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

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.

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: (BP;𝑢)−𝑥^′′(𝑡)(𝑡)𝑓((𝑡)(𝑡(𝑡),()⧵𝑡=𝑎𝑡,𝑥,𝑥))+𝑢𝑡∈0,1₁,𝑡₂,…,𝑡_𝑚,Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘,𝑥𝑡_𝑘,𝑥(0)𝑘=1,2,…,𝑚,=𝑏₀,𝑥^′(1)=0. (1) 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<⋯<𝑡_𝑚<1,Δ𝑥|_{𝑡=𝑡𝑘}denotes the jump of 𝑥(𝑡) at 𝑡=𝑡_𝑘 and Δ𝑥|_{𝑡=𝑡𝑘}=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), where 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) represent the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively. Also, 𝐼_𝑘 is a given function in 𝐶[𝑅⁺×𝑅⁺,𝑅⁺],𝑘=1,2,…,𝑚. Furthermore, 𝑏₀>0 is a given constant.

For convenience, we put 𝐽₀=[0,𝑡₁],𝐽₁=(𝑡₁,𝑡₂],…,𝐽_𝑚−1=(𝑡_𝑚−1,𝑡_𝑚],𝐽_𝑚=(𝑡_𝑚,1], and 𝐽^′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑚}.

Then, we study the existence and uniqueness of positive solutions to initial value problem as follows: (IP;𝑢)−𝑥^′′(𝑡)(𝑡)𝑓((𝑡)(𝑡(𝑡),()⧵𝑡=𝑎𝑡,𝑥,𝑥))+𝑢𝑡∈0,1₁,𝑡₂,…,𝑡_𝑚,Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘,𝑥𝑡_𝑘,𝑥(0)𝑘=1,2,…,𝑚,=𝑏₀,𝑥^′(0)=𝑏₁, (2) 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^∗∈𝒰_𝑀 such that 𝜋𝑢^∗=inf_{𝑢∈𝒰𝑀}𝜋(𝑢). (3) Here, 𝒰_𝑀 is a control space defined by 𝒰_𝑀∶=𝑢∈𝐿²(0,1)∣−𝑀⩽𝑢(𝑡)⩽0a.e.𝑡∈[0,1], (4) where 𝑀 is a fixed positive number and 𝜋(𝑢) is the cost functional defined by 𝜋(𝑢)1∶=2¹₀𝑥−𝑥_𝑑(𝑡)²(1)+1𝑑𝑡+𝑥2¹₀|𝑢(𝑡)|²𝑑𝑡, (5) where 𝑢∈𝒰_𝑀 is the control, function 𝑥 is a unique positive solution to the state problem (IP; 𝑢), and 𝑥_𝑑 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: 𝑢^′(𝑡)((𝑡)(𝑡[],𝑢(0)=𝑓𝑡,𝑢,𝑢)),𝑡∈0,𝑎=𝑥₀. (6) 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: 𝑢^′′(𝑥)−𝑓(𝑥,𝑢,𝑢)=𝜃,𝑥∈𝐽,𝑥≠𝑥_𝑖|,𝑖=1,2,…,𝑚,Δ𝑢_{𝑥=𝑥𝑖}=𝐼_𝑖𝑢𝑥_𝑖,𝑖=1,2,…,𝑚,Δ𝑢^′_{𝑥=𝑥𝑖}=𝐼̃_𝑖𝑢𝑥_𝑖𝑢(0),𝑖=1,2,…,𝑚,=𝑤₀,𝑢^′(0)=𝑤₁. (7) 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: −𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡))+𝑢(𝑡),𝑡∈(0,𝑇)⧵𝑡₁,𝑡₂,…,𝑡_𝑚,Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘,𝑘=1,2,…,𝑚,Δ𝑥^′_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑎,𝑥^′(0)=𝑏. (8)

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 Section 2, 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.

Let 𝑃𝐶[𝐽,𝑅]∶={𝑥∣𝑥∶𝐽→𝑅,𝑥(𝑡) is continuous at 𝑡≠𝑡_𝑘 and left continuous at 𝑡=𝑡_𝑘,𝑥(𝑡⁺_𝑘) 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 put 𝐻∶=𝐿²(𝐽) with the usual real Hilbert structure and denote by |⋅|_𝐻 the norm in 𝐻, for simplicity, and 𝑊^2,1(𝐽,𝑅) is a usual Sobolev space, namely, 𝑊^2,1(𝐽,𝑅)=𝑓∈𝐿¹(𝐽)∶𝐷^𝑘𝑓∈𝐿¹(𝐽),𝑘=1,2, (9) where 𝐷^𝑘𝑓 denoted the 𝑘th derivative of 𝑓.

Also, 𝑁_𝑖 and 𝑁^′_𝑖,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.

