Control Systems Described by a Class of Fractional Semilinear Evolution Equations and Their Relaxation Property

Abstract

We consider a control system described by a class of fractional semilinear evolution equations in a separable reflexive Banach space. The constraint on the control is a multivalued map with nonconvex values which is lower semicontinuous with respect to the state variable. Along with the original system we also consider the system in which the constraint on the control is the upper semicontinuous convex-valued regularization of the original constraint. We obtain the existence results for the control systems and the relaxation property between the solution sets of these systems.

1. Introduction

Let 𝐽=[0,𝑏] and 0<𝛼<1. We consider the following control system described by a class of fractional semilinear evolution equations of the form: ^𝐶𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)((𝑡((𝑡𝑢(𝑡)𝑥(0)=𝐴𝑥+ℎ𝑡,𝑥))+𝑔𝑡,𝑥)),𝑡∈𝐽,=𝑥₀, (1.1) with the mixed nonconvex constraint on the control 𝑢(𝑡)∈𝑈(𝑡,𝑥(𝑡))a.e.on𝐽. (1.2) Here ^𝐶𝐷^𝛼_𝑡 is the Caputo fractional derivative of order 𝛼, 𝑏>0 is a finite real number, 𝐴 is the infinitesimal generator of a strongly continuous semigroup {𝑇(𝑡),𝑡≥0} in a separable reflexive Banach space 𝑋, 𝑔∶𝐽×𝑋→ℒ(𝑌,𝑋) (ℒ(𝑌,𝑋) is the space of continuous linear operators from 𝑌 into 𝑋), ℎ∶𝐽×𝑋→𝑋 is a nonlinear function, and 𝑈∶𝐽×𝑋→2^𝑌⧵{∅} is a multivalued mapping with closed values that is not necessarily convex. The space 𝑌 is a separable, reflexive Banach space modeling the control space.

We denote by 𝐶(𝐽,𝑋) the space of all continuous functions from 𝐽 into 𝑋 with the supremum norm given by ‖𝑥‖_𝐶=sup_𝑡∈𝐽‖𝑥(𝑡)‖_𝑋 for 𝑥∈𝐶(𝐽,𝑋). Let 𝐵_𝑋⊆𝑋 be the open unit ball centered at zero. Consider the multivalued map 𝑈_𝛿(𝑡,𝑥)=co𝑈(𝑡,𝑧)∶𝑧∈𝑥+𝛿𝐵_𝑋𝑉()=,𝛿>0,𝑡,𝑥_𝛿↓0𝑈_𝛿(𝑡,𝑥), (1.3)(1.4) here co stands for the closed convex hull of a set. The map (1.4) is usually called the convex upper semicontinuous regularization of 𝑈(𝑡,𝑥).

Along with the constraint (1.2) on the control we also consider the constraint 𝑢(𝑡)∈𝑉(𝑡,𝑥(𝑡))a.e.on𝐽 (1.5) on the control. Note that usually we have co𝑈(𝑡,𝑥)⊆𝑉(𝑡,𝑥).

Definition 1.1. A solution of the control system (1.1), (1.2) is defined to be a pair (𝑥(⋅),𝑢(⋅)) consisting of a trajectory 𝑥∈𝐶(𝐽,𝑋) and a control 𝑢∈𝐿¹(𝐽,𝑌) satisfying (1.1) and the inclusion (1.2) a.e.

A solution of the control system (1.1), (1.5) is defined similarly. We denote by ℛ_𝑈, 𝒯𝑟_𝑈 (ℛ_𝑉, 𝒯𝑟_𝑉) the sets of all solutions, all admissible trajectories of the control system (1.1) and (1.2) (the control system (1.1) and (1.5)).

Relaxation property [1] has important ramifications in control theory. There are many papers dealing with the verification of the relaxation property for various classes of control systems. For example, we refer to [2–5] for nonlinear evolution inclusions or equations, [6, 7] for control problems of subdifferential type and the references therein. In this paper, we investigate this property for control systems described by fractional semilinear evolution equations. We will prove that 𝒯𝑟_𝑉 is a compact set in 𝐶(𝐽,𝑋) and 𝒯𝑟_𝑉=𝒯𝑟_𝑈, (1.6) where the bar stands for the closure in 𝐶(𝐽,𝑋).

Fractional calculus has recently gained much attentions due to its numerous applications in science and engineering. Examples can be found in various disciplines such as mechanics, electrophysics, signal and image processing, thermodynamics, biophysics, aerodynamics, and economics, (see [8–12] for instance). For some recent results on fractional differential equations, we can refer to [13–17]. As for the study of fractional semilinear differential equations, we can refer to Zhou and Jiao [18, 19], Wang and Zhou [20] for the existence results. The issue of approximate controllability was considered by Kumar and Sukavanam [21], Sakthivel et al. [22]. Wang and Zhou in [23] were concerned with the optimal control settings.

2. Preliminaries and Assumptions

Let 𝐽=[0,𝑏] be a closed interval of the real line with the Lebesgue measure 𝜇 and the 𝜎-algebra Σ of 𝜇 measurable sets. The norm of the space 𝑋 (or 𝑌) will be denoted by ‖⋅‖_𝑋 (or ‖⋅‖_𝑌). For any Banach space 𝑉 the symbol 𝜔−𝑉 stands for 𝑉 equipped with the weak 𝜎(𝑉,𝑉^∗) topology. The same notation will be used for subsets of 𝑉. In all other cases we assume that 𝑉 and its subsets are equipped with the strong (normed) topology.

We first recall the following known definitions from fractional differential theory. For more details, please see [11, 12].

Definition 2.1. The fractional integral of order 𝛼 with the lower limit zero for a function 𝑓 is defined as 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀𝑓(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,𝑡>0,𝛼>0 (2.1) provided the right hand side is point-wise defined on [0,∞), where Γ(⋅) is the gamma function.

Definition 2.2. The Riemann-Liouville derivative of order 𝛼 with the lower limit zero for a function 𝑓 is defined as ^𝐿𝐷^𝛼𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡₀𝑓(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠,𝑡>0,𝑛−1<𝛼<𝑛. (2.2)

Definition 2.3. The Caputo derivative of order 𝛼 with the lower limit zero for a function 𝑓 is defined as ^𝐶𝐷^𝛼𝑓(𝑡)=^𝐿𝐷^𝛼𝑓(𝑡)−^𝑛−1_𝑘=0𝑡^𝑘𝑓𝑘!^(𝑘)(0),𝑡>0,𝑛−1<𝛼<𝑛. (2.3)

If 𝑓 is an abstract function with values in 𝑋, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner's sense.

We now proceed to some basic definitions and results from multivalued analysis. For more details on multivalued analysis, see the books [24, 25].

We use the following notations: 𝑃_𝑓(𝑌) is the set of all nonempty closed subsets of 𝑌, 𝑃_𝑓𝑏(𝑌) is the set of all nonempty, closed and bounded subsets of 𝑌, and 𝑃_𝑓𝑐(𝑌) is the set of all nonempty, closed, and convex subsets of 𝑌.

On 𝑃_𝑏𝑓(𝑌), we have a metric known as the “Hausdorff metric” and defined by ℎ(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴), (2.4) where 𝑑(𝑥,𝐶) is the distance from a point 𝑥 to a set 𝐶. We say a multivalued map is ℎ-continuous if it is continuous in the Hausdorff metric ℎ(⋅,⋅).

We say that a multivalued map 𝐹∶𝐽→𝑃_𝑓(𝑌) is measurable if 𝐹⁻¹(𝐸)={𝑡∈𝐽∶𝐹(𝑡)∩𝐸≠∅}∈Σ for every closed set 𝐸⊆𝑌. If 𝐹∶𝐽×𝑋→𝑃_𝑓(𝑌), then measurability of 𝐹 means that 𝐹⁻¹(𝐸)∈Σ⊗ℬ_𝑋, where Σ⊗ℬ_𝑋 is the 𝜎-algebra of subsets in 𝐽×𝑋 generated by the sets 𝐴×𝐵, 𝐴∈Σ, 𝐵∈ℬ_𝑋, and ℬ_𝑋 is the 𝜎-algebra of the Borel sets in 𝑋.

