# Positive Solutions for Impulsive Differential Equations with Mixed Monotonicity and Optimal Control

Authors: Lingling Zhang; Noriaki Yamazaki; Rui Guo
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition,  related examples will be given for illustrations.


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## Body

## 1. Introduction

Mixed monotone operators have been introduced by Guo and Lakshmikantham [1] in 1987. Recently, many authors have investigated those kinds of operators in Banach spaces and obtained a lot of interesting and important results (see [2–9]). In this work, by using a fixed point theorem of mixed monotone operator, we study the existence and uniqueness of positive solutions to the boundary value problem of impulsive differential equations with mixed monotone terms:
(1)(BP;u){-x′′(t)=a(t)f(t,x(t),x(t))+u(t),t∈(0,1)∖{t1,t2,…,tm},Δx|t=tk=Ik(x(tk),x(tk)),k=1,2,…,m,x(0)=b0,x′(1)=0.
Here, J=[0,1],R+=[0,+∞),f∈C[J×R+×R+,R+], and a∈C[J,R+] with min⁡t∈Ja(t)>0 on any subinterval of J. A function u is given on [0,1], 0<t1<t2<⋯<tm<1,Δx|t=tkdenotes the jump of x(t) at t=tk and Δx|t=tk=x(tk+)-x(tk-), where x(tk+) and x(tk-) represent the right and left limits of x(t) at t=tk, respectively. Also, Ik is a given function in C[R+×R+,R+],k=1,2,…,m. Furthermore, b0>0 is a given constant.For convenience, we putJ0=[0,t1],J1=(t1,t2],…,Jm-1=(tm-1,tm],Jm=(tm,1], and J′=J∖{t1,t2,…,tm}.Then, we study the existence and uniqueness of positive solutions to initial value problem as follows:(2)(IP;u){-x′′(t)=a(t)f(t,x(t),x(t))+u(t),t∈(0,1)∖{t1,t2,…,tm},Δx|t=tk=Ik(x(tk),x(tk)),k=1,2,…,m,x(0)=b0,x′(0)=b1,
where f∈C[J×R+×R+,R] and b0>0 and b1⩾0 are given constants. Moreover, we consider the related optimal control problem (OP) of (2) as follows:Problem (OP). Find an optimal control u*∈UM such that
(3)π(u*)=inf⁡u∈UMπ(u).
Here, UM is a control space defined by
(4)UM:={u∈L2(0,1)∣-M⩽u(t)⩽0a.e.t∈[0,1]},
where M is a fixed positive number and π(u) is the cost functional defined by
(5)π(u):=12∫01‍|(x-xd)(t)|2dt+x(1)+12∫01|u(t)|2‍dt,
where u∈UM is the control, function x is a unique positive solution to the state problem (IP; u), and xd is the given desired target profiles in L2(0,1).The existence and uniqueness of solutions for boundary value problem have been discussed by many authors, and the boundary value problem of impulsive differential equation is a new and important branch of the differential equation theory, which has an extensive physical, chemical, biological, and engineering background, realistic mathematical model, and so forth (see [10–14]). The theory on mixed monotone operators has attracted much attention and has been widely studied, such as Guo and Lakshmikantham [1] have applied the monotone iterative technique to discuss an initial value problem of differential equations without impulse:
(6)u′(t)=f(t,u(t),u(t)),t∈[0,a],u(0)=x0.
They obtained the existence of the coupled quasisolutions by mixed monotone sequence of coupled quasi upper and lower solutions. Zhai and Zhang [7] showed a new fixed point theorem for differential equations with mixed monotone term. Jinli and Yihai [15] considered the following problem:
(7)u′′(x)-f(x,u,u)=θ,x∈J,x≠xi,i=1,2,…,m,Δu|x=xi=Ii(u(xi)),i=1,2,…,m,Δu′|x=xi=I~i(u(xi)),i=1,2,…,m,u(0)=w0,u′(0)=w1.
They used the coupled fixed point theorem for mixed monotone condensing operators to obtain the existence and uniqueness of solutions.Also, there is a vast literature on optimal control problems (see [16–19]). For instance, with a fixed point theorem of generalized concave operator, the authors [19] have studied the optimal control problem of positive solutions to the following second order impulsive differential equation:
(8)-x′′(t)=f(t,x(t))+u(t),t∈(0,T)∖{t1,t2,…,tm},Δx|t=tk=Ik(x(tk)),k=1,2,…,m,Δx′|t=tk=I¯k(x(tk)),k=1,2,…,m,x(0)=a,x′(0)=b.In this paper, we investigate impulsive differential equations with mixed monotone terms, which have variable coefficient nonlinear terms. Then, we prove the existence and uniqueness of positive solutions to (BP;u) and (IP; u). Moreover, we prove the existence of an optimal control to (OP) and its stability.The plan of this paper is as follows. In Section2, we recall the fundamentals of a fixed point theorem of mixed monotone operators. In Section 3, we deal with the existence and uniqueness of positive solutions to (BP; u). In Section 4, we show the existence and uniqueness of positive solutions to (IP; u). In Section 5, we prove the existence of an optimal control to (OP) and its stability. In the final Section 6, related examples on the main results are given.