Putting 𝑃^∘={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 all 𝑥,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 𝑃_ℎ the set 𝑃_ℎ={𝑥∈𝐸∣𝑥∼ℎ}. It is easy to see that 𝑃_ℎ⊂𝑃 is convex and 𝜆𝑃_ℎ=𝑃_ℎ for all 𝜆>0. If 𝑃^∘≠∅ and ℎ∈𝑃^∘, it is clear that 𝑃_ℎ=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, 𝑢_𝑖,𝑣_𝑖∈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]). Let 𝑃 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 exists ℎ∈𝑃 with ℎ≠𝜃 such that 𝐴(ℎ,ℎ)∈𝑃_ℎ; (A₂) for any 𝑢,𝑣∈𝑃 and 𝑡∈(0,1), there exists 𝜑(𝑡)∈(𝑡,1] such that 𝐴(𝑡𝑢,𝑡⁻¹𝑣)⩾𝜑(𝑡)𝐴(𝑢,𝑣).Then operator 𝐴 has a unique fixed point 𝑥^∗ in 𝑃_ℎ. Moreover, for any initial 𝑥₀,𝑦₀∈𝑃_ℎ, constructing successively the sequences 𝑥_𝑛=𝐴𝑥_𝑛−1,𝑦_𝑛−1,𝑦_𝑛=𝐴𝑦_𝑛−1,𝑥_𝑛−1,𝑛=1,2,…, (10) one has ‖𝑥_𝑛−𝑥^∗‖→0 and ‖𝑦_𝑛−𝑥^∗‖→0 as 𝑛→∞.

By applying Proposition 2, 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 each 𝑘=1,2,…,𝑚, 𝐼_𝑘∶𝑅⁺×𝑅⁺→𝑅⁺,𝐼_𝑘(𝑥,𝑦) is nondecreasing in 𝑥 for each 𝑦∈𝑅⁺ and is nonincreasing in 𝑦 for each 𝑥∈𝑅⁺.(H₃)For all 𝛾∈(0,1), there exists a constant 𝜑₁(𝛾),𝜑₂(𝛾)∈(𝛾,1] such that 𝑓𝑡,𝛾𝑥,𝛾⁻¹𝑦⩾𝜑₁𝛾𝑓,𝐼𝑡,𝑥,𝑦_𝑘𝛾𝑥,𝛾⁻¹𝑦⩾𝜑₂𝛾𝐼_𝑘𝑥,𝑦. (11) 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. Let 𝑢∈𝐻, and let 𝑏₀ be a given constant. Then, a function 𝑥∈𝑃𝐶[𝐽,𝑅]∩𝑊^2,1(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 Theorem 4. Then, 𝑥∈𝑃𝐶[𝐽,𝑅]∩𝑊^2,1(𝐽,𝑅) is a solution to (BP; 𝑢) on 𝐽 if and only if 𝑥∈𝑃𝐶[𝐽,𝑅] satisfies the following integral equation: 𝑥(𝑡)=𝑏₀+¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝑠))+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑥𝑡_𝑘,∀𝑡∈𝐽, (12) where 𝐺(𝑡,𝑠)=𝑡,0⩽𝑡⩽𝑠⩽1,𝑠,0⩽𝑠⩽𝑡⩽1. (13)

Proof. Firstly, integrating −𝑥^′′(𝑠)=𝑎(𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝑠))+𝑢(𝑠) from 0 to 𝑡, we obtain 𝑥^′(𝑡)=𝑥^′(0)−^𝑡₀𝑎(𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝑠))+𝑢(𝑠)𝑑𝑠,∀𝑡∈𝐽. (14) Again, integrating (14), we have 𝑥(𝑡)=𝑥(0)+𝑥^′(0)𝑡−^𝑡₀(𝑡−𝑠)𝑎(𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝑠))+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑥𝑡⁺_𝑘−𝑥𝑡_𝑘,∀𝑡∈𝐽. (15) From (14) with 𝑥^′(1)=0, we infer that 𝑥^′(0)=¹₀𝑎(𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝑠))+𝑢(𝑠)𝑑𝑠. (16) Hence, from 𝑥(0)=𝑏₀, (15), and (16), we can find that 𝑥(𝑡)=𝑏₀+¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝑠))+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑥𝑡_𝑘,∀𝑡∈𝐽, (17) where 𝐺(𝑡,𝑠) is the function defined as in (13). Thus, the proof is completed.