Suppose 𝑉,𝑍 are two Hausdorff topological spaces and 𝐹∶𝑉→2^𝑍⧵{∅}. We say that 𝐹 is lower semicontinuous in the sense of Vietoris (l.s.c. for short) at a point 𝑥₀∈𝑉 if for any open set 𝑊⊆𝑍, 𝐹(𝑥₀)∩𝑊≠∅, there is a neighborhood 𝑂(𝑥₀) of 𝑥₀ such that 𝐹(𝑥)∩𝑊≠∅ for all 𝑥∈𝑂(𝑥₀). 𝐹 is said to be upper semicontinuous in the sense of Vietoris (u.s.c. for short) at a point 𝑥₀∈𝑉 if for any open set 𝑊⊆𝑍, 𝐹(𝑥₀)⊆𝑊, there is a neighborhood 𝑂(𝑥₀) of 𝑥₀ such that 𝐹(𝑥)⊆𝑊 for all 𝑥∈𝑂(𝑥₀). For the properties of l.s.c and u.s.c, see the book [24].

Besides the standard norm on 𝐿^𝑞(𝐽,𝑌) (here 𝑌 is a separable, reflexive Banach space), 1<𝑞<∞, we also consider the so called weak norm: ‖𝑢(⋅)‖_𝜔=sup_{0≤𝑡1≤𝑡2≤𝑏}^𝑡2_𝑡1𝑢(𝑠)𝑑𝑠_𝑌,for𝑢∈𝐿^𝑞(𝐽,𝑌). (2.5) The space 𝐿^𝑞(𝐽,𝑌) furnished with this norm will be denoted by 𝐿^𝑞_𝜔(𝐽,𝑌). The following result establishes a relation between convergence in 𝜔−𝐿^𝑞(𝐽,𝑌) and convergence in 𝐿^𝑞_𝜔(𝐽,𝑌).

Lemma 2.4 (see [5]). If a sequence {𝑢_𝑛}_𝑛≥1⊆𝐿^𝑞(𝐽,𝑌) is bounded and converges to 𝑢 in 𝐿^𝑞_𝜔(𝐽,𝑌), then it converges to 𝑢 in 𝜔−𝐿^𝑞(𝐽,𝑌).

We assume the following assumptions on the data of our problems in the whole paper.

𝐇(𝐀): The operator 𝐴 generates a strongly continuous semigroup 𝑇(𝑡), 𝑡≥0 in 𝑋, and there exists a constant 𝑀_𝐴≥1 such that sup_{𝑡∈[0,∞)}‖𝑇(𝑡)‖≤𝑀_𝐴. For any 𝑡>0, 𝑇(𝑡) is compact.

Remark 2.5. Let us take 𝑋=𝐿²[0,𝜋] and define the operator 𝐴 by 𝐴𝜔=𝜔^′′ with the domain 𝐷(𝐴)={𝜔∈𝑋:𝜔,𝜔^′ are absolutely continuous, 𝜔^′′∈𝑋, and 𝜔(0)=𝜔(𝜋)=0}. Then 𝐴𝜔=−∑^∞_𝑛=1𝑛²⟨𝜔,𝑒_𝑛⟩𝑒_𝑛, 𝜔∈𝐷(𝐴), where 𝑒_𝑛(𝑧)=(2/𝜋)^1/2sin𝑛𝑧, 0≤𝑧≤𝜋, 𝑛=1,2,…. Clearly 𝐴 generates a compact semigroup {𝑇(𝑡)∶𝑡>0} in 𝑋 and it is given by 𝑇(𝑡)𝜔=∑^∞_𝑛=1𝑒^−𝑛2𝑡⟨𝜔,𝑒_𝑛⟩𝑒_𝑛, 𝜔∈𝑋. In such a case, it is easy to see that H(A) holds [22].

𝐇(𝐠): The operator 𝑔∶𝐽×𝑋→ℒ(𝑌,𝑋) is such that

𝐇(𝐡): The function ℎ∶𝐽×𝑋→𝑋 of Carathéodory type satisfies: there exists a constant 0<𝛽<𝛼 such that for a.e. 𝑡∈𝐽 and all 𝑥∈𝑋, ‖ℎ(𝑡,𝑥)‖_𝑋≤𝑎₁(𝑡)+𝑐₁‖𝑥‖_𝑋, where 𝑎₁∈𝐿^1/𝛽(𝐽,ℝ⁺) and 𝑐₁>0.

𝐇(𝐔): The multivalued map 𝑈∶𝐽×𝑋→𝑃_𝑓(𝑌) is such that:

𝐇(𝐌): For any 𝑀>0, there exists a function 𝑙_𝑀∈𝐿^∞(𝐽,ℝ⁺) such that for a.e. 𝑡∈𝐽 and for any 𝑥₁,𝑥₂∈𝑋, ‖𝑥₁‖_𝑋≤𝑀, ‖𝑥₂‖_𝑋≤𝑀 and 𝑢₁∈𝑈(𝑡,𝑥₁), there is a 𝑢₂∈𝑈(𝑡,𝑥₂) such that 𝑔𝑡,𝑥₁𝑢₁+ℎ𝑡,𝑥₁−𝑔𝑡,𝑥₂𝑢₂−ℎ𝑡,𝑥_2𝑋≤𝑙_𝑀(𝑡)𝑥₁−𝑥_2𝑋. (2.6) We note that the condition similar to H(M) was also assumed in [6, 7].

From the Definitions 2.1 and 2.2 and the results obtained in [18, 19], Definition 1.1 can be rewritten in the following form.

Definition 2.6. A function 𝑥∈𝐶(𝐽,𝑋) is a (mild) solution of the system (1.1), (1.2) if 𝑥(0)=𝑥₀ and there exists 𝑢∈𝐿¹(𝐽,𝑌) such that 𝑢(𝑡)∈𝑈(𝑡,𝑥(𝑡)) a.e. on 𝑡∈𝐽 and 𝑥(𝑡)=𝑃_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑢(𝑠)+ℎ(𝑠,𝑥(𝑠))𝑑𝑠. (2.7)
A similar definition can be introduced for the system (1.1) and (1.5). Here 𝑃_𝛼(𝑡)=^∞₀𝜉_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝑑𝜃,𝑄_𝛼(𝑡)=𝛼^∞₀𝜃𝜉_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝜉𝑑𝜃,_𝛼(𝜃)=1𝛼𝜃^{−1−(1/𝛼)}𝜛_𝛼𝜃^−(1/𝛼)𝜛≥0,_𝛼(𝜃)=1𝜋^∞_𝑛=1(−1)^(𝑛−1)(𝜃)^{(−𝑛𝛼−1)}Γ(𝑛𝛼+1)()(),𝑛!sin𝑛𝜋𝛼,𝜃∈0,∞ (2.8) and 𝜉_𝛼 is a probability density function defined on (0,∞) [26], that is 𝜉_𝛼(𝜃)≥0,𝜃∈(0,∞),^∞₀𝜉_𝛼(𝜃)𝑑𝜃=1. (2.9) It is not difficult to verify that ^∞₀𝜃𝜉_𝛼(𝜃)1𝑑𝜃=Γ().1+𝛼 (2.10)

Lemma 2.7 (see [18, 19]). Let H(A) hold, then the operators 𝑃_𝛼 and 𝑄_𝛼 have the following properties.(1) For any fixed 𝑡≥0, 𝑃_𝛼(𝑡) and 𝑄_𝛼(𝑡) are linear and bounded operators, that is, for any 𝑥∈𝑋, 𝑃_𝛼(𝑡)𝑥_𝑋≤𝑀_𝐴‖𝑥‖_𝑋,𝑄_𝛼(𝑡)𝑥_𝑋≤𝛼𝑀_𝐴Γ()‖𝑥‖1+𝛼^𝑋. (2.11)(2){𝑃_𝛼(𝑡),𝑡≥0} and {𝑄_𝛼(𝑡),𝑡≥0} are strongly continuous.(3) For every 𝑡>0, 𝑃_𝛼(𝑡) and 𝑄_𝛼(𝑡) are compact operators.

Lemma 2.8 (see [27, Theorem 3.1]). Let 𝑥(𝑡) be continuous and nonnegative on [0,𝑏]. If 𝑥(𝑡)≤𝜓(𝑡)+𝑀^𝑡₀𝑥(𝑠)(𝑡−𝑠)^𝛾𝑑𝑠,0≤𝑡≤𝑏, (2.12) where 0≤𝛾<1, 𝜓(𝑡) is a non-negative, monotonic increasing continuous function on [0,𝑏] and 𝑀 is a positive constant, then 𝑥(𝑡)≤𝜓(𝑡)𝐸_1−𝛾𝑀Γ1−𝛾𝑡^1−𝛾,0≤𝑡≤𝑏, (2.13) where 𝐸_1−𝛾(𝑧) is the Mittag-Leffler function defined for all 𝛾<1 by 𝐸_1−𝛾(𝑧)=^∞_𝑛=0𝑧^𝑛Γ𝑛.1−𝛾+1 (2.14)

3. Auxiliary Results

In this section, we will give some auxiliary results needed in the proof of the main results. We begin with the a priori estimation of the trajectory of the control systems.