### 1.1. Notations

Throughout this paper, we use the following notations.LetPC[J,R]:={x∣x:J→R,x(t) is continuous at t≠tk and left continuous at t=tk,x(tk+) exists, k=1,2,…,m}. Then, we can easily find that PC[J,R] is a Banach space with the norm |x|PC:=sup⁡t∈J|x|.We putH:=L2(J) with the usual real Hilbert structure and denote by |·|H the norm in H, for simplicity, and W2,1(J,R) is a usual Sobolev space, namely,
(9)W2,1(J,R)={f∈L1(J):Dkf∈L1(J),k=1,2},
where Dkf denoted the kth derivative of f.Also,Ni and Ni′,i=1,2,3,…, denote positive (or nonnegative) constants only depending on their arguments.


## 1.1. Notations

Throughout this paper, we use the following notations.LetPC[J,R]:={x∣x:J→R,x(t) is continuous at t≠tk and left continuous at t=tk,x(tk+) exists, k=1,2,…,m}. Then, we can easily find that PC[J,R] is a Banach space with the norm |x|PC:=sup⁡t∈J|x|.We putH:=L2(J) with the usual real Hilbert structure and denote by |·|H the norm in H, for simplicity, and W2,1(J,R) is a usual Sobolev space, namely,
(9)W2,1(J,R)={f∈L1(J):Dkf∈L1(J),k=1,2},
where Dkf denoted the kth derivative of f.Also,Ni and Ni′,i=1,2,3,…, denote positive (or nonnegative) constants only depending on their arguments.


## 2. A Fixed Point Theorem of Mixed Monotone Operator

In this section, we recall the fundamentals of a fixed point theorem of mixed monotone operator.Suppose that(E,∥·∥) is a real Banach space which is partially ordered by a cone P⊂E, that is, x⩽y if and only if y-x∈P. If x⩽y and x≠y, then we denote x<y or y>x. By θ we denote the zero element of E. Recall that a nonempty closed convex set P⊂E is a cone if it satisfies(i)
x∈P,λ≥0⇒λx∈P;(ii)
x∈P,-x∈P⇒x=θ.PuttingP°={x∈P∣x is an interior point of P}, a cone P is said to be solid if its interior P° is nonempty. Moreover, P is called normal if there exists a constant M>0 such that, for all x,y∈E,θ⩽x⩽y implies ∥x∥⩽M∥y∥; in this case M is called the normality constant of P. If x1,x2∈E, the set [x1,x2]={x∈E∣x1⩽x⩽x2} is called the order interval between x1 and x2.For allx,y∈E, the notation x~y means that there exist λ>0 and μ>0 such that λx⩾y⩾μx. Clearly,~ is an equivalence relation. Giving h>θ (i.e., h⩾θ and h≠θ), we denote by Ph the set Ph={x∈E∣x~h}. It is easy to see that Ph⊂P is convex and λPh=Ph for all λ>0. If P°≠∅ and h∈P°, it is clear that Ph=P. For other detailed properties of cones, we refer to the monograph by Guo and Lakshmikantham [20].Definition 1 (cf. [1, 2]).
A
:
P
×
P
→
P is said to be a mixed monotone operator if A(x,y) is increasing in x and decreasing in y, that is, ui,vi∈P(i=1,2),u1⩽u2,v1⩾v2 implies that A(u1,v1)⩽A(u2,v2). Element x∈P is called a fixed point of A if A(x,x)=x.Here, one recalls the following fixed point theorem of mixed monotone operator which has been established by Zhai and Zhang [7].Proposition 2 (cf. [7, Theorem 2.1]).
LetP be a normal cone of a real Banach space E. Also, let A:P×P→P be a mixed monotone operator. Assume that(A1)
there existsh∈Pwithh≠θsuch thatA(h,h)∈Ph;(A2)
for anyu,v∈P and t∈(0,1),there existsφ(t)∈(t,1]such thatA(tu,t-1v)⩾φ(t)A(u,v).Then operator A has a unique fixed point x* in Ph. Moreover, for any initial x0,y0∈Ph, constructing successively the sequences
(10)xn=A(xn-1,yn-1),yn=A(yn-1,xn-1),in=1,2,…,
one has ∥xn-x*∥→0 and ∥yn-x*∥→0 as n→∞.By applying Proposition2, one shows the existence and uniqueness of the positive solution to (BP; u) and (IP; u) on J.


