On the Continuity Properties of the Attainable Sets of Nonlinear Control Systems with Integral Constraint on Controls

Abstract

The attainable sets of the nonlinear control systems with integral constraint on the control functions are considered. It is assumed that the behavior of control system is described by differential equation which is nonlinear with respect to phase-state vector and control vector. The admissible control functions are chosen from the closed ball centered at the origin with radius 𝜇₀ in 𝐿_𝑝([𝑡₀,𝜃];ℝ^𝑚)(𝑝∈(1,+∞)). Precompactness of the solutions set is specified, and dependence of the attainable sets on the initial conditions and on the parameters of the control system is studied.

1. Introduction

Control problems with integral constraints on control arise in various problems of mathematical modeling. For example, the motion of flying apparatus with variable mass is described in the form of controllable system, where the control function has integral constraints (see, e.g., [1–3]). One of the important constructions of the control systems theory is the attainable set notion. Attainable set is the set of all points to which the system can be steered at the instant of given time. Attainable sets of control systems are very useful tools in the study of various problems of optimization, dynamical systems and differential game theory.

In [4–10], topological properties and numerical construction methods of the attainable sets of linear control systems with integral constraint on control functions are investigated. The attainable sets of affine control systems, that is, the attainable sets of control systems which are nonlinear with respect to the phase-state vector, but are linear with respect to the control vector have been considered in [11–14]. The properties of the attainable sets of the nonlinear control systems have been studied in [15–18].

Approximation method for the construction of attainable sets of affine control systems with integral constraints on the control is given in [11, 13]. In [14], using the topological properties of attainable sets of affine control systems, the continuity properties of minimum time and minimum energy functions are discussed.

The dependence of the attainable set on 𝑝 is studied in [8, 12, 15]. In [15], it is proved that attainable set of affine control system depends on 𝑝 continuously. In [15], the same property is shown for nonlinear control systems.

In [17], if the control resource is sufficiently small, then under some suitable assumptions on the right-hand side of the system, it is proved that the attainable set of the nonlinear control system with integral constraints on control is convex.

The value function of nonlinear optimal control problem with generalized integral constraints on control and phase-state vectors is investigated in [16, 18].

In this article, we consider the attainable sets of the control systems the behavior of which is described by nonlinear differential equations. It is assumed that the admissible control functions are chosen from the closed ball centered at the origin with radius 𝜇₀ in 𝐿_𝑝𝑡⁰,𝜃;ℝ^𝑚𝑝∈(1,+∞).

In Section 2, it is illustrated that, in general, the attainable set is not closed (Example 2.5) and it is shown that the set of solutions generated by all possible admissible control functions is precompact in the space of continuous functions (Corollary 2.4). In Section 3, the diameter of the attainable set is evaluated (Proposition 3.1) and it is proved that the attainable set is Hölder continuous with respect to time variable (Proposition 3.3). In Section 4, it is shown that the attainable set of the control system is continuous with respect to initial condition (Proposition 4.1). In Section 5, it is proved that the attainable set is Lipschitz continuous with respect to a parameter of the system which define the resource of the control effort (Proposition 5.1).

Consider the control system the behavior of which is described by the differential equation 𝑥^⋅(𝑡)=𝑓𝑡,𝑥(𝑡),𝑢(𝑡),𝑥(𝑡₀)∈𝑋₀,(1.1) where 𝑥∈ℝ^𝑛 is the phase-state vector of the system, 𝑢∈ℝ^𝑚 is the control vector, 𝑡∈[𝑡₀,𝜃] is the time, and 𝑋₀⊂ℝ^𝑛 is a compact set.

For 𝑝∈(1,∞) and 𝜇₀>0, we set 𝑈^𝜇0_𝑝=𝑢(⋅)∈𝐿_𝑝𝑡⁰,𝜃,ℝ^𝑚∶‖‖𝑢(⋅)‖‖_𝑝≤𝜇₀,(1.2) where ‖𝑢(⋅)‖_𝑝=(∫^𝜃𝑡₀‖𝑢(𝑡)‖^𝑝𝑑𝑡)^1/𝑝 and ‖⋅‖ denotes the Euclidian norm.