Lemma 3.1. For any admissible trajectory 𝑥 of the control system (1.1) and (1.5), that is, 𝑥∈𝒯𝑟_𝑉, there is a constant L such that ‖𝑥‖_𝐶≤𝐿. (3.1)

Proof. Let any 𝑥∈𝒯𝑟_𝑉, from Definition 2.6, we know that there exists a 𝑢 with 𝑢(𝑡)∈𝑉(𝑡,𝑥(𝑡)) a.e. and 𝑥(𝑡)=𝑃_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑢(𝑠)+ℎ(𝑠,𝑥(𝑠))𝑑𝑠. (3.2) Then by Lemma 2.7, we get ‖𝑥(𝑡)‖_𝑋≤𝑀_𝐴𝑥_0𝑋+𝛼𝑀_𝐴Γ(1+𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1‖ℎ(𝑠,𝑥(𝑠))‖_𝑋+𝑑𝑠𝛼𝑀_𝐴Γ(1+𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠,𝑥(𝑠))𝑢(𝑠)_𝑋𝑑𝑠. (3.3) From H(h) and Hölder inequality, we have ^𝑡₀(𝑡−𝑠)^𝛼−1‖ℎ(𝑠,𝑥(𝑠))‖_𝑋𝑑𝑠≤^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₁(𝑠)+𝑐₁‖𝑥(𝑠)‖_𝑋≤𝑑𝑠1−𝛽𝑏𝛼−𝛽^{(𝛼−𝛽)/(1−𝛽)1−𝛽}𝑎_{1𝐿1/𝛽(𝐽)}+𝑐₁^𝑡₀(𝑡−𝑠)^𝛼−1‖𝑥(𝑠)‖_𝑋𝑑𝑠. (3.4) Similarly, by H(g)(3) and H(U)(3), ^𝑡₀(𝑡−𝑠)^𝛼−1𝑔(𝑠,𝑥(𝑠))𝑢(𝑠)_𝑋𝑑𝑠≤𝑑^𝑡₀(𝑡−𝑠)^𝛼−1𝑎₂(𝑠)+𝑐₂‖𝑥(𝑠)‖_𝑋𝑑𝑠≤𝑑1−𝛽𝑏𝛼−𝛽^{(𝛼−𝛽)/(1−𝛽)1−𝛽}𝑎_{2𝐿1/𝛽(𝐽)}+𝑑𝑐₂^𝑡₀(𝑡−𝑠)^𝛼−1‖𝑥(𝑠)‖_𝑋𝑑𝑠. (3.5) Combining (3.4), (3.5) with (3.3), we obtain ‖𝑥(𝑡)‖_𝑋≤𝑀_𝐴𝑥_0𝑋+𝛼𝑀_𝐴Γ(1+𝛼)𝑑𝑐₂+𝑐₁^𝑡₀(𝑡−𝑠)^𝛼−1‖𝑥(𝑠)‖_𝑋+𝑑𝑠𝛼𝑀_𝐴Γ(1+𝛼)1−𝛽𝑏𝛼−𝛽^{(𝛼−𝛽)/(1−𝛽)1−𝛽}𝑎_{1𝐿1/𝛽(𝐽)}+𝑑𝑎_{2𝐿1/𝛽(𝐽)}. (3.6) From the above inequality, using the well-known singular version of the Gronwall inequality (see Lemma 2.8), we can deduce that there exists a constant 𝐿>0 such that ‖𝑥‖_𝐶≤𝐿.
Let 𝑄={ℎ∈𝑋∶‖ℎ‖_𝑋≤𝐿}. Let pr_𝐿∶𝑋→𝑋 be the 𝐿-radial retraction, that is, pr_𝐿(𝑥)=𝑥,‖𝑥‖_𝑋≤𝐿,𝐿𝑥‖𝑥‖_𝑋,‖𝑥‖_𝑋>𝐿. (3.7) This map is Lipschitz continuous. We define 𝑈₁(𝑡,𝑥)=𝑈(𝑡,pr_𝐿𝑥). Obviously, 𝑈₁ satisfies H(U)(1) and H(U)(2). Moreover, by the properties of pr_𝐿, we have for a.e. 𝑡∈𝐽, all 𝑥∈𝑋 and all 𝑢∈𝑈₁(𝑡,𝑥) the estimates ‖𝑢‖_𝑌≤𝑎₂(𝑡)+𝑐₂𝐿,‖𝑢‖_𝑌≤𝑎₂(𝑡)+𝑐₂‖𝑥‖_𝑋. (3.8) Hence, Lemma 3.1 is still valid with 𝑈(𝑡,𝑥) substituted by 𝑈₁(𝑡,𝑥). Therefore, we assume without any loss of generality that for a.e. 𝑡∈𝐽, and all 𝑥∈𝑋sup‖𝑣‖_𝑌∶𝑣∈𝑈(𝑡,𝑥)≤𝜑(𝑡)=𝑎₂(𝑡)+𝑐₂𝐿,with𝜑∈𝐿^1/𝛽𝐽,ℝ⁺. (3.9) Similarly, we can assume that for a.e. 𝑡∈𝐽 and all 𝑥∈𝑋‖ℎ(𝑡,𝑥)‖_𝑋≤𝛾(𝑡)=𝑎₁(𝑡)+𝑐₁𝐿,with𝛾∈𝐿^1/𝛽𝐽,ℝ⁺. (3.10) Let 𝑌_𝜑=𝑢∈𝐿^1/𝛽(𝐽,𝑌)∶‖𝑢(𝑡)‖_𝑌(𝑡),𝑋≤𝜑a.e.𝑡∈𝐽_𝜑=𝑓∈𝐿^1/𝛽(𝐽,𝑋)∶𝑓(𝑡)_𝑋≤𝑑𝜑(𝑡)+𝛾(𝑡)a.e.𝑡∈𝐽. (3.11)(3.12) It follows from assumption H(g) that for any ℎ∈𝑋^∗, the function ⟨ℎ,𝑔(𝑡,𝑥)𝑢⟩=⟨𝑔^∗(𝑡,𝑥)ℎ,𝑢⟩ is measurable in 𝑡 and continuous in (𝑥,𝑢) almost everywhere. Hence, for any measurable functions 𝑥∶𝐽→𝑋 and 𝑢∶𝐽→𝑌, the function 𝑡→𝑔(𝑡,𝑥(𝑡))𝑢(𝑡) is scalarly measurable [28]. The separability of the space 𝑋 implies that the function 𝑡→𝑔(𝑡,𝑥(𝑡))𝑢(𝑡) is measurable. Therefore, according to H(g) and H(h), for any 𝑥∈𝐿^1/𝛽(𝐽,𝑋) and 𝑢∈𝐿^1/𝛽(𝐽,𝑌), the function 𝑡→𝑔(𝑡,𝑥(𝑡))𝑢(𝑡)+ℎ(𝑡,𝑥(𝑡)) is an element of the space 𝐿^1/𝛽(𝐽,𝑋). Hence we can consider the operator 𝒜∶𝐿^1/𝛽(𝐽,𝑋)×𝐿^1/𝛽(𝐽,𝑌)→𝐿^1/𝛽(𝐽,𝑋) defined by the rule 𝒜(𝑥,𝑢)(𝑡)=𝑔(𝑡,𝑥(𝑡))𝑢(𝑡)+ℎ(𝑡,𝑥(𝑡)). (3.13)

Lemma 3.2. The operator (𝑥,𝑢)→𝒜(𝑥,𝑢) is sequentially continuous as an operator from 𝐿^1/𝛽(𝐽,𝑋)×𝜔−𝐿^1/𝛽(𝐽,𝑌) into 𝜔−𝐿^1/𝛽(𝐽,𝑋).