## 3. Boundary Value Problem (BP;u)

In this section, we show the existence and uniqueness of the positive solution to (BP;u) by applying a fixed point theorem of mixed monotone operator (Proposition 2).Throughout this section, we assume the following conditions (H1)–(H3):(H1)
f:J×R+×R+→R+,f(t,x,y) is nondecreasing in x for each t∈J and y∈R+, and is nonincreasing in y for each t∈J and x∈R+. Also, f(t,(1/2),1)>0 for all t∈J.(H2)
For eachk=1,2,…,m, Ik:R+×R+→R+,Ik(x,y) is nondecreasing in x for each y∈R+ and is nonincreasing in y for each x∈R+.(H3)
For allγ∈(0,1), there exists a constant φ1(γ),φ2(γ)∈(γ,1] such that
(11)f(t,γx,γ-1y)⩾φ1(γ)f(t,x,y),Ik(γx,γ-1y)⩾φ2(γ)Ik(x,y).
for any x,y∈R+, any t∈J, and any k=1,2,…,m.We give the definition of solutions to (BP;u).Definition 3.
Letu∈H, and let b0 be a given constant. Then, a function x∈PC[J,R]∩W2,1(J,R) is called a solution to (BP; u) on J if it satisfies (1).Now, we mention our first main theorem in this paper, which is concerned with the existence-uniqueness of the positive solution to (BP;u) on J.Theorem 4.
Assuming the conditions (H1)–(H3), and having M has a fixed positive constant, then for each function u∈H with 0⩽u(t)⩽M a.e. t∈J, there exists a unique positive solution to (BP; u) on J.Here, we give the key lemma, which is concerned with the characterization of solutions to (BP;u).Lemma 5.
Assume the same conditions as in Theorem4. Then, x∈PC[J,R]∩W2,1(J,R) is a solution to (BP; u) on J if and only if x∈PC[J,R] satisfies the following integral equation:
(12)x(t)=b0+∫01‍G(t,s){a(s)f(s,x(s),x(s))+u(s)}ds+∑0<tk<t‍Ik(x(tk),x(tk)),∀t∈J,
where
(13)G(t,s)={t,0⩽t⩽s⩽1,s,0⩽s⩽t⩽1.Proof.
Firstly, integrating-x′′(s)=a(s)f(s,x(s),x(s))+u(s) from 0 to t, we obtain
(14)x′(t)=x′(0)-∫0t‍{a(s)f(s,x(s),x(s))+u(s)}ds,{a(s)f(s,x(s)WWWIx(s))+u(s)}ds∀t∈J.
Again, integrating (14), we have
(15)x(t)=x(0)+x′(0)t-∫0t‍(t-s){a(s)f(s,x(s),x(s))+u(s)}ds+∑0<tk<t‍{x(tk+)-x(tk)},∀t∈J.
From (14) with x′(1)=0, we infer that
(16)x′(0)=∫01‍{a(s)f(s,x(s),x(s))+u(s)}ds.
Hence, from x(0)=b0, (15), and (16), we can find that
(17)x(t)=b0+∫01‍G(t,s){a(s)f(s,x(s),x(s))+u(s)}ds+∑0<tk<t‍Ik(x(tk),x(tk)),∀t∈J,
where G(t,s) is the function defined as in (13). Thus, the proof is completed.By Lemma5, we can show the solvability of (BP; u). In fact, we define an operator A:PC[J,R]×PC[J,R]→PC[J,R] by
(18)A(x,y)(t)=b0+∫01‍G(t,s){a(s)f(s,x(s),y(s))+u(s)}ds+∑0<tk<t‍Ik(x(tk),y(tk))∀x,y∈PC[J,R],t∈J,
where G(t,s) is the function defined as in (13).We can easily find that the following lemma holds.Lemma 6.