By Lemma 5, 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 𝐴𝑥,𝑦(𝑡)=𝑏₀+¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑥(𝑠),𝑦(𝑠)+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑦𝑡_𝑘∀𝑥,𝑦∈𝑃𝐶[𝐽,𝑅],𝑡∈𝐽, (18) 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 Theorem 4. Then, 𝑥∈𝑃𝐶[𝐽,𝑅]∩𝑊^2,1(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 Proposition 2 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 Theorem 4. By applying a fixed point theorem of mixed monotone operator (Proposition 2), we show the existence and uniqueness of the positive solution to (BP; 𝑢) on 𝐽.
To do so, set 𝑃∶={𝑥∈𝑃𝐶[𝐽,𝑅];𝑥(𝑡)⩾0,∀𝑡∈𝐽}. (19) Clearly, 𝑃 is a normal cone in 𝑃𝐶[𝐽,𝑅] and the normality constant is 1.
Let 𝐴∶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 𝐴𝑥,𝑦(𝑡)⩾0∀𝑥,𝑦∈𝑃,𝑡∈𝐽. (20) 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 𝜑𝛾=min𝜑₁𝛾,𝜑₂𝛾,𝛾∈(0,1). (21) 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 𝐴𝛾𝑥,𝛾⁻¹𝑦(𝑡)=𝑏₀+¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝛾𝑥(𝑠),𝛾⁻¹𝑦(𝑠(𝑠+)+𝑢)𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝛾𝑥𝑡_𝑘,𝛾⁻¹𝑦𝑡_𝑘⩾𝑏₀+𝜑₁𝛾¹₀𝐺(𝑎(𝑠)𝑓(𝑠)(𝑠)(𝑠𝑡,𝑠)𝑠,𝑥,𝑦+𝑢)}𝑑𝑠+𝜑₂𝛾_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑦𝑡_𝑘⩾𝜑𝛾𝐴𝑥,𝑦(𝑡),∀𝑡∈𝐽, (22) which implies that 𝐴𝛾𝑥,𝛾⁻¹𝑦⩾𝜑𝛾𝐴𝑥,𝑦∀𝑥,𝑦∈𝑃,𝛾∈(0,1). (23) Thus, the condition (A₂) holds.
Now, we show (A₁), defining a function ℎ by ℎ(𝑡)1∶=2+¹₀𝐺(𝑡,𝑠)𝑑𝑠,∀𝑡∈𝐽; (24) 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 that 𝐴(ℎ,ℎ)∈𝑃_ℎ. Set 𝑟₁=min_𝑡∈𝐽𝑓1𝑡,2,1,𝑟₂=max_𝑡∈𝐽𝑓1𝑡,1,2, (25) then, 0<𝑟₁⩽𝑟₂.
Note that 𝑎(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 𝑎_min=min_𝑡∈𝐽𝑎(𝑡),𝑎_max=max_𝑡∈𝐽𝑎(𝑡). (26)
Here, put 𝑟₃∶=min{2𝑏₀,𝑟₁𝑎_min}. Then, from (H₁), (H₂), (13), and 𝑢(𝑡)⩾0 a.e. 𝑡∈𝐽, it follows that 𝐴(ℎ,ℎ)(𝑡)=𝑏₀+¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓(𝑠,ℎ(𝑠),ℎ(𝑠))+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘,ℎ𝑡_𝑘⩾𝑏₀+𝑎_min¹₀𝐺()𝑓1𝑡,𝑠𝑠,2,1𝑑𝑠⩾𝑏₀+𝑟₁𝑎_min¹₀𝐺(𝑡,𝑠)𝑑𝑠⩾𝑟₃ℎ(𝑡),∀𝑡∈𝐽. (27)
Also, we have 𝐴(ℎ,ℎ)(𝑡)=𝑏₀+¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓(𝑠,ℎ(𝑠),ℎ(𝑠))+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘,ℎ𝑡_𝑘⩽𝑏₀+𝑎_max¹₀𝐺()𝑓1𝑡,𝑠𝑠,1,2𝑑𝑠+𝑀¹₀𝐺(𝑡,𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘11,2⩽2𝑏₀+𝑟₂𝑎_max+𝑀+2^𝑚_𝑘=1𝐼_𝑘11,2ℎ(𝑡),∀𝑡∈𝐽. (28) Thus, we observe that 𝑟₃ℎ⩽𝐴(ℎ,ℎ)⩽2𝑏₀+𝑟₂𝑎_max+𝑀+2^𝑚_𝑘=1𝐼_𝑘11,2ℎ, (29) which implies that 𝐴(ℎ,ℎ)∈𝑃_ℎ.
By arguments as above, we see that the operator 𝐴∶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 all 𝛾∈(0,1), there exists a constant 𝜑₁(𝛾),𝜑₂(𝛾)∈(𝛾,1] such that 𝑓𝑡,𝛾𝑥,𝛾⁻¹𝑦⩽𝜑₁𝛾𝑓,𝐼𝑡,𝑥,𝑦_𝑘𝛾𝑥,𝛾⁻¹𝑦⩾𝜑₂𝛾𝐼_𝑘𝑥,𝑦, (30) 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. Let 𝑢∈𝐻 and 𝑏₀ and 𝑏₁ as given constants. Then, a function 𝑥∈𝑃𝐶[𝐽,𝑅]∩𝑊^2,1(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 Lemma 5 (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 Theorem 8. Then, 𝑥∈𝑃𝐶[𝐽,𝑅]∩𝑊^2,1(𝐽,𝑅) is a solution to (IP; 𝑢) on 𝐽 if and only if 𝑥∈𝑃𝐶[𝐽,𝑅] satisfies the following integral equations: 𝑥(𝑡)=𝑏₀+𝑏₁𝑡−^𝑡₀(𝑎(𝑠)𝑓((𝑠)(𝑠(𝑠+𝑡−𝑠)𝑠,𝑥,𝑥))+𝑢)𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑥𝑡_𝑘,∀𝑡∈𝐽. (31)

By Lemma 9 and Proposition 2, one can show Theorem 8 concerning the existence-uniqueness of the positive solution to (IP; 𝑢) on 𝐽.