Proof. Suppose that 𝑥_𝑛→𝑥 in 𝐿^1/𝛽(𝐽,𝑋) and 𝑢_𝑛→𝑢 in 𝜔−𝐿^1/𝛽(𝐽,𝑌). Take an arbitrary 𝑢∈𝑌 and any ℎ∈𝐿^1/(1−𝛽)(𝐽,𝑋^∗). H(g) and the equality ℎ(𝑡),𝑔𝑡,𝑥_𝑛(𝑡)𝑢=𝑔^∗𝑡,𝑥_𝑛(𝑡)ℎ(𝑡),𝑢 (3.14) imply that 𝑡→𝑔^∗(𝑡,𝑥_𝑛(𝑡))ℎ(𝑡) is a scalarly measurable function from 𝐽 to 𝑌^∗. Hence it is measurable. Consider a subsequence 𝑥_𝑛𝑘, 𝑘≥1, of the sequence 𝑥_𝑛, 𝑛≥1, converging to 𝑥 a.e. in 𝑡∈𝐽. By H(g), H(h), and (3.10), we have 𝑔^∗𝑡,𝑥_𝑛𝑘(𝑡)ℎ(𝑡)→𝑔^∗(𝑡,𝑥(𝑡))ℎ(𝑡)a.e.𝑡∈𝐽in𝑌^∗,𝑔^∗(𝑡,𝑥_𝑛𝑘(𝑡))ℎ(𝑡)_𝑌∗≤𝑑‖ℎ(𝑡)‖_𝑋^∗ℎa.e.𝑡∈𝐽,𝑡,𝑥_𝑛𝑘(𝑡)→ℎ(𝑡,𝑥(𝑡))a.e.𝑡∈𝐽in𝑋,ℎ(𝑡,𝑥_𝑛𝑘(𝑡))_𝑋≤𝛾(𝑡). (3.15) Using the preceding four formulae and Lebesgue's theorem of dominated convergence, we obtain 𝑔^∗𝑡,𝑥_𝑛𝑘(𝑡)ℎ(𝑡)→𝑔^∗(𝑡,𝑥(𝑡))ℎ(𝑡)in𝐿^1/(1−𝛽)𝐽,𝑌^∗,ℎ𝑡,𝑥_𝑛𝑘(𝑡)→ℎ(𝑡,𝑥(𝑡))in𝐿^1/𝛽(𝐽,𝑋). (3.16)(3.17) Then it follows from (3.16) that _𝐽𝑔^∗𝑡,𝑥_𝑛𝑘(𝑡)ℎ(𝑡),𝑢_𝑛𝑘(𝑡)𝑑𝑡→_𝐽𝑔^∗(𝑡,𝑥(𝑡))ℎ(𝑡),𝑢(𝑡)𝑑𝑡. (3.18) Since ⟨ℎ(𝑡),𝑔(𝑡,𝑥(𝑡))𝑢(𝑡)⟩=⟨𝑔^∗(𝑡,𝑥(𝑡))ℎ(𝑡),𝑢(𝑡)⟩ and ℎ∈𝐿^1/(1−𝛽)(𝐽,𝑋^∗) is arbitrary, by (3.17) and (3.18), we deduce that 𝒜𝑥_𝑛𝑘,𝑢_𝑛𝑘→𝒜(𝑥,𝑢)in𝜔−𝐿^1/𝛽(𝐽,𝑋). (3.19) It follows from (3.9), (3.10), and (3.12) that {𝒜(𝑥_𝑛,𝑢_𝑛)}_𝑛≥1 is a subset of 𝑋_𝜑 which is a metrizable compact set in 𝜔−𝐿^1/𝛽(𝐽,𝑋). If the sequence 𝒜(𝑥_𝑛,𝑢_𝑛), 𝑛≥1, does not converge to 𝒜(𝑥,𝑢) in 𝜔−𝐿^1/𝛽(𝐽,𝑋), then it has a subsequence 𝒜(𝑥_𝑛𝑖,𝑢_𝑛𝑖), 𝑖≥1, such that none of its subsequences converges to 𝒜(𝑥,𝑢) in 𝜔−𝐿^1/𝛽(𝐽,𝑋). Applying the above arguments to this very subsequence (𝑥_𝑛𝑖,𝑢_𝑛𝑖), 𝑖≥1, we obtain a contradiction. The lemma is proved.

Lemma 3.3. For a.e. 𝑡∈𝐽, the multivalued map 𝑥→𝑉(𝑡,𝑥) defined by (1.4) from 𝑋 to 2^𝑌 is u.s.c. with convex closed values.

Proof. From the definition of 𝑉(𝑡,𝑥), it is clear that 𝑉(𝑡,𝑥) is closed convex valued. Since 𝛿→𝑈_𝛿(𝑡,𝑥) is increasing (in the sense of inclusion), and letting 𝑈_1/𝑛(𝑡,𝑥)=𝑈()1co𝑡,𝑧∶𝑧∈𝑥+𝑛𝐵_𝑋, (3.20) we obtain 𝑉(𝑡,𝑥)=_𝑛≥1𝑈_1/𝑛(𝑡,𝑥). (3.21) Let 𝑥₀∈𝑋 and 𝑊 be an open set in 𝑌 such that 𝑉(𝑡,𝑥₀)⊆𝑊. By (3.21), we can find an 𝑛₀≥1 such that 𝑈_1/𝑛0𝑡,𝑥₀⊆𝑊. (3.22) For an arbitrary 𝑦∈𝑥₀+(1/𝑛₀)𝐵_𝑋, we can find a 𝛿>0 such that 𝑦+𝛿𝐵_𝑋⊆𝑥₀+(1/𝑛₀)𝐵_𝑋. Therefore we obtain 𝑈_𝛿𝑡,𝑦⊆𝑈_1/𝑛0𝑡,𝑥₀⊆𝑊. (3.23) Then it is clear that 𝑉(𝑡,𝑦)⊆𝑊, for all 𝑦∈𝑥₀+(1/𝑛₀)𝐵_𝑋. This means that 𝑥→𝑉(𝑡,𝑥) is u.s.c.

Let 𝐶_𝑋={𝑧_𝑘}_𝑘≥1 be a dense countable subset of the ball 𝐵_𝑋. We put 𝑈^𝑘_1/𝑛()1𝑡,𝑥=𝑈𝑡,𝑥+𝑛𝑧_𝑘,𝑧_𝑘∈𝐶_𝑋. (3.24)

Lemma 3.4. For a.e. 𝑡∈𝐽, let 𝑈_1/𝑛(𝑡,𝑥) be defined by (3.20), then we have 𝑈_1/𝑛(𝑡,𝑥)=co^∞_𝑘=1𝑈^𝑘_1/𝑛(𝑡,𝑥), (3.25) where the closure is taken in 𝑌.

Proof. We recall that co𝐴=co𝐴 for any subset 𝐴⊆𝑌. Hence it is sufficient to prove that for a.e. 𝑡∈𝐽, ^∞_𝑘=1𝑈^𝑘_1/𝑛(𝑡,𝑥)=𝑈()1𝑡,𝑧∶𝑧∈𝑥+𝑛𝐵_𝑋. (3.26) That the left hand side of (3.26) is contained in its right hand side is obvious. Let 𝑤∈{⋃𝑈(𝑡,𝑧)∶𝑧∈𝑥+(1/𝑛)𝐵_𝑋}, then 𝑤∈𝑈(𝑡,𝑥+(1/𝑛)𝑧_∗) for some 𝑧_∗∈𝐵_𝑋. Now let 𝑧_𝑚→𝑧_∗, {𝑧_𝑚}_𝑚≥1⊆𝐶_𝑋. Since a.e. 𝑡∈𝐽, 𝑥→𝑈(𝑡,𝑥) is l.s.c. at 𝑥+(1/𝑛)𝑧_∗ and 𝑥+(1/𝑛)𝑧_𝑚→𝑥+(1/𝑛)𝑧_∗, there is a sequence 𝑤_𝑚∈𝑈(𝑡,𝑥+(1/𝑛)𝑧_𝑚), 𝑚≥1 converging to 𝑤 (Proposition 1.2.6 [24]). Due to {𝑤_𝑚}_𝑚≥1⊆⋃^∞_𝑘=1𝑈^𝑘_1/𝑛(𝑡,𝑥), we have 𝑤∈⋃^∞_𝑘=1𝑈^𝑘_1/𝑛(𝑡,𝑥). Since 𝑤 is arbitrary, we can get {⋃𝑈(𝑡,𝑧)∶𝑧∈𝑥+(1/𝑛)𝐵_𝑋}⊆⋃^∞_𝑘=1𝑈^𝑘_1/𝑛(𝑡,𝑥). Therefore (3.26) holds. The lemma is proved.

Now we consider the following auxiliary problem: ^𝐶𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)(𝑡)[],𝑥(0)=𝐴𝑥+𝑓,𝑡∈𝐽=0,𝑏=𝑥₀. (3.27) It is clear that for every 𝑓∈𝐿^1/𝛽(𝐽,𝑋), (3.27) has a unique (mild) solution 𝑆(𝑓)∈𝐶(𝐽,𝑋) which is given by 𝑆𝑓(𝑡)=𝑃_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (3.28)

The following lemma describes a property of the solution map 𝑆 which is crucial in our investigation.

Lemma 3.5. The solution map 𝑆∶𝑋_𝜑→𝐶(𝐽,𝑋) is continuous from 𝜔−𝑋_𝜑 to 𝐶(𝐽,𝑋).