Assume the same conditions as in Theorem4. Then, x∈PC[J,R]∩W2,1(J,R) is a solution to (BP; u) on J if and only if x∈PC[J,R] is the fixed point of the operator A:PC[J,R]×PC[J,R]→PC[J,R] defined by (18).Taking account of Proposition2 and Lemmas 5 and 6, one can prove Theorem 4 concerning the existence and uniqueness of the positive solution to (BP; u) on J.Proof of Theorem4.
By applying a fixed point theorem of mixed monotone operator (Proposition2), we show the existence and uniqueness of the positive solution to (BP; u) on J.
To do so, set(19)P~:={x∈PC[J,R];x(t)⩾0,∀t∈J}.
Clearly, P~ is a normal cone in PC[J,R] and the normality constant is 1.
LetA:PC[J,R]×PC[J,R]→PC[J,R] be the operator defined by (18). Then, we infer from (H1), (H2), (13), and u(t)⩾0 a.e. t∈J that
(20)A(x,y)(t)⩾0∀x,y∈P~,t∈J.
Thus, we see that A:P~×P~→P~.
Firstly, by (H1), (H2), and (18), we can easily prove that A:P~×P~→P~ is a mixed monotone operator.
Next, we show(A2). Put
(21)φ(γ)=min⁡{φ1(γ),φ2(γ)},γ∈(0,1).
Then, we see from (H3) that φ(γ)∈(γ,1]. Therefore, for any γ∈(0,1) and x,y∈P~, we observe from (H1)–(H3), (13), and u(t)⩾0 a.e. t∈J that
(22)A(γx,γ-1y)(t)=b0+∫01‍G(t,s){a(s)f(s,γx(s),γ-1y(s))+u(s)γ-1}ds+∑0<tk<t‍Ik(γx(tk),γ-1y(tk))⩾b0+φ1(γ)∫01‍G(t,s){a(s)f(s,x(s),y(s))+u(s)}ds+φ2(γ)∑0<tk<t‍Ik(x(tk),y(tk))⩾φ(γ)A(x,y)(t),∀t∈J,
which implies that
(23)A(γx,γ-1y)⩾φ(γ)A(x,y)∀x,y∈P~,γ∈(0,1).
Thus, the condition (A2) holds.
Now, we show (A1), defining a function h by
(24)h(t):=12+∫01‍G(t,s)ds,∀t∈J;
hence, h(t)=-(t2/2)+t+(1/2) for all t∈J, then we can easily see that (1/2)⩽h(t)⩽1 for all t∈J.
Now we show thatA(h,h)∈P~h. Set
(25)r1=min⁡t∈J⁡f(t,12,1),r2=max⁡t∈J⁡f(t,1,12),
then, 0<r1⩽r2.
Note thata(t) has maximum and minimum on J, since a(t)∈C[J,R+] with min⁡t∈Ja(t)>0 on any subinterval of J. So, let
(26)amin⁡=min⁡t∈J⁡a(t),amax⁡=max⁡t∈J⁡a(t).
Here, putr3:=min⁡{2b0,r1amin⁡}. Then, from (H1), (H2), (13), and u(t)⩾0 a.e. t∈J, it follows that
(27)A(h,h)(t)=b0+∫01‍G(t,s){a(s)f(s,h(s),h(s))+u(s)}ds+∑0<tk<t‍Ik(h(tk),h(tk))⩾b0+amin⁡∫01‍G(t,s)f(s,12,1)ds⩾b0+r1amin⁡∫01‍G(t,s)ds⩾r3h(t),∀t∈J.
Also, we have(28)A(h,h)(t)=b0+∫01‍G(t,s){a(s)f(s,h(s),h(s))+u(s)}ds+∑0<tk<t‍Ik(h(tk),h(tk))⩽b0+amax⁡∫01‍G(t,s)f(s,1,12)ds+M∫01‍G(t,s)ds+∑0<tk<t‍Ik(1,12)⩽[2b0+r2amax⁡+M+2∑k=1m‍Ik(1,12)]h(t),ppppppppppppppppppppppppppppppp∀t∈J.
Thus, we observe that
(29)r3h⩽A(h,h)⩽[2b0+r2amax⁡+M+2∑k=1m‍Ik(1,12)]h,
which implies that A(h,h)∈P~h.
By arguments as above, we see that the operatorA:P~×P~→P~ defined by (18) satisfies conditions (A1) and (A2) in Proposition 2. Therefore, by applying Proposition 2, we conclude that an operator equation x=A(x,x) has a unique solution in P~h; hence there exists a unique positive solution to (BP; u) on J.