Proof of Theorem 8. Now, we define an operator 𝐴∶𝑃𝐶[𝐽,𝑅]×𝑃𝐶[𝐽,𝑅]→𝑃𝐶[𝐽,𝑅] by 𝐴𝑥,𝑦(𝑡)=𝑏₀+𝑏₁𝑡−^𝑡₀(𝑡−𝑠)𝑎(𝑠)𝑓𝑠,𝑥(𝑠),𝑦(𝑠)+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑦𝑡_𝑘∀𝑥,𝑦∈𝑃𝐶[𝐽,𝑅],𝑡∈𝐽. (32) 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.
Let 𝑃 be the same space defined by (19). Then, we infer from (H₁)^′, (H₂), (32), and 𝑢(𝑡)⩽0 a.e. 𝑡∈𝐽 that 𝐴𝑥,𝑦(𝑡)⩾0∀𝑥,𝑦∈𝑃,𝑡∈𝐽. (33) 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 𝜑𝛾=min𝜑₁𝛾,𝜑₂𝛾,𝛾∈(0,1). (34) 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 𝐴𝛾𝑥,𝛾⁻¹𝑦(𝑡)=𝑏₀+𝑏₁𝑡−^𝑡₀(𝑡−𝑠)𝑎(𝑠)𝑓𝑠,𝛾𝑥(𝑠),𝛾⁻¹𝑦(𝑠(𝑠+)+𝑢)𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝛾𝑥𝑡_𝑘,𝛾⁻¹𝑦𝑡_𝑘⩾𝑏₀+𝑏₁𝑡−𝜑₁𝛾^𝑡₀(𝑎(𝑠)𝑓(𝑠)(𝑠)(𝑠𝑡−𝑠)𝑠,𝑥,𝑦+𝑢)}𝑑𝑠+𝜑₂𝛾_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑦𝑡_𝑘⩾𝜑𝛾𝐴𝑥,𝑦(𝑡),∀𝑡∈𝐽, (35) which implies that 𝐴𝛾𝑥,𝛾⁻¹𝑦⩾𝜑𝛾𝐴𝑥,𝑦∀𝑥,𝑦∈𝑃,𝛾∈(0,1). (36) Thus, the condition (A₂) holds.
Now, we show (A₁), defining a function ℎ by ℎ(𝑡)1∶=2+^𝑡₀(𝑡−𝑠)𝑑𝑠,∀𝑡∈𝐽; (37) 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 that 𝐴(ℎ,ℎ)∈𝑃_ℎ. Set 𝑟₁=min_𝑡∈𝐽1−𝑓𝑡,2,1,𝑟₂=max_𝑡∈𝐽1−𝑓𝑡,1,2, (38) then, 0<𝑟₁⩽𝑟₂.
From (H₁)^′, (H₂), (32), and 𝑢(𝑡)⩽0 a.e. 𝑡∈𝐽, it follows that 𝐴(ℎ,ℎ)(𝑡)=𝑏₀+𝑏₁𝑡−^𝑡₀(𝑡−𝑠)𝑎(𝑠)𝑓(𝑠,ℎ(𝑠),ℎ(𝑠))+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘,ℎ𝑡_𝑘⩾𝑏₀+𝑎_min−^𝑡₀()𝑓1𝑡−𝑠𝑠,2,1𝑑𝑠⩾𝑏₀+𝑟₁𝑎_min^𝑡₀(𝑡−𝑠)𝑑𝑠⩾𝑟₃ℎ(𝑡),∀𝑡∈𝐽. (39)
Also, we have 𝐴(ℎ,ℎ)(𝑡)=𝑏₀+𝑏₁𝑡−^𝑡₀(𝑡−𝑠)𝑎(𝑠)𝑓(𝑠,ℎ(𝑠),ℎ(𝑠))+𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘ℎ𝑡_𝑘,ℎ𝑡_𝑘⩽𝑏₀+𝑏₁+𝑎_max−^𝑡₀()𝑓1𝑡−𝑠𝑠,1,2𝑑𝑠+𝑀^𝑡₀(𝑡−𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘11,2⩽2𝑏₀+𝑏₁+𝑟₂𝑎_max+𝑀+^𝑚_𝑘=1𝐼_𝑘11,2ℎ(𝑡),∀𝑡∈𝐽. (40) Thus, we observe that 𝑟₃ℎ⩽𝐴(ℎ,ℎ)⩽2𝑏₀+𝑏₁+𝑟₂𝑎_max+𝑀+^𝑚_𝑘=1𝐼_𝑘11,2ℎ, (41) which implies that 𝐴(ℎ,ℎ)∈𝑃_ℎ.
By arguments as above, we see that the operator 𝐴∶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 constant 𝐶_𝑓>0 such that 𝑓(𝑡,𝑥,𝑥)−𝑓𝑡,𝑦,𝑦⩽𝐶_𝑓𝑥−𝑦∀𝑡∈𝐽,𝑥,𝑦∈𝑅⁺. (42) Also, for each 𝑘=1,2,…,𝑚, there exists a positive constant 𝐶_𝑘>0 such that 𝐼_𝑘(𝑥,𝑥)−𝐼_𝑘𝑦,𝑦⩽𝐶_𝑘𝑥−𝑦∀𝑥,𝑦∈𝑅⁺. (43)(H₅)𝑥_𝑑 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 {𝑢_𝑛}⊂𝐻, and let 𝑄∶𝐻→𝐶[𝐽,𝑅] be an operator given by (𝑄𝑧)(𝑡)∶=^𝑡₀(𝑡−𝑠)𝑧(𝑠)𝑑𝑠,∀𝑧∈𝐻,∀𝑡∈𝐽. (44) Assume that 𝑢_𝑛→𝑢 weakly in 𝐻 as 𝑛→∞ for some 𝑢∈𝐻. Then 𝑄𝑢_𝑛⟶𝑄𝑢𝑖𝑛𝐶[𝐽,𝑅]𝑎𝑠𝑛⟶∞. (45)