Proof. Consider the operator 𝐻∶𝐿^1/𝛽(𝐽,𝑋)→𝐶(𝐽,𝑋) defined by 𝐻𝑓(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (3.29) We know 𝐻 is linear. From simple calculation, one has 𝐻(𝑓)_𝐶≤𝛼𝑀_𝐴Γ(1+𝛼)1−𝛽𝑏𝛼−𝛽^{(𝛼−𝛽)/(1−𝛽)1−𝛽}𝑓_{𝐿1/𝛽(𝐽,𝑋)}, (3.30) that is, the operator 𝐻 is continuous from 𝐿^1/𝛽(𝐽,𝑋) into 𝐶(𝐽,𝑋), hence 𝐻 is also continuous from 𝜔−𝐿^1/𝛽(𝐽,𝑋) into 𝜔−𝐶(𝐽,𝑋).
Let any 𝐵∈𝑃_𝑏(𝐿^1/𝛽(𝐽,𝑋)) and suppose that for any 𝑓∈𝐵, ‖𝑓‖_{𝐿1/𝛽(𝐽,𝑋)}≤𝐾 (𝐾>0 is a constant). Next we will show that 𝐻 is completely continuous.(a) From (3.30), we know that ‖𝐻(𝑓)(𝑡)‖_𝑋 is uniformly bounded for any 𝑡∈𝐽 and 𝑓∈𝐵.(b)𝐻 is equicontinuous on 𝐵. Let 0≤𝑡₁<𝑡₂≤𝑏 and any 𝑓∈𝐵, we get 𝐻(𝑓)(𝑡₂)−𝐻(𝑓)(𝑡₁)_𝑋=^𝑡2₀(𝑡₂−𝑠)^𝛼−1𝑄_𝛼(𝑡₂−𝑠)𝑓(𝑠)𝑑𝑠−^𝑡1₀(𝑡₁−𝑠)^𝛼−1𝑄_𝛼(𝑡₁−𝑠)𝑓(𝑠)𝑑𝑠_𝑋≤^𝑡2_𝑡1(𝑡₂−𝑠)^𝛼−1𝑄_𝛼(𝑡₂−𝑠)𝑓(𝑠)𝑑𝑠_𝑋⋯denotedby𝐼₁+^𝑡1₀((𝑡₂−𝑠)^𝛼−1−(𝑡₁−𝑠)^𝛼−1)𝑄_𝛼(𝑡₂−𝑠)𝑓(𝑠)𝑑𝑠_𝑋⋯denotedby𝐼₂+^𝑡1₀(𝑡₁−𝑠)^𝛼−1(𝑄_𝛼(𝑡₂−𝑠)−𝑄_𝛼(𝑡₁−𝑠))𝑓(𝑠)𝑑𝑠_𝑋⋯denotedby𝐼₃. (3.31) By using analogous arguments as in Lemma 3.1, we find 𝐼₁≤𝛼𝑀_𝐴Γ(1+𝛼)1−𝛽𝛼−𝛽^1−𝛽𝐾𝑡₂−𝑡₁^𝛼−𝛽,𝐼₂≤𝛼𝑀_𝐴Γ(1+𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^{𝛼−11/(1−𝛽)}𝑑𝑠^1−𝛽𝐾≤𝛼𝑀_𝐴Γ(1+𝛼)^𝑡1₀𝑡₁−𝑠^{(𝛼−1)/(1−𝛽)}−𝑡₂−𝑠^{(𝛼−1)/(1−𝛽)}𝑑𝑠^1−𝛽𝐾=𝛼𝑀_𝐴Γ(1+𝛼)1−𝛽𝛼−𝛽^1−𝛽𝑡₁^{(𝛼−𝛽)/(1−𝛽)}−𝑡₂^{(𝛼−𝛽)/(1−𝛽)}+𝑡₂−𝑡₁^{(𝛼−𝛽)/(1−𝛽)1−𝛽}𝐾≤2𝛼𝑀_𝐴Γ(1+𝛼)1−𝛽𝛼−𝛽^1−𝛽𝑡₂−𝑡₁^𝛼−𝛽𝐾. (3.32) For 𝑡₁=0, 0<𝑡₂≤𝑏, it is easy to see that 𝐼₃=0. For 𝑡₁>0 and 𝜖>0 be enough small, we have 𝐼₃≤^𝑡1₀^−𝜖(𝑡₁−𝑠)^𝛼−1𝑄_𝛼𝑡₂−𝑠−𝑄_𝛼𝑡₁−𝑠𝑓(𝑠)𝑑𝑠_𝑋+^𝑡1_𝑡1−𝜖(𝑡₁−𝑠)^𝛼−1𝑄_𝛼𝑡₂−𝑠−𝑄_𝛼𝑡₁−𝑠𝑓(𝑠)𝑑𝑠_𝑋≤sup_{𝑠∈0,𝑡1−𝜖}𝑄_𝛼𝑡₂−𝑠−𝑄_𝛼𝑡₁−𝑠1−𝛽𝛼−𝛽^1−𝛽×𝑡₁^{(𝛼−𝛽)/(1−𝛽)}−𝜖^{(𝛼−𝛽)/(1−𝛽)1−𝛽}𝐾+2𝛼𝑀_𝐴Γ(1+𝛼)1−𝛽𝛼−𝛽^1−𝛽𝜖^𝛼−𝛽𝐾. (3.33) Combining the estimations for 𝐼₁,𝐼₂, and 𝐼₃, and letting 𝑡₂→𝑡₁ and 𝜖→0 in 𝐼₃, we obtain that 𝐻 is equicontinuous. For more details, please see [19].(c) The set Π(𝑡)={𝐻(𝑓)(𝑡)∶𝑓∈𝐵} is relatively compact in 𝑋. Clearly, Π(0)={0} is compact, and hence, it is only necessary to consider 𝑡>0. For each ℎ∈(0,𝑡), 𝑡∈(0,𝑏], 𝑓∈𝐵 and 𝛿>0 be arbitrary, we define Π_ℎ,𝛿(𝑡)=𝐻_ℎ,𝛿𝑓(𝑡)∶𝑓∈𝐵, (3.34)where 𝐻_ℎ,𝛿𝑓(𝑡)=𝛼₀^{𝑡−ℎ∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃𝑓(𝑠)𝑑𝜃𝑑𝑠=𝛼₀^{𝑡−ℎ∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇ℎ^𝛼𝛿𝑇(𝑡−𝑠)^𝛼𝜃−ℎ^𝛼𝛿𝑓(𝑠)ℎ𝑑𝜃𝑑𝑠=𝛼𝑇^𝛼𝛿₀^{𝑡−ℎ∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃−ℎ^𝛼𝛿𝑓(𝑠)𝑑𝜃𝑑𝑠. (3.35) From the compactness of 𝑇(ℎ^𝛼𝛿) (ℎ^𝛼𝛿>0), we obtain that the set Π_ℎ,𝛿(𝑡) is relatively compact in 𝑋 for any ℎ∈(0,𝑡) and 𝛿>0. Moreover, we have 𝐻𝑓(𝑡)−𝐻_ℎ,𝛿𝑓(𝑡)_𝑋=𝛼^𝑡₀^𝛿₀𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃𝑓(𝑠)+𝑑𝜃𝑑𝑠_𝑡0∞𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃𝑓(𝑠)𝑑𝜃𝑑𝑠−₀^{𝑡−ℎ∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃𝑓(𝑠)𝑑𝜃𝑑𝑠_𝑋≤𝛼^𝑡₀^𝛿₀𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇(𝑡−𝑠)^𝛼𝜃𝑓(𝑠)𝑑𝜃𝑑𝑠_𝑋+𝛼^𝑡_𝑡−ℎ^∞_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜉_𝛼(𝜃)𝑇((𝑡−𝑠)^𝛼𝜃)𝑓(𝑠)𝑑𝜃𝑑𝑠_𝑋≤𝑀_𝐴𝛼^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝛽)}𝑑𝑠^1−𝛽𝑓_{𝐿1/𝛽(𝐽,𝑋)}^𝛿₀𝜃𝜉_𝛼(𝜃)𝑑𝜃+𝑀_𝐴𝛼^𝑡_𝑡−ℎ(𝑡−𝑠)^{(𝛼−1)/(1−𝛽)}𝑑𝑠^1−𝛽𝑓_{𝐿1/𝛽(𝐽,𝑋)}^∞_𝛿𝜃𝜉_𝛼(𝜃)𝑑𝜃≤𝑀_𝐴𝐾𝛼1−𝛽𝛼−𝛽^1−𝛽𝑏^{𝛼−𝛽𝛿}₀𝜃𝜉_𝛼(𝜃)1𝑑𝜃+Γ()ℎ1+𝛼^𝛼−𝛽. (3.36) In virtue of (2.10), the last term of the preceding inequality tends to zero as ℎ→0 and 𝛿→0. Therefore, there exist relatively compact sets arbitrarily close to the set Π(𝑡), 𝑡>0. Hence the set Π(𝑡), 𝑡>0 is also relatively compact in 𝑋.
Since 𝑋_𝜑 is a convex compact metrizable subset of 𝜔−𝐿^1/𝛽(𝐽,𝑋), it suffices to prove the sequential continuity of the map 𝑆. Now let {𝑓_𝑛}_𝑛≥1⊆𝑋_𝜑 such that 𝑓_𝑛→𝑓in𝜔−𝐿^1/𝛽(𝐽,𝑋),𝑓∈𝑋_𝜑. (3.37)
By the property of the operator 𝐻, we have 𝐻(𝑓_𝑛)→𝐻(𝑓) in 𝜔−𝐶(𝐽,𝑋). Since {𝑓_𝑛}_𝑛≥1 is bounded, there is a subsequence {𝑓_𝑛𝑘}_𝑘≥1 of the sequence {𝑓_𝑛}_𝑛≥1 such that 𝐻(𝑓_𝑛𝑘)→𝑧 in 𝐶(𝐽,𝑋) for some 𝑧∈𝐶(𝐽,𝑋). From the facts that 𝐻𝑓_𝑛→𝐻𝑓in𝜔−𝐶(𝐽,𝑋),𝐻𝑓_𝑛𝑘→𝑧in𝐶(𝐽,𝑋), (3.38) we obtain that 𝑧=𝐻(𝑓) and 𝐻(𝑓_𝑛)→𝐻(𝑓) in 𝐶(𝐽,𝑋).
From the definitions of the operators 𝑆 and 𝐻, we have that 𝑆(𝑓)(𝑡)=𝑃_𝛼(𝑡)𝑥₀+𝐻(𝑓)(𝑡). Then due to the arguments above, we have 𝑆(𝑓_𝑛)→𝑆(𝑓) in 𝐶(𝐽,𝑋). This completes the proof of the lemma.