## 4. Initial Value Problem (IP;u)

In this section, we show the existence-uniqueness of the positive solution to (IP;u) on J by arguments similar to (BP; u).Throughout this section, we assume the following conditions(H1)′, (H3)′:(H1)′
f:J×R+×R+→R, such that f(t,x,y)⩽0 for all t∈J and x,y∈R+. Also, f(t,x,y) is nonincreasing in x for each t∈J and y∈R+ and is nondecreasing in y for each t∈J and x∈R+. Moreover, f(t,(1/2),1)<0 for all t∈J.(H3)′
For allγ∈(0,1), there exists a constant φ1(γ),φ2(γ)∈(γ,1] such that
(30)f(t,γx,γ-1y)⩽φ1(γ)f(t,x,y),Ik(γx,γ-1y)⩾φ2(γ)Ik(x,y),
for any x,y∈R+, any t∈J, and any k=1,2,…,m.Here, we give the definition of solutions to (IP;u).Definition 7.
Letu∈H and b0 and b1 as given constants. Then, a function x∈PC[J,R]∩W2,1(J,R) is called a solution to (IP; u) on J if it satisfies (2).Now, we mention our second main theorem in this paper, which is concerned with the existence-uniqueness of the positive solution to (IP;u) on J.Theorem 8.
Assuming the conditions(H2), (H1)′, and (H3)′ and having M has a fixed positive constant. Then, for each function u∈H with -M⩽u(t)⩽0 a.e. t∈J, there exists a unique positive solution to (IP; u) on J.Based on the proof of Lemma5 (cf. (15)), one can get the following key lemma concerning the characterization of solutions to (IP; u).Lemma 9.
Assume the same conditions as in Theorem8. Then, x∈PC[J,R]∩W2,1(J,R) is a solution to (IP; u) on J if and only if x∈PC[J,R] satisfies the following integral equations:
(31)x(t)=b0+b1t-∫0t‍(t-s){a(s)f(s,x(s),x(s))+u(s)}ds+∑0<tk<t‍Ik(x(tk),x(tk)),∀t∈J.By Lemma9 and Proposition 2, one can show Theorem 8 concerning the existence-uniqueness of the positive solution to (IP; u) on J.Proof of Theorem8.
Now, we define an operatorA:PC[J,R]×PC[J,R]→PC[J,R] by
(32)A(x,y)(t)=b0+b1t-∫0t‍(t-s){a(s)f(s,x(s),y(s))+u(s)}ds+∑0<tk<t‍Ik(x(tk),y(tk))∀x,y∈PC[J,R],t∈J.
Then, we have to find the fixed point of the operator A in order to show the existence-uniqueness of the solution to (IP; u) on J.
LetP~ be the same space defined by (19). Then, we infer from (H1)′, (H2), (32), and u(t)⩽0 a.e. t∈J that
(33)A(x,y)(t)⩾0∀x,y∈P~,t∈J.
Thus, we see that A:P~×P~→P~.
Also, we observe from(H1)′, (H2), and (32) that A:P~×P~→P~ is a mixed monotone operator.
Next, we show(A2). Put
(34)φ(γ)=min⁡{φ1(γ),φ2(γ)},γ∈(0,1).
Then, we see from (H3)′ that φ(γ)∈(γ,1]. Therefore, for any γ∈(0,1) and x,y∈P~, we observe from (H2), (H1)′, (H3)′, (32), and u(t)⩽0 a.e. t∈J that
(35)A(γx,γ-1y)(t)=b0+b1t-∫0t‍(t-s){a(s)f(s,γx(s),γ-1y(s))+u(s)γ-1}ds+∑0<tk<t‍Ik(γx(tk),γ-1y(tk))⩾b0+b1t-φ1(γ)∫0t‍(t-s){a(s)f(s,x(s),y(s))+u(s)}ds+φ2(γ)∑0<tk<t‍Ik(x(tk),y(tk))⩾φ(γ)A(x,y)(t),∀t∈J,
which implies that
(36)A(γx,γ-1y)⩾φ(γ)A(x,y)∀x,y∈P~,γ∈(0,1).
Thus, the condition (A2) holds.
Now, we show (A1), defining a function h by
(37)h(t):=12+∫0t‍(t-s)ds,∀t∈J;
hence, h(t)=(t2/2)+(1/2) for all t∈J. Then, we can easily see that (1/2)⩽h(t)⩽1 for all t∈J.
Now we show thatA(h,h)∈P~h. Set
(38)r1=min⁡t∈J[-f(t,12,1)],r2=max⁡t∈J[-f(t,1,12)],
then, 0<r1⩽r2.
From(H1)′, (H2), (32), and u(t)⩽0 a.e. t∈J, it follows that
(39)A(h,h)(t)=b0+b1t-∫0t‍(t-s){a(s)f(s,h(s),h(s))+u(s)}ds+∑0<tk<t‍Ik(h(tk),h(tk))⩾b0+amin⁡[-∫0t‍(t-s)f(s,12,1)ds]⩾b0+r1amin⁡∫0t‍(t-s)ds⩾r3h(t),∀t∈J.
Also, we have(40)A(h,h)(t)=b0+b1t-∫0t‍(t-s){a(s)f(s,h(s),h(s))+u(s)}ds+∑0<tk<t‍Ik(h(tk),h(tk))⩽b0+b1+amax⁡[-∫0t‍(t-s)f(s,1,12)ds]+M∫0t‍(t-s)ds+∑0<tk<t‍Ik(1,12)⩽2[b0+b1+r2amax⁡+M+∑k=1m‍Ik(1,12)]h(t),pppppppppppppppppppppppppppppppppp∀t∈J.
Thus, we observe that
(41)r3h⩽A(h,h)⩽2[b0+b1+r2amax⁡+M+∑k=1m‍Ik(1,12)]h,
which implies that A(h,h)∈P~h.
By arguments as above, we see that the operatorA:P~×P~→P~ defined by (32) satisfies conditions (A1) and (A2) in Proposition 2. Therefore, by applying Proposition 2, we conclude that an operator equation x=A(x,x) has a unique solution in P~h; hence there exists a unique positive solution to (IP; u) on J.