For the detailed proof of Lemma 10, we refer to [19, Lemma 5.1].

Taking account of Lemma 10, 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 Theorem 8, (𝐻₄), and (𝐻₅). Let {𝑢_𝑛}⊂𝒰_𝑀 and 𝑢∈𝒰_𝑀, where 𝒰_𝑀 is the control space defined by (4). Assume 𝑢_𝑛→𝑢 weakly in 𝐻 as 𝑛→∞. Then, the unique positive solution 𝑥_𝑛 to (IP; 𝑢_𝑛) on 𝐽 converges to one 𝑥 to (IP; 𝑢) on 𝐽 in the sense that 𝑥_𝑛⟶𝑥𝑖𝑛𝑃𝐶[𝐽,𝑅]𝑎𝑠𝑛⟶∞. (46)

Proof. By arguments similar to [19, Proposition 5.2], we can prove (46). In fact, note from Lemma 9 that 𝑥_𝑛 is a solution to (IP;𝑢_𝑛) on 𝐽 if and only if 𝑥_𝑛(𝑡)=𝑏₀+𝑏₁𝑡−^𝑡₀(𝑡−𝑠)𝑎(𝑠)𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠)−𝑑𝑠_𝑡0(𝑡−𝑠)𝑢_𝑛(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡_𝑘,𝑥_𝑛𝑡_𝑘,∀𝑡∈𝐽. (47)
Now, let 𝑡∈𝐽₀=[0,𝑡₁]⊂𝐽, then, we obtain from (H₄) that 𝑥_𝑛(𝑡)(𝑡⩽−𝑥)_𝑡0(𝑡−𝑠)𝑎(𝑠)𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠)−𝑑𝑠_𝑡0()𝑎(𝑠)𝑓((𝑠)(𝑠+𝑡−𝑠𝑠,𝑥,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)𝑢_𝑛(𝑠)𝑑𝑠−^𝑡₀(𝑡−𝑠)𝑢(𝑠)𝑑𝑠⩽𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}^𝑡₀𝑥_𝑛(𝑠)−𝑥(𝑠)𝑑𝑠+𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},∀𝑡∈𝐽₀ (48) 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 ^𝑡₀𝑥_𝑛(𝑠)−𝑥(𝑠)𝑑𝑠⩽𝑒^{𝐶𝑓|𝑎|𝐶[𝐽,𝑅]}𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},∀𝑡∈𝐽₀, (49) for all 𝑛=1,2,…. Therefore, it follows from (48) and (49) that 𝑥_𝑛(𝑡)−𝑥(𝑡)⩽𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}𝑒^{𝐶𝑓|𝑎|𝐶[𝐽,𝑅]}𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]}+𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]}≡𝑁₁𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},∀𝑡∈𝐽₀=0,𝑡₁, (50) for all 𝑛=1,2,….
By (50) and the assumption (H₄), we also have 𝑥_𝑛𝑡⁺₁−𝑥𝑡⁺₁=𝑥_𝑛𝑡₁+𝐼₁𝑥_𝑛𝑡₁,𝑥_𝑛𝑡₁−𝑥𝑡₁−𝐼₁𝑥𝑡₁,𝑥𝑡₁⩽1+𝐶₁𝑥_𝑛𝑡₁−𝑥𝑡₁⩽1+𝐶₁𝑁₁𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]}≡𝑁^′₁𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]}, (51) for all 𝑛=1,2,….