4. Existence Results for the Control Systems

In the present section, we are interested in the existence results for the control systems (1.1), (1.2) and (1.1), (1.5).

Let Λ=𝑆(𝑋_𝜑), from Lemma 3.5, we have that Λ is a compact subset of 𝐶(𝐽,𝑋). It follows from formulae (3.9), (3.10), and (3.12) that 𝒯𝑟_𝑈⊆𝒯𝑟_𝑉⊆Λ. Let 𝑈∶𝐶(𝐽,𝑋)→2^{𝐿1/𝛽(𝐽,𝑌)} be defined by 𝑈(𝑥)={ℎ∶𝐽→𝑌measurable∶ℎ(𝑡)∈𝑈(𝑡,𝑥(𝑡))a.e.},𝑥∈𝐶(𝐽,𝑋). (4.1)

Theorem 4.1. The set ℛ_𝑈 is nonempty and the set ℛ_𝑉 is a compact subset of the space 𝐶(𝐽,𝑋)×𝜔−𝐿^1/𝛽(𝐽,𝑌).

Proof. By the hypothesis H(U)(1), we have that for any measurable function 𝑥∶𝐽→𝑋, the map 𝑡→𝑈(𝑡,𝑥(𝑡)) is measurable and has closed values. Therefore it has measurable selectors [29]. So the operator 𝑈 is well defined and its values are closed decomposable subsets of 𝐿^1/𝛽(𝐽,𝑌). We claim that 𝑥→𝑈(𝑥) is l.s.c. Let 𝑥_∗∈𝐶(𝐽,𝑋), ℎ_∗∈𝑈(𝑥_∗) and let {𝑥_𝑛}_𝑛≥1⊆𝐶(𝐽,𝑋) be a sequence converging to 𝑥_∗. It follows from Lemma 3.2 in [30] that there exists a sequence ℎ_𝑛∈𝑈(𝑥_𝑛) such that ℎ_∗(𝑡)−ℎ_𝑛(𝑡)_𝑌≤𝑑_𝑌ℎ_∗(𝑡),𝑈𝑡,𝑥_𝑛(𝑡+1)𝑛,a.e.𝑡∈𝐽. (4.2) Since the map 𝑦→𝑈(𝑡,𝑦) is l.s.c., by the Proposition 1.2.26 in [24], the function 𝑦→𝑑_𝑌(ℎ_∗(𝑡),𝑈(𝑡,𝑦)) is u.s.c. for a.e. 𝑡∈𝐽. It follows from (4.2) that for a.e. 𝑡∈𝐽lim_𝑛→∞ℎ_∗(𝑡)−ℎ_𝑛(𝑡)_𝑌≤lim_𝑛→∞𝑑_𝑌ℎ_∗(𝑡),𝑈𝑡,𝑥_𝑛(𝑡)≤𝑑_𝑌ℎ_∗(𝑡),𝑈𝑡,𝑥_∗(𝑡)=0. (4.3) This together with (3.9) implies that ℎ_𝑛→ℎ_∗ in 𝐿^1/𝛽(𝐽,𝑌). Therefore the map 𝑥→𝑈(𝑥) is l.s.c. By Proposition 2.2 in [31], there is a continuous function 𝑚∶Λ→𝐿^1/𝛽(𝐽,𝑌) such that 𝑚(𝑥)∈𝑈(𝑥),∀𝑥∈Λ. (4.4) Consider the map 𝒫∶𝐿^1/𝛽(𝐽,𝑋)→𝐿^1/𝛽(𝐽,𝑌) defined by 𝒫(𝑓)=𝑚(𝑆(𝑓)). Due to Lemma 3.5 and the continuity of 𝑚, the map 𝒫 is continuous from 𝜔−𝑋_𝜑 into 𝐿^1/𝛽(𝐽,𝑌). Then by Lemma 3.2, we deduce that the map 𝑓→𝒜(𝑆(𝑓),𝒫(𝑓)) is continuous from 𝜔−𝑋_𝜑 into 𝜔−𝐿^1/𝛽(𝐽,𝑋). It follows from (3.9), (3.10), (3.12), and (3.13) that 𝒜(𝑆(𝑓),𝒫(𝑓))∈𝑋_𝜑 for every 𝑓∈𝑋_𝜑. Therefore, the map 𝑓→𝒜(𝑆(𝑓),𝒫(𝑓)) is continuous from 𝜔−𝑋_𝜑 into 𝜔−𝑋_𝜑. Since 𝜔−𝑋_𝜑 is a convex metrizable compact set in 𝜔−𝐿^1/𝛽(𝐽,𝑋), Schauder's fixed point theorem implies that this map has a fixed point 𝑓_∗∈𝑋_𝜑, that is, 𝑓_∗=𝒜(𝑆(𝑓_∗),𝒫(𝑓_∗)). Let 𝑢_∗=𝒫(𝑓_∗) and 𝑥_∗=𝑆(𝑓_∗), then we have 𝑢_∗=𝑚(𝑥_∗) and 𝑓_∗=𝒜(𝑥_∗,𝑢_∗). That means 𝑥_∗(𝑡)=𝑃_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑔𝑠,𝑥_∗(𝑠)𝑢_∗(𝑠)+ℎ𝑠,𝑥_∗(𝑠)𝑢𝑑𝑠,_∗(𝑡)∈𝑈𝑡,𝑥_∗(𝑡),a.e.𝑡∈𝐽, (4.5) which implies that (𝑥_∗(⋅),𝑢_∗(⋅)) is a solution of the control system (1.1) and (1.2). Hence ℛ_𝑈 is nonempty.
It is easy to see that ℛ_𝑉⊆Λ×𝑌_𝜑. Since Λ is compact in 𝐶(𝐽,𝑋) and 𝑌_𝜑 is metrizable convex compact in 𝜔−𝐿^1/𝛽(𝐽,𝑌), we have that ℛ_𝑉 is relatively compact in 𝐶(𝐽,𝑋)×𝜔−𝐿^1/𝛽(𝐽,𝑌). Hence to complete the proof of this theorem, it is sufficient to prove that ℛ_𝑉 is sequentially closed in 𝐶(𝐽,𝑋)×𝜔−𝐿^1/𝛽(𝐽,𝑌).
Let {(𝑥_𝑛(⋅),𝑢_𝑛(⋅))}_𝑛≥1⊆ℛ_𝑉 be a sequence converging to (𝑥(⋅),𝑢(⋅)) in 𝐶(𝐽,𝑋)×𝜔−𝐿^1/𝛽(𝐽,𝑌). Denote 𝑓_𝑛(𝑡)=𝑔𝑡,𝑥_𝑛(𝑡)𝑢_𝑛(𝑡)+ℎ𝑡,𝑥_𝑛(𝑡),𝑓(𝑡)=𝑔(𝑡,𝑥(𝑡))𝑢(𝑡)+ℎ(𝑡,𝑥(𝑡)). (4.6) According to Lemma 3.2, 𝑓_𝑛→𝑓 in 𝜔−𝐿^1/𝛽(𝐽,𝑋). Since 𝑓_𝑛∈𝑋_𝜑 and 𝑥_𝑛=𝑆(𝑓_𝑛), 𝑛≥1, Lemma 3.5 implies that 𝑥=𝑆𝑓. (4.7) Hence, to prove that (𝑥(⋅),𝑢(⋅))∈ℛ_𝑉, we only need to verify that 𝑢(𝑡)∈𝑉(𝑡,𝑥(𝑡)) a.e. 𝑡∈𝐽.
Since 𝑢_𝑛→𝑢 in 𝜔−𝐿^1/𝛽(𝐽,𝑌), by Mazur's theorem we have 𝑢(𝑡)∈^∞_𝑛=1co^∞_𝑘=𝑛𝑢_𝑘(𝑡),fora.e.𝑡∈𝐽. (4.8) From Lemma 3.3, we have that for a.e. 𝑡∈𝐽, the map 𝑥→𝑉(𝑡,𝑥)∈𝑃_𝑓𝑐(𝑌) is u.s.c., then by Proposition 1.2.61 in [24], the map 𝑥→𝑉(𝑡,𝑥)∈𝑃_𝑓𝑐(𝑌) is ℎ-upper semicontinuous. Therefore from assertion (b) of Proposition 1.2.86 in [24], the map 𝑥→𝑉(𝑡,𝑥) has property 𝑄. Hence we have ^∞_𝑛=1co^∞_𝑘=𝑛𝑉𝑡,𝑥_𝑘(𝑡)⊆𝑉(𝑡,𝑥(𝑡)),fora.e.𝑡∈𝐽. (4.9) In virtue of (4.8) and (4.9), and for a.e. 𝑡∈𝐽, 𝑢_𝑛(𝑡)∈𝑉(𝑡,𝑥_𝑛(𝑡)), 𝑛≥1, we obtain that 𝑢(𝑡)∈𝑉(𝑡,𝑥(𝑡)) a.e. 𝑡∈𝐽. This means that ℛ_𝑉 is compact in 𝐶(𝐽,𝑋)×𝜔−𝐿^1/𝛽(𝐽,𝑌). The proof is complete.