## 5. Optimal Control Problem (OP)

In this section, we consider an optimal control problem (OP) to (IP;u). Throughout this section, we assume all the conditions of Theorem 8. Also, we assume the following additional conditions.(H4)
There is a constantCf>0 such that
(42)|f(t,x,x)-f(t,y,y)|⩽Cf|x-y|∀t∈J,x,y∈R+.
Also, for each k=1,2,…,m, there exists a positive constant Ck>0 such that
(43)|Ik(x,x)-Ik(y,y)|⩽Ck|x-y|∀x,y∈R+.(H5)
xd is a given desired target profile in H.At first, we give the key lemma in order to show the result of continuous dependence of positive solutions to (IP;u).Lemma 10 (cf. [19, Lemma 5.1]).
Let{un}⊂H, and let Q:H→C[J,R] be an operator given by
(44)(Qz)(t):=∫0t‍(t-s)z(s)ds,∀z∈H,∀t∈J.
Assume that un→u weakly in H as n→∞ for some u∈H. Then
(45)Qun⟶QuinC[J,R]asn⟶∞.For the detailed proof of Lemma10, we refer to [19, Lemma 5.1].Taking account of Lemma10, one can show the following proposition concerning the result of continuous dependence of positive solutions to (IP; u).Proposition 11 (cf. [19, Proposition 5.2]).
Assume the same conditions as in Theorem8, (H4), and (H5). Let {un}⊂UM and u∈UM, where UM is the control space defined by (4). Assume un→u weakly in H as n→∞. Then, the unique positive solution xn to (IP; un) on J converges to one x to (IP; u) on J in the sense that
(46)xn⟶xinPC[J,R]asn⟶∞.Proof.
By arguments similar to [19, Proposition 5.2], we can prove (46). In fact, note from Lemma 9 that xn is a solution to (IP;un) on J if and only if
(47)xn(t)=b0+b1t-∫0t‍(t-s)a(s)f(s,xn(s),xn(s))ds-∫0t‍(t-s)un(s)ds+∑0<tk<t‍Ik(xn(tk),xn(tk)),(t-s)a(s)f(s,xn(s),xn(s))pdppppppis∀t∈J.
Now, lett∈J0=[0,t1]⊂J, then, we obtain from (H4) that
(48)|xn(t)-x(t)|⩽|∫0t‍(t-s)a(s)f(s,xn(s),xn(s))ds-∫0t‍(t-s)a(s)f(s,x(s),x(s))ds|+|∫0t‍(t-s)un(s)ds-∫0t‍(t-s)u(s)ds|⩽Cf|a|C[J,R]∫0t‍|xn(s)-x(s)|ds+|Qun-Qu|C[J,R],∀t∈J0
for all n=1,2,…, where Q is a function defined in (44).
Applying a Gronwall-type inequality (e.g., [21, Proposition 0.4.1]) to (48), we obtain
(49)∫0t‍|xn(s)-x(s)|ds⩽eCf|a|C[J,R]|Qun-Qu|C[J,R],∀t∈J0,
for all n=1,2,…. Therefore, it follows from (48) and (49) that
(50)|xn(t)-x(t)|⩽Cf|a|C[J,R]eCf|a|C[J,R]|Qun-Qu|C[J,R]+|Qun-Qu|C[J,R]≡N1|Qun-Qu|C[J,R],∀t∈J0=[0,t1],
for all n=1,2,….
By (50) and the assumption (H4), we also have
(51)|xn(t1+)-x(t1+)|=|xn(t1)+I1(xn(t1),xn(t1))-x(t1)-I1(x(t1),x(t1))|⩽(1+C1)|xn(t1)-x(t1)|⩽(1+C1)N1|Qun-Qu|C[J,R]≡N1′|Qun-Qu|C[J,R],
for all n=1,2,….
Next, we consider the time intervalJ1=(t1,t2]. Then, we see from (50) and (H4) that
(52)|xn(t)-x(t)|⩽|∫0t‍(t-s)a(s)f(s,xn(s),xn(s))ds-∫0t‍(t-s)a(s)f(s,x(s),x(s))ds|+|∫0t‍(t-s)un(s)ds-∫0t‍(t-s)u(s)ds|+|I1(xn(t1),xn(t1))-I1(x(t1),x(t1))|⩽Cf|a|C[J,R]∫0t‍|xn(s)-x(s)|ds+|Qun-Qu|C[J,R]+C1|xn(t1)-x(t1)|⩽Cf|a|C[J,R]∫0t‍|xn(s)-x(s)|ds+(1+C1N1)|Qun-Qu|C[J,R],
for any t∈J1 and n=1,2,…. By the same arguments as before (cf. (49) and (50)), we can take some constants N2>0 so that
(53)|xn(t)-x(t)|⩽N2|Qun-Qu|C[J,R],∀t∈J1=(t1,t2],
for all n=1,2,….
Also, we obtain from (H4) and (53) that
(54)|xn(t2+)-x(t2+)|⩽|xn(t2)-x(t2)|+|I2(xn(t2),xn(t2))-I2(x(t2),x(t2))|⩽(1+C2)|xn(t2)-x(t2)|⩽N2′|Qun-Qu|C[J,R],∀n=1,2,…,
for some positive constants N2′>0.
By repeating this procedure, we can take positive constantsNk>0 and Nk′>0 such that
(55)|xn(t)-x(t)|⩽Nk|Qun-Qu|C[J,R],POPOI∀t∈Jk-1,k=1,2,…,m+1,|xn(tk+)-x(tk+)|⩽Nk′|Qun-Qu|C[J,R],k=1,2,…,m,
for all n=1,2,….
Here, putN:=max⁡{N1,N1′,N2,N2′,…,Nm,Nm′,Nm+1}. Then, we infer from (55) that
(56)|xn-x|PC⩽N|Qun-Qu|C[J,R],∀n=1,2,….
Sinceun→u weakly in H as n→∞, we observe from Lemma 10 that
(57)Qun⟶QuinC[J,R]asn⟶∞.