Next, we consider the time interval 𝐽₁=(𝑡₁,𝑡₂]. Then, we see from (50) and (H₄) that 𝑥_𝑛(𝑡)(𝑡⩽−𝑥)_𝑡0(𝑡−𝑠)𝑎(𝑠)𝑓𝑠,𝑥_𝑛(𝑠),𝑥_𝑛(𝑠)−𝑑𝑠_𝑡0()𝑎(𝑠)𝑓((𝑠)(𝑠+𝑡−𝑠𝑠,𝑥,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)𝑢_𝑛(𝑠)𝑑𝑠−^𝑡₀()𝑢(𝑠)+𝐼𝑡−𝑠𝑑𝑠₁𝑥_𝑛𝑡₁,𝑥_𝑛𝑡₁−𝐼₁𝑥𝑡₁,𝑥𝑡₁⩽𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}^𝑡₀𝑥_𝑛(𝑠)(𝑠+−𝑥)𝑑𝑠𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]}+𝐶₁𝑥_𝑛𝑡₁−𝑥𝑡₁⩽𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}^𝑡₀𝑥_𝑛(𝑠)(𝑠+−𝑥)𝑑𝑠1+𝐶₁𝑁₁𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]}, (52) 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 𝑥_𝑛(𝑡)−𝑥(𝑡)⩽𝑁₂𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},∀𝑡∈𝐽₁=𝑡₁,𝑡₂, (53) for all 𝑛=1,2,….
Also, we obtain from (H₄) and (53) that 𝑥_𝑛𝑡⁺₂−𝑥𝑡⁺₂⩽𝑥_𝑛𝑡₂−𝑥𝑡₂+𝐼₂𝑥_𝑛𝑡₂,𝑥_𝑛𝑡₂−𝐼₂𝑥𝑡₂,𝑥𝑡₂⩽1+𝐶₂𝑥_𝑛𝑡₂−𝑥𝑡₂⩽𝑁^′₂𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},∀𝑛=1,2,…, (54) for some positive constants 𝑁^′₂>0.
By repeating this procedure, we can take positive constants 𝑁_𝑘>0 and 𝑁^′_𝑘>0 such that 𝑥_𝑛(𝑡)−𝑥(𝑡)⩽𝑁_𝑘𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},∀𝑡∈𝐽_𝑘−1𝑥,𝑘=1,2,…,𝑚+1,_𝑛𝑡⁺_𝑘−𝑥𝑡⁺_𝑘⩽𝑁^′_𝑘𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},𝑘=1,2,…,𝑚, (55) for all 𝑛=1,2,….
Here, put 𝑁∶=max{𝑁₁,𝑁^′₁,𝑁₂,𝑁^′₂,…,𝑁_𝑚,𝑁^′_𝑚,𝑁_𝑚+1}. Then, we infer from (55) that 𝑥_𝑛−𝑥_𝑃𝐶⩽𝑁𝑄𝑢_𝑛−𝑄𝑢_{𝐶[𝐽,𝑅]},∀𝑛=1,2,…. (56)
Since 𝑢_𝑛→𝑢 weakly in 𝐻 as 𝑛→∞, we observe from Lemma 10 that 𝑄𝑢_𝑛⟶𝑄𝑢in𝐶[𝐽,𝑅]as𝑛⟶∞. (57) Hence, we see from (56) and (57) that 𝑥_𝑛⟶𝑥in𝑃𝐶[𝐽,𝑅]as𝑛⟶∞. (58) 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 Theorem 8, (𝐻₄), and (𝐻₅). Then, the problem (OP) has at least one optimal control 𝑢^∗∈𝒰_𝑀 such that 𝜋𝑢^∗=inf_{𝑢∈𝒰𝑀}𝜋(𝑢), (59) where 𝒰_𝑀 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 Theorem 12. In fact, let {𝑢_𝑛}⊂𝒰_𝑀 be a minimizing sequence so that lim_𝑛→∞𝜋𝑢_𝑛=inf_{𝑢∈𝒰𝑀}𝜋(𝑢). (60) By the definition (5) of 𝜋(⋅), we see that {𝑢_𝑛} is bounded in 𝐻. Hence, there is a subsequence {𝑛_𝑘}⊂{𝑛} and a function 𝑢^∗∈𝒰_𝑀 such that 𝑛_𝑘→∞ and 𝑢_𝑛𝑘⟶𝑢^∗weaklyin𝐻as𝑘⟶∞. (61)
For any 𝑘∈ℕ, let 𝑥_𝑛𝑘 be a unique positive solution to (IP;𝑢_𝑛𝑘) on 𝐽. Then, from (61) and Proposition 11, we observe that 𝑥_𝑛𝑘⟶𝑥in𝑃𝐶[𝐽,𝑅]as𝑘⟶∞, (62) 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 𝜋𝑢^∗⩽lim_𝑘→∞𝜋𝑢_𝑛𝑘=inf_{𝑢∈𝒰𝑀}𝜋(𝑢), (63) which implies that 𝑢^∗∈𝒰_𝑀 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 Theorem 12. Let 𝑢∈𝒰_𝑀 and 𝑢+𝜀𝑢₀∈𝒰_𝑀 for some 𝑢₀∈𝐻 and small positive constant 𝜀. Also, let 𝑥 and 𝑥_𝜀 be unique positive solutions to (IP; 𝑢) and (IP;𝑢+𝜀𝑢₀) on 𝐽, respectively. Then, 𝑥_𝜀−𝑥_𝑃𝐶=𝒪(𝜀),(𝜀⟶0). (64)