5. Main Results

In this section, we will prove the relaxation result. But first, we give a lemma which is important in the proof of our relaxation theorem.

Lemma 5.1. For any pair (𝑥_∗(⋅),𝑢_∗(⋅))∈ℛ_𝑉, there exists a sequence of simple functions 𝑦_𝑛∶𝐽→𝑋 and a sequence 𝑣_𝑛∈𝐿^1/𝛽(𝐽,𝑌), 𝑛≥1, such that 𝑦_𝑛(𝑡)_𝑋≤1𝑛𝑣,𝑡∈𝐽,𝑛≥1,_𝑛(𝑡)∈𝑈𝑡,𝑥_∗(𝑡)+𝑦_𝑛(𝑡)𝑣,𝑡∈𝐽,_𝑛→𝑢_∗in𝜔−𝐿^1/𝛽(𝐽,𝑌). (5.1)(5.2)(5.3)

Proof. Let (𝑥_∗(⋅),𝑢_∗(⋅))∈ℛ_𝑉. From Lemma 3.4, we have that for a.e. 𝑡∈𝐽, 𝑛≥1𝑢_∗(𝑡)∈𝑉𝑡,𝑥_∗(𝑡)⊆𝑈_1/𝑛𝑡,𝑥_∗(𝑡)=co^∞_𝑘=1𝑈^𝑘_1/𝑛𝑡,𝑥_∗(𝑡). (5.4) The map 𝑡→⋃^∞_𝑘=1𝑈^𝑘_1/𝑛(𝑡,𝑥_∗(𝑡)) is measurable (see Propositions 2.3 and 2.6 in [29]) and, by (3.9), is integrally bounded. Therefore, from (5.4) and Theorem 2.2 in [32], we have that there exists an 𝑓_𝑛∈𝐿^1/𝛽(𝐽,𝑌) such that 𝑓_𝑛(𝑡)∈^∞_𝑘=1𝑈^𝑘_1/𝑛𝑡,𝑥_∗(𝑡)a.e.𝑡∈𝐽,𝑢_∗−𝑓_𝑛𝜔≤1𝑛. (5.5) We know that the map 𝑡→𝑈^𝑘_1/𝑛(𝑡,𝑥_∗(𝑡)) is measurable and its value are closed, then following Theorem 5.6 in [29] (also Proposition 2.2.3 in [24]), there exists a sequence of measurable selectors 𝑓^𝑚_𝑘(𝑡)∈𝑈^𝑘_1/𝑛(𝑡,𝑥_∗(𝑡)), 𝑡∈𝐽, 𝑚≥1 such that 𝑈^𝑘_1/𝑛𝑡,𝑥_∗(𝑡)=𝑈𝑡,𝑥_∗(𝑡)+1𝑛𝑧_𝑘=^∞_𝑚=1𝑓^𝑚_𝑘(𝑡),for𝑡∈𝐽,𝑘≥1. (5.6) Therefore we have ^∞_𝑘=1𝑈^𝑘_1/𝑛𝑡,𝑥_∗(𝑡)=^∞_𝑘=1𝑈𝑡,𝑥_∗(𝑡)+1𝑛𝑧_𝑘=^∞∞_𝑘=1𝑚=1𝑓^𝑚_𝑘(𝑡),for𝑡∈𝑇. (5.7) From (5.5) and (5.7), according to Lemma 1.3 in [33] (also see Proposition 2.3.6 [24]), there is a finite measurable partition 𝐽₁,𝐽₂,…,𝐽_𝑙(𝑛) of 𝐽 such that _𝐽𝑓_𝑛(𝑡)−^𝑙(𝑛)_𝑖=1𝜒𝐽_𝑖𝑓^𝑚𝑖_𝑘𝑖(𝑡)_𝑌^1/𝛽𝑑𝑡^𝛽≤1𝑛, (5.8) where 𝜒(𝐽_𝑖) is the characteristic function of the set 𝐽_𝑖. Now let 𝑣_𝑛(𝑡)=^𝑙(𝑛)_𝑖=1𝜒𝐽_𝑖𝑓^𝑚𝑖_𝑘𝑖(𝑡),𝑦_𝑛(𝑡)=^𝑙(𝑛)_𝑖=1𝜒𝐽_𝑖1𝑛𝑧_𝑘𝑖. (5.9) Formula (5.9) implies that 𝑦_𝑛 is a simple function, 𝑣_𝑛∈𝐿^1/𝛽(𝐽,𝑌) and (5.1), (5.2) hold. By Lemma 2.4, (5.5) and (5.8), we obtain that (5.3) holds. The lemma is proved.

Now we are ready to present our main result.

Theorem 5.2. The set 𝒯𝑟_𝑉 is compact in 𝐶(𝐽,𝑋) and the following relation holds 𝒯𝑟_𝑉=𝒯𝑟_𝑈, (5.10) where the bar stands for the closure in 𝐶(𝐽,𝑋).