Hence, we see from (56) and (57) that
(58)xn⟶xinPC[J,R]asn⟶∞.
Thus, the proof of Proposition 11 has been completed.Now, we mention our main result concerning the existence of an optimal control to (OP).Theorem 12.
Assume the same conditions as in Theorem8, (H4), and (H5). Then, the problem (OP) has at least one optimal control u*∈UM such that
(59)π(u*)=inf⁡u∈UMπ(u),
where UM is a control space defined by (4) and π(·) is the cost functional defined in (5).Proof.
By the quite standard method, we can prove Theorem12. In fact, let {un}⊂UM be a minimizing sequence so that
(60)lim⁡n→∞π(un)=inf⁡u∈UMπ(u).
By the definition (5) of π(·), we see that {un} is bounded in H. Hence, there is a subsequence {nk}⊂{n} and a function u*∈UM such that nk→∞ and
(61)unk⟶u*weaklyinHask⟶∞.
For anyk∈N, let xnk be a unique positive solution to (IP;unk) on J. Then, from (61) and Proposition 11, we observe that
(62)xnk⟶xinPC[J,R]ask⟶∞,
where x is a unique positive solution to (IP;u) on J.
Hence, it follows from (61), (62), and the weak lower semicontinuity of H-norm that
(63)π(u*)⩽lim⁡k→∞π(unk)=inf⁡u∈UMπ(u),
which implies that u*∈UM is an optimal control to (OP).Now, we mention our final main result in this paper, which is concerned with the stability of the optimal control to (OP).Theorem 13.
Assume the same conditions as in Theorem12. Let u∈UM and u+εu0∈UM for some u0∈H and small positive constant ε. Also, let x and xε be unique positive solutions to (IP; u) and (IP;u+εu0) on J, respectively. Then,
(64)|xε-x|PC=O(ε),(ε⟶0).Proof.
Note from Lemma9 that x is a solution of (IP;u) on J if and only if
(65)x(t)=b0+b1t-∫0t‍(t-s)a(s)f(s,x(s),x(s))ds-∫0t‍(t-s)u(s)ds+∑0<tk<t‍Ik(x(tk),x(tk)),∀t∈J.
Now, lett∈J0=[0,t1]⊂J. Then, we obtain from (H4) that
(66)|xε(t)-x(t)|⩽|∫0t‍(t-s)a(s)f(s,xε(s),xε(s))ds-∫0t‍(t-s)a(s)f(s,x(s),x(s))ds|+|∫0t‍(t-s)(u+εu0)(s)ds-∫0t‍(t-s)u(s)ds|⩽Cf|a|C[J,R]∫0t‍|xε(s)-x(s)|ds+ε|u0|H,∀t∈J0.
Applying a Gronwall-type inequality (e.g., [21, Proposition 0.4.1]) to (66), we obtain
(67)∫0t‍|xε(s)-x(s)|ds⩽εeCf|a|C[J,R]|u0|H,∀t∈J0.
Therefore, it follows from (66) and (67) that
(68)|xε(t)-x(t)|⩽εCf|a|C[J,R]eCf|a|C[J,R]|u0|H+ε|u0|H≡εN~1|u0|H,∀t∈J0=[0,t1].
By (68) and the assumption (H4), we also have
(69)|xε(t1+)-x(t1+)|=|xε(t1)+I1(xε(t1),xε(t1))-x(t1)-I1(x(t1),x(t1))|⩽(1+C1)|xε(t1)-x(t1)|⩽(1+C1)εN~1|u0|H≡εN~1′|u0|H.
Next, we consider the time intervalJ1=(t1,t2]. Then, we see from (68) and (H4) that
(70)|xε(t)-x(t)|⩽|∫0t‍(t-s)a(s)f(s,xε(s),xε(s))ds-∫0t‍(t-s)a(s)f(s,x(s),x(s))ds|+|∫0t‍(t-s)(u+εu0)(s)ds-∫0t‍(t-s)u(s)ds|+|I1(xε(t1),xε(t1))-I1(x(t1),x(t1))|⩽Cf|a|C[J,R]∫0t‍|xε(s)-x(s)|ds+ε|u0|H+C1|xε(t1)-x(t1)|⩽Cf|a|C[J,R]∫0t‍|xε(s)-x(s)|ds+ε(1+C1N~1)|u0|H,
for any t∈J1. By the same arguments as before (cf. (67) and (68)), we can take some constant N~2>0 so that
(71)|xε(t)-x(t)|⩽εN~2|u0|H,∀t∈J1=(t1,t2].
Also, we obtain from (H4) and (71) that
(72)|xε(t2+)-x(t2+)|⩽|xε(t2)-x(t2)|+|I2(xε(t2),xε(t2))-I2(x(t2),x(t2))|⩽(1+C2)|xε(t2)-x(t2)|⩽εN~2′|u0|H,
for some positive constant N~2′>0.
By repeating this procedure, we can take positive constantsN~k>0 and N~k′>0 such that
(73)|xε(t)-x(t)|⩽εN~k|u0|H,∀t∈Jk-1,k=1,2,…,m+1,|xε(tk+)-x(tk+)|⩽εN~k′|u0|H,k=1,2,…,m.
Here, putN~:=max⁡{N~1,N~1′,N~2,N~2′,…,N~m,N~m′,N~m+1}. Then, we infer from (73) that
(74)|xε-x|PC⩽εN~|u0|H.
Thus, the proof of Theorem 13 has been completed.By Theorem13 and the definition of π (cf. (5)), we easily see that the following corollary holds.Corollary 14.
Assume the same conditions as in Theorem12. Let u∈UM and u+εu0∈UM for some u0∈H and small positive constant ε. Then,
(75)|π(u+εu0)-π(u)|=O(ε),(ε⟶0).


