψ   s i  dτ. (4.20) Repeating the procedure till the time interval which is expanded, and combining all of the solutions on [t i ,t i+1 ](i = 0, 1, ,n), we obtain classical solution of (4.10)givenby ψ(s) =  s 0 S ∗ (s − τ)  f 0∗ ˙ x (T − τ)ψ(τ)+F(τ)  dτ +  0<s i <s  S ∗  s − s i  J 10∗ i ˙ x  t i  ψ  s i  +  s s i S ∗ (s − τ)g i  ψ  s i  ,ψ   s i  dτ  . (4.21) Further , (4.9) has a unique classical solution ϕ ∈ PC 1 (I,X ∗ )  PC(I,D(A ∗ )) given by (4.8).  Using the assumption [F ∗ ], [3, Corollary 3.2], and [2,Theorem2],{A ∗ (t) = A ∗ + f 0∗ ˙ x (t) | t ∈ I} generates a strongly continuous evolution operator U ∗ (t,s), 0 ≤ s ≤ t ≤ T. For simplicity, we have the following result. Remark 4.3. The PC-mild solution ϕ of (4.6)canberewrittenas ϕ(t) =  T t U ∗ (τ,t)   T τ  f 0∗ x (s)ϕ(s)+l 0 x (s) − l 0  ˙ x (s)  ds+ l 0 ˙ x (T)  dτ +  t i >t U ∗  t i ,t  J 10∗ i ˙ x  t i  ϕ  t i  +  t i >t  t i t U ∗ (τ,t)G i  ϕ  t i  ,ϕ   t i  dτ. (4.22) Now we can give the necessary conditions of optimality for Lagrange problem (P). [...]... control for a class of strongly nonlinear impulsive equations in Banach spaces,” Nonlinear Analysis, vol 63, no 5–7, pp e53–e63, 2005 [9] X Xiang and N U Ahmed, Necessary conditions of optimality for differential inclusions on Banach space,” Nonlinear Analysis, vol 30, no 8, pp 5437–5445, 1997 [10] X Xiang, W Wei, and Y Jiang, “Strongly nonlinear impulsive system and necessary conditions of optimality, ”... Dynamics of Continuous, Discrete & Impulsive Systems A, vol 12, no 6, pp 811– 824, 2005 [11] Y Peng and X Xiang, “Second order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls,” to appear in Optimization [12] Y Peng and X Xiang, Necessary conditions of optimality for second order nonlinear evolution equations on Banach spaces,” in Proceedings of the... for impulsive systems in Banach spaces,” International Journal of Differential Equations and Applications, vol 1, no 1, pp 37–52, 2000 [3] N U Ahmed, Necessary conditions of optimality for impulsive systems on Banach spaces,” Nonlinear Analysis, vol 51, no 3, pp 409–424, 2002 Y Peng et al 17 [4] A G Butkovski˘, “The maximum principle for optimum systems with distributed parameters,” ı Avtomatika i... Wei, “A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces,” Discrete and Continuous Dynamical Systems, vol 2005, supplement, pp 911–919, 2005 [17] T Yang, Impulsive Control Theory, vol 272 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2001 [18] E J Balder, Necessary and sufficient conditions for L1 -strong-weak... Science Foundation Of China under Grant no 10661004 and the Science and Technology Committee of Guizhou Province under Grant no 20052001 References [1] L S Pontryagin, “The maximum principle in the theory of optimal processes,” in Proceedings of the 1st International Congress of the IFAC on Automatic Control, Moscow, Russia, June-July 1960 [2] N U Ahmed, “Optimal impulse control for impulsive systems... 15 for λk ∈ ρ(A∗ ) > 0 Taking the limit k → ∞, we find that T 0 ϕ(t),B(t) u(t) − u0 (t) T = X ∗ ,X dt 0 0 lx (t) − l˙x (t), y(t) ˙ 0 n X ∗ ,X dt + 0 lx (T), y(T) ˙ X ∗ ,X 0 lx ti ,Δl y ti ˙ − i =1 X ∗ ,X (4.49) Further, T 0 lu t,x0 (t),u0 (t) + B∗ (t)ϕ(t),u(t) − u0 (t) Y ∗ ,Y dt ≥ 0, ∀u ∈ Uad (4.50) Thus, we have proved all the necessary conditions of optimality given by (4.23)–(4.25) At the end of. .. (4.29) n 0 + lx (T), y(T) ˙ X ∗ ,X 0 lx ti ,Δl y ti ˙ − X ∗ ,X i=1 ≥ 0 Due to the reflexivity of Banach space X, we have the Yosida approximation λk R(λk , A∗ ) → I ∗ as λk → ∞, where R(λk ,A∗ ) is the resolvent of A∗ for λk ∈ ρ(A∗ ) and I ∗ stands 0 0 for the identity operator in X ∗ Consider the Yosida approximation of fx0∗ , fx0∗ , lx , lx , ˙ ˙ 0 00∗ 10∗ lx (T), Jix (ti ), Jix (ti ) given by ˙ ˙ fxk∗... International Conference on Impulsive and Hyprid Dynamical Systems, pp 433–437, Nanning, China, 2007 [13] S Hu and N S Papageorgiou, Handbook of Multivalued Analysis Vol I: Theory, vol 419 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997 [14] N U Ahmed, Semigroup Theory with Applications to Systems and Control, vol 246 of Pitman Research Notes in Mathematics... (4.32) Similar to the proof of Lemma 4.2, one can show that (4.31) has a unique class solution ϕk given by ϕk (t) = T t U ∗ (τ,t) T τ k k k fxk∗ (s)ϕk (s) + lx (s) − lx (s) ds + lx (T) dτ ˙ ˙ U ∗ ti ,t Jik˙∗ ti ϕk ti + x + ti >t ti t U ∗ (τ,t)Gk ϕk ti ,ϕk ti dτ i (4.33) Y Peng et al 13 Next, show that 1 in PCr I,X ∗ as λk −→ ∞ → ϕk − ϕ (4.34) Employing the method of proof for Lemma 4.2, there exists... -strong-weak lower semicontinuity of integral functionals,” Nonlinear Analysis, vol 11, no 12, pp 1399–1404, 1987 Y Peng: Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China Email address: pengyf0803@163.com X Xiang: Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China Email address: xxl3621070@yahoo.com.cn W Wei: Department of Mathematics, Guizhou University, . Difference Equations Volume 2007, Article ID 40160, 17 pages doi:10.1155/2007/40160 Research Article Necessary Conditions of Optimality for Second-Order Nonlinear Impulsive Differential Equations Y control for a class of strongly nonlinear impulsive equations in Banach spaces,” Nonlinear Analysis, vol. 63, no. 5–7, pp. e53–e63, 2005. [9] X. Xiang and N. U. Ahmed, Necessary conditions of optimality. that J attains its minimum at u 0 ∈ U ad .  4. Necessary conditions of optimality In this section, we present necessar y conditions of optimality for Lagrange problem (P). Let (x 0 ,u 0 ) be an