Proof. Let (𝑥_∗(⋅),𝑢_∗(⋅))∈ℛ_𝑉 and {𝑣_𝑛}_𝑛≥1, {𝑦_𝑛}_𝑛≥1 be given as in Lemma 5.1. Put 𝑄={ℎ∈𝑋∶‖ℎ‖_𝑋≤𝐿}, for fixed 𝑛≥1, we consider the function defined by 𝑟_𝑛(𝑡,𝑥,𝑢)=𝑔𝑡,𝑥_∗(𝑡)+𝑦_𝑛(𝑡)𝑣_𝑛(𝑡)+ℎ𝑡,𝑥_∗(𝑡)+𝑦_𝑛(𝑡)−𝑔(𝑡,𝑥)𝑢−ℎ(𝑡,𝑥)_𝑋−𝑙_𝐿+1(𝑡)𝑥_∗(𝑡)+𝑦_𝑛(𝑡)−𝑥_𝑋. (5.11) It is clear that the function 𝑡→𝑟_𝑛(𝑡,𝑥,𝑢) is measurable and the function (𝑥,𝑢)→𝑟_𝑛(𝑡,𝑥,𝑢) is continuous (in view of H(g) and the fact that if 𝑥∶𝐽→𝑋 is a measurable function, ‖𝑥(𝑡)‖_𝑋 is a measurable real-valued function). According to the Theorem 2.4 in [34], there exists a sequence of nested (in the sense of inclusion) closed sets 𝐽_𝑘⊆𝐽, 𝑘≥1, 𝜇(𝐽⧵∪^∞_𝑘=1𝐽_𝑘)=0 such that the map (𝑡,𝑥)→𝑈(𝑡,𝑥) is l.s.c. on 𝐽_𝑘×𝑄 and 𝑟_𝑛(𝑡,𝑥,𝑢) is continuous on 𝐽_𝑘×𝑄×𝑌. Let the multivalued map 𝐻_𝑛∶𝐽×𝑄→𝑌 be defined by 𝐻_𝑛(𝑡,𝑥)=𝑢∈𝑌∶𝑟_𝑛()−1𝑡,𝑥,𝑢𝑛<0. (5.12) For every 𝑘≥1 the graph of the map 𝐻_𝑛(𝑡,𝑥) is an open subset of 𝐽_𝑘×𝑄×𝑌. Let the map 𝑈_𝑛∶𝐽×𝑄→𝑌 be defined by 𝑈_𝑛(𝑡,𝑥)=𝐻_𝑛(𝑡,𝑥)∩𝑈(𝑡,𝑥). (5.13) Hypothesis H(M) together with (5.12) implies that 𝑈_𝑛(𝑡,𝑥) is nonempty for a.e. 𝑡∈𝐽 and all 𝑥∈𝑄. Since the map 𝑈(𝑡,𝑥) is l.s.c. on 𝐽_𝑘×𝑄, 𝑘≥1, and the graph of the map 𝐻_𝑛(𝑡,𝑥) is an open subset of 𝐽_𝑘×𝑄×𝑌, 𝑘≥1, then according to Proposition 1.2.47 in [24], we obtain that the map 𝑈_𝑛(𝑡,𝑥) is l.s.c. on 𝐽_𝑘×𝑄, 𝑘≥1. Hence the map 𝑈_𝑛(𝑡,𝑥)=𝑈_𝑛(𝑡,𝑥) is l.s.c. on 𝐽_𝑘×𝑄, 𝑘≥1. Therefore, for every continuous function 𝑥∶𝐽→𝑄 the map 𝑡→𝑈_𝑛(𝑡,𝑥(𝑡)) is measurable and the map 𝑥→𝑈_𝑛(𝑡,𝑥) is l.s.c. on 𝑄 for a.e. 𝑡∈𝐽.
It is clear that 𝑈_𝑛(𝑡,𝑥)⊆𝑈(𝑡,𝑥). Consider the system (1.1) with the constraint 𝑈_𝑛(𝑡,𝑥(𝑡)) on the control. The arguments used in the proof of the Theorem 4.1 enable us to obtain the existence of a solution (𝑥_𝑛(⋅),𝑢_𝑛(⋅))∈ℛ_𝑈 and 𝑔𝑡,𝑥_∗(𝑡)+𝑦_𝑛(𝑡)𝑣_𝑛(𝑡)+ℎ𝑡,𝑥_∗(𝑡)+𝑦_𝑛(𝑡)−𝑔𝑡,𝑥_𝑛(𝑡)𝑢_𝑛(𝑡)−ℎ𝑡,𝑥_𝑛(𝑡)_𝑋−𝑙_𝐿+1(𝑡)𝑥_∗(𝑡)+𝑦_𝑛(𝑡)−𝑥_𝑛(𝑡)_𝑋−1𝑛≤0. (5.14) Now by (𝑥_∗(⋅),𝑢_∗(⋅))∈ℛ_𝑉 and (𝑥_𝑛(⋅),𝑢_𝑛(⋅))∈ℛ_𝑈, 𝑛≥1, we have 𝑥_∗(𝑡)=𝑃_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑔𝑠,𝑥_∗(𝑠)𝑢_∗(𝑠)+ℎ𝑠,𝑥_∗(𝑠)𝑥𝑑𝑠,_𝑛(𝑡)=𝑃_𝛼(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑔𝑠,𝑥_𝑛(𝑠)𝑢_𝑛(𝑠)+ℎ𝑠,𝑥_𝑛(𝑠)𝑑𝑠. (5.15)(5.16) Theorem 4.1 and ℛ_𝑈⊆ℛ_𝑉 imply that we can assume, possibly up to a subsequence, that the sequence (𝑥_𝑛(⋅),𝑢_𝑛(⋅))→(𝑥(⋅),𝑢(⋅))∈ℛ_𝑉 in 𝐶(𝐽,𝑋)×𝜔−𝐿^1/𝛽(𝐽,𝑌). Subtracting (5.15) from (5.16), we have 𝑥_𝑛(𝑡)−𝑥_∗(𝑡)_𝑋=𝐻𝑔⋅,𝑥_𝑛(⋅)𝑢_𝑛(⋅)+ℎ⋅,𝑥_𝑛(⋅)(𝑡)−𝐻𝑔⋅,𝑥_∗(⋅)𝑢_∗(⋅)−ℎ⋅,𝑥_∗(⋅)(𝑡)_𝑋≤𝐻𝑔⋅,𝑥_∗(⋅)+𝑦_𝑛(⋅)𝑣_𝑛(⋅)+ℎ⋅,𝑥_∗(⋅)+𝑦_𝑛(⋅)−𝑔⋅,𝑥_∗(⋅)𝑢_∗(⋅)−ℎ⋅,𝑥_∗(⋅)(𝑡)_𝑋⋯denotedby𝐷₁+𝐻𝑔⋅,𝑥_𝑛(⋅)𝑢_𝑛(⋅)+ℎ⋅,𝑥_𝑛(⋅)−𝑔⋅,𝑥_∗(⋅)+𝑦_𝑛(⋅)𝑣_𝑛(⋅)−ℎ⋅,𝑥_∗(⋅)+𝑦_𝑛(⋅)(𝑡)_𝑋⋯denotedby𝐷₂. (5.17) Here the linear operator 𝐻 is defined by (3.29). In virtue of (5.1), we have 𝑥_∗+𝑦_𝑛→𝑥_∗ in 𝐿^1/𝛽(𝐽,𝑋). Since 𝑣_𝑛→𝑢_∗ in 𝜔−𝐿^1/𝛽(𝐽,𝑌) (see (5.3)), from Lemma 3.2 we have that 𝒜(𝑥_∗+𝑦_𝑛,𝑣_𝑛)→𝒜(𝑥_∗,𝑢_∗) in 𝜔−𝐿^1/𝛽(𝐽,𝑋). By the property of the operator 𝐻 in Lemma 3.5, we obtain 𝐷₁→0forevery𝑡∈𝐽. (5.18) Due to (5.14), we have 𝐷₂≤𝛼𝑀_𝐴Γ(1+𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑙_𝐿+1(𝑠)𝑥_∗(𝑠)+𝑦_𝑛(𝑠)−𝑥_𝑛(𝑠)_𝑋+1𝑛≤𝑑𝑠𝛼𝑀_𝐴𝑙_{𝐿+1𝐿∞(𝐽)}Γ(1+𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑥_∗(𝑠)+𝑦_𝑛(𝑠)−𝑥_𝑛(𝑠)_𝑋+𝑑𝑠𝛼𝑀_𝐴𝑏^𝛼().𝑛𝛼Γ1+𝛼 (5.19) Note that ‖𝑥_∗(𝑡)‖_𝑋≤𝐿,‖𝑥_𝑛(𝑡)‖_𝑋≤𝐿 for any 𝑛≥1, 𝑡∈𝐽. Combining (5.18), (5.19) with (5.17), let 𝑛→∞, we get 𝑥(𝑡)−𝑥_∗(𝑡)_𝑋≤𝛼𝑀_𝐴𝑙_{𝐿+1𝐿∞(𝐽)}Γ(1+𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑄_𝛼(𝑡−𝑠)𝑥_∗(𝑠)−𝑥(𝑠)_𝑋𝑑𝑠. (5.20) This together with Lemma 2.8 implies that 𝑥_∗=𝑥. Hence we have that the sequence 𝑥_𝑛∈𝒯𝑟_𝑈, 𝑛≥1, converges to 𝑥_∗∈𝒯𝑟_𝑉 in 𝐶(𝐽,𝑋). From Theorem 4.1, we know that 𝒯𝑟_𝑉 is compact in 𝐶(𝐽,𝑋). Now it follows from 𝒯𝑟_𝑈⊆𝒯𝑟_𝑉 and the proof above that the relation (5.10) holds. The proof is complete.

Acknowledgment

This project is supported by Tianyuan foundation 11226096.