Proof. Note from Lemma 9 that 𝑥 is a solution of (IP;𝑢) on 𝐽 if and only if 𝑥(𝑡)=𝑏₀+𝑏₁𝑡−^𝑡₀()𝑎(𝑠)𝑓((𝑠)(𝑠−𝑡−𝑠𝑠,𝑥,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)𝑢(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘,𝑥𝑡_𝑘,∀𝑡∈𝐽. (65)
Now, let 𝑡∈𝐽₀=[0,𝑡₁]⊂𝐽. Then, we obtain from (H₄) that 𝑥_𝜀(𝑡)(𝑡⩽−𝑥)_𝑡0(𝑡−𝑠)𝑎(𝑠)𝑓𝑠,𝑥_𝜀(𝑠),𝑥_𝜀(𝑠)−𝑑𝑠_𝑡0()𝑎(𝑠)𝑓((𝑠)(𝑠+𝑡−𝑠𝑠,𝑥,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)𝑢+𝜀𝑢₀(𝑠)𝑑𝑠−^𝑡₀(𝑡−𝑠)𝑢(𝑠)𝑑𝑠⩽𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}^𝑡₀𝑥_𝜀(𝑠)−𝑥(𝑠)𝑑𝑠+𝜀𝑢_0𝐻,∀𝑡∈𝐽₀. (66)
Applying a Gronwall-type inequality (e.g., [21, Proposition 0.4.1]) to (66), we obtain ^𝑡₀𝑥_𝜀(𝑠)−𝑥(𝑠)𝑑𝑠⩽𝜀𝑒^{𝐶𝑓|𝑎|𝐶[𝐽,𝑅]}𝑢_0𝐻,∀𝑡∈𝐽₀. (67) Therefore, it follows from (66) and (67) that 𝑥_𝜀(𝑡)−𝑥(𝑡)⩽𝜀𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}𝑒^{𝐶𝑓|𝑎|𝐶[𝐽,𝑅]}𝑢_0𝐻+𝜀𝑢_0𝐻≡𝜀𝑁₁𝑢_0𝐻,∀𝑡∈𝐽₀=0,𝑡₁. (68)
By (68) and the assumption (H₄), we also have 𝑥_𝜀𝑡⁺₁−𝑥𝑡⁺₁=𝑥_𝜀𝑡₁+𝐼₁𝑥_𝜀𝑡₁,𝑥_𝜀𝑡₁−𝑥𝑡₁−𝐼₁𝑥𝑡₁,𝑥𝑡₁⩽1+𝐶₁𝑥_𝜀𝑡₁−𝑥𝑡₁⩽1+𝐶₁𝜀𝑁₁𝑢_0𝐻≡𝜀𝑁^′₁𝑢_0𝐻. (69)
Next, we consider the time interval 𝐽₁=(𝑡₁,𝑡₂]. Then, we see from (68) and (H₄) that 𝑥_𝜀(𝑡)(𝑡⩽−𝑥)_𝑡0(𝑡−𝑠)𝑎(𝑠)𝑓𝑠,𝑥_𝜀(𝑠),𝑥_𝜀(𝑠)−𝑑𝑠_𝑡0()𝑎(𝑠)𝑓((𝑠)(𝑠+𝑡−𝑠𝑠,𝑥,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)𝑢+𝜀𝑢₀(𝑠)𝑑𝑠−^𝑡₀()𝑢(𝑠)+𝐼𝑡−𝑠𝑑𝑠₁𝑥_𝜀𝑡₁,𝑥_𝜀𝑡₁−𝐼₁𝑥𝑡₁,𝑥𝑡₁⩽𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}^𝑡₀𝑥_𝜀(𝑠)−𝑥(𝑠)𝑑𝑠+𝜀𝑢_0𝐻+𝐶₁𝑥_𝜀𝑡₁−𝑥𝑡₁⩽𝐶_𝑓|𝑎|_{𝐶[𝐽,𝑅]}^𝑡₀𝑥_𝜀(𝑠)−𝑥(𝑠)𝑑𝑠+𝜀1+𝐶₁𝑁₁𝑢_0𝐻, (70) for any 𝑡∈𝐽₁. By the same arguments as before (cf. (67) and (68)), we can take some constant 𝑁₂>0 so that 𝑥_𝜀(𝑡)−𝑥(𝑡)⩽𝜀𝑁₂𝑢_0𝐻,∀𝑡∈𝐽₁=𝑡₁,𝑡₂. (71)
Also, we obtain from (H₄) and (71) that 𝑥_𝜀𝑡⁺₂−𝑥𝑡⁺₂⩽𝑥_𝜀𝑡₂−𝑥𝑡₂+𝐼₂𝑥_𝜀𝑡₂,𝑥_𝜀𝑡₂−𝐼₂𝑥𝑡₂,𝑥𝑡₂⩽1+𝐶₂𝑥_𝜀𝑡₂−𝑥𝑡₂⩽𝜀𝑁^′₂𝑢_0𝐻, (72) for some positive constant 𝑁^′₂>0.
By repeating this procedure, we can take positive constants 𝑁_𝑘>0 and 𝑁^′_𝑘>0 such that 𝑥_𝜀(𝑡)−𝑥(𝑡)⩽𝜀𝑁_𝑘𝑢_0𝐻,∀𝑡∈𝐽_𝑘−1𝑥,𝑘=1,2,…,𝑚+1,_𝜀𝑡⁺_𝑘−𝑥𝑡⁺_𝑘⩽𝜀𝑁^′_𝑘𝑢_0𝐻,𝑘=1,2,…,𝑚. (73)
Here, put 𝑁∶=max{𝑁₁,𝑁^′₁,𝑁₂,𝑁^′₂,…,𝑁_𝑚,𝑁^′_𝑚,𝑁_𝑚+1}. Then, we infer from (73) that 𝑥_𝜀−𝑥_𝑃𝐶⩽𝜀𝑁𝑢_0𝐻. (74) Thus, the proof of Theorem 13 has been completed.