## 6. Examples

In this section, we give an example of the main results.Example 1.
Consider the following boundary value problem of second order impulsive differential equation:(76)-x′′(t)=(2t+1)[(1+x(t))(1/2)+(1+x(t))-(1/4)]+u(t),[(1+x(t))(1/2)+(1+x(t))-(1/4)]+t∈(0,1),t≠13,Δx|t=(1/3)=(1+x(13))(1/2)+(1+x(13))-(1/4),x(0)=1,x′(1)=0.Conclusion. The boundary value problem (76) admits a unique positive solution, which is continuously differentiable on [0,(1/3))∪((1/3),1].Proof.
LetJ=[0,1],t1=(1/3),f(t,x,y):=f(x,y)=(1+x)(1/2)+(1+y)-(1/4), a(t)=2t+1, and I1(x,y)=(1+x)(1/2)+(1+y)-(1/4). Evidently, f(x,y) and I1(x,y) are increasing in x for y⩾0 and are decreasing in y for x⩾0.
Setφ(γ)=γ(1/2), γ∈(0,1), then,
(77)f(γx,γ-1y)=(1+γx)(1/2)+(1+γ-1y)-(1/4)⩾φ(γ)f(x,y)∀x,y⩾0,I1(γx,γ-1y)⩾φ(γ)I1(x,y)∀x,y⩾0.
Therefore, we easily see that (H1)–(H5) hold. Hence, applying Theorem 4 to (76), we get a unique positive solution to (76) on J for each u∈H with 0⩽u(t)⩽M a.e. t∈J, where M>0 is a given constant.Example 2.
Consider the following initial value problem of the second order impulsive differential equation:(78)-x′′(t)=(2t+1)[(1+x(t))(1/2)+(1+x(t))-(1/4)]+u(t),t∈(0,1),t≠13,Δx|t=(1/3)=(1+x(13))(1/2)+(1+x(13))-(1/4),x(0)=1,x′(0)=0.Conclusion. The initial value problem (78) admits a unique positive solution, which is continuously differentiable on [0,(1/3))∪((1/3),1]. Moreover, the problem (OP) to (78) has at least one optimal control, and the stability result of optimal control holds.Proof.
LetJ=[0,1], t1=(1/3), f(t,x,y):=f(x,y)=-(1+x)(1/2)-(1+y)-(1/4), and a(t)=2t+1. Clearly, f(x,y) is decreasing in x for y⩾0 and is increasing in y for x⩾0.
Also, letI1(x,y)=(1+x)(1/2)+(1+y)-(1/4), evidently, I1(x,y) is increasing in x for y⩾0 and is decreasing in y for x⩾0.
Setφ(γ)=γ(1/2), γ∈(0,1), then,
(79)f(γx,γ-1y)=-(1+γx)(1/2)+(1+γ-1y)-(1/4)⩽φ(γ)f(x,y)∀x,y⩾0,I1(γx,γ-1y)⩾φ(γ)I1(x,y)∀x,y⩾0.
Therefore, we easily see that conditions (H2), (H4), (H1)′, (H3)′, and (H5)′ hold. Hence, applying Theorem 8 to (78), we get a unique positive solution to (78) on J for each u∈H with -M⩽u(t)⩽0 a.e. t∈J, where M>0 is a given constant.
In addition, letCf=C1=1. Then, we easily see that (H6) holds. Hence, applying Theorem 12, we see that Problem (OP) to (78) has at least one optimal control for each desired target profile xd in H. Also, applying Theorem 13, we get the result on the stability of optimal control to (OP).



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*Source: 974968-2014-07-07.xml*