By Theorem 13 and the definition of 𝜋 (cf. (5)), we easily see that the following corollary holds.

Corollary 14. Assume the same conditions as in Theorem 12. Let 𝑢∈𝒰_𝑀 and 𝑢+𝜀𝑢₀∈𝒰_𝑀 for some 𝑢₀∈𝐻 and small positive constant 𝜀. Then, 𝜋𝑢+𝜀𝑢₀−𝜋(𝑢)=𝒪(𝜀),(𝜀⟶0). (75)

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: −𝑥^′′(𝑡)=(2𝑡+1)(1+𝑥(𝑡))^(1/2)+(1+𝑥(𝑡))^−(1/4)(𝑡),()1+𝑢𝑡∈0,1,𝑡≠3,Δ𝑥|_𝑡=(1/3)=11+𝑥3^(1/2)+11+𝑥3^−(1/4),𝑥(0)=1,𝑥^′(1)=0. (76)

Conclusion. The boundary value problem (76) admits a unique positive solution, which is continuously differentiable on [0,(1/3))∪((1/3),1].

Proof. Let 𝐽=[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.
Set 𝜑(𝛾)=𝛾^(1/2), 𝛾∈(0,1), then, 𝑓𝛾𝑥,𝛾⁻¹𝑦=1+𝛾𝑥^(1/2)+1+𝛾⁻¹𝑦^−(1/4)𝛾𝑓𝐼⩾𝜑𝑥,𝑦∀𝑥,𝑦⩾0,₁𝛾𝑥,𝛾⁻¹𝑦⩾𝜑𝛾𝐼₁𝑥,𝑦∀𝑥,𝑦⩾0. (77) 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: −𝑥^′′(𝑡)=(2𝑡+1)(1+𝑥(𝑡))^(1/2)+(1+𝑥(𝑡))^−(1/4)(𝑡),()1+𝑢𝑡∈0,1,𝑡≠3,Δ𝑥|_𝑡=(1/3)=11+𝑥3^(1/2)+11+𝑥3^−(1/4),𝑥(0)=1,𝑥^′(0)=0. (78)

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. Let 𝐽=[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, let 𝐼₁(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.
Set 𝜑(𝛾)=𝛾^(1/2), 𝛾∈(0,1), then, 𝑓𝛾𝑥,𝛾⁻¹𝑦=−1+𝛾𝑥^(1/2)+1+𝛾⁻¹𝑦^−(1/4)𝛾𝑓𝐼⩽𝜑𝑥,𝑦∀𝑥,𝑦⩾0,₁𝛾𝑥,𝛾⁻¹𝑦⩾𝜑𝛾𝐼₁𝑥,𝑦∀𝑥,𝑦⩾0. (79) 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, let 𝐶_𝑓=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 𝑥_𝑑 in 𝐻. Also, applying Theorem 13, we get the result on the stability of optimal control to (OP).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for their valuable suggestions and comments. The research was partially supported by the National Natural Science Foundation of China (no. 61250011), the Postdoctoral Science Foundation of China (no.2012M510786) and the Science Foundation of Shanxi Province (no. 2012011004-4).