A function 𝑢(⋅)∈𝑈^𝜇0_𝑝 is said to be an admissible control function. It is obvious that the set of all admissible control functions 𝑈^𝜇0_𝑝 is the closed ball centered at the origin with the radius 𝜇₀ in 𝐿_𝑝([𝑡₀,𝜃];ℝ^𝑚).

It is assumed that the right-hand side of the system (1.1) satisfies the following conditions.

(a)The function 𝑓(⋅)∶[𝑡₀,𝜃]×ℝ^𝑛×ℝ^𝑚→ℝ^𝑛 is continuous.(b)For any bounded set 𝐷⊂[𝑡₀,𝜃]×ℝ^𝑛, there exist constants 𝐿₁=𝐿₁(𝐷)>0, 𝐿₂=𝐿₂(𝐷)>0, and 𝐿₃=𝐿₃(𝐷)>0 such that ‖‖𝑓𝑡,𝑥₁,𝑢₁−𝑓𝑡,𝑥²,𝑢₂‖‖≤𝐿¹+𝐿₂‖‖𝑢₂‖‖‖‖𝑥₁−𝑥₂‖‖+𝐿₃‖‖𝑢₁−𝑢₂‖‖(1.3) for any (𝑡,𝑥₁)∈𝐷, (𝑡,𝑥₂)∈𝐷, 𝑢₁∈ℝ^𝑚, and 𝑢₂∈ℝ^𝑚.(c) There exists a constant 𝑐>0 such that ‖‖𝑓(𝑡,𝑥,𝑢)‖‖≤𝑐1+‖‖𝑥‖‖1+‖‖𝑢‖‖(1.4) for every (𝑡,𝑥,𝑢)∈[𝑡₀,𝜃]×ℝ^𝑛×ℝ^𝑚.

If the right-hand side of the system (1.1) is affine, that is, if 𝑓(𝑡,𝑥,𝑢)=𝜑(𝑡,𝑥)+𝐵(𝑡,𝑥)𝑢 and the functions 𝜑(⋅)∶[𝑡₀,𝜃]×ℝ^𝑛→ℝ^𝑛,𝐵(⋅)∶[𝑡₀,𝜃]×ℝ^𝑛→ℝ^𝑚 satisfy the assumptions given in [11–14], then, under these assumptions, the conditions (a), (b), and (c) are also fulfilled.

Let 𝑢_∗(⋅)∈𝑈^𝜇0_𝑝. The absolutely continuous function 𝑥_∗(⋅)∶[𝑡₀,𝜃]→ℝ^𝑛, which satisfies the equation 𝑥̇_∗(𝑡)=𝑓(𝑡,𝑥_∗(𝑡),𝑢_∗(𝑡)) a.e. in [𝑡₀,𝜃], and the initial condition 𝑥_∗(𝑡₀)=𝑥₀∈𝑋₀ is said to be a solution of the system (1.1) with initial condition 𝑥_∗(𝑡₀)=𝑥₀, generated by the admissible control function 𝑢_∗(⋅). By the symbol 𝑥(⋅;𝑡₀,𝑥₀,𝑢(⋅)), we denote the solution of the system (1.1) with initial condition 𝑥(𝑡₀)=𝑥₀, which is generated by the admissible control function 𝑢(⋅). Note that the conditions (a)–(c) guarantee the existence, uniqueness, and extendability of the solutions up to the instant of time 𝜃 for every given 𝑢_∗(⋅)∈𝑈^𝜇0_𝑝 and 𝑥₀∈𝑋₀.

Let us define

the sets 𝑋𝑋_𝑝𝑝𝑡⁰,𝑋₀,𝜇₀=𝑥⋅;𝑡⁰,𝑥₀,𝑢(⋅)∶[𝑡₀,𝜃]⟶ℝ^𝑛∣𝑥₀∈𝑋₀,𝑢(⋅)∈𝑈^𝜇0_𝑝,𝑡;𝑡⁰,𝑋₀,𝜇₀=𝑥(𝑡)∈ℝ^𝑛∶𝑥(⋅)∈𝑋_𝑝𝑡⁰,𝑋₀,𝜇₀,(1.5) where 𝑡∈[𝑡₀,𝜃].

The set 𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀) is called the attainable set of the system (1.1) at the instant of time 𝑡. It is obvious that the set 𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀) consists of all 𝑥∈ℝ^𝑛 to which the system (1.1) can be steered at the instant of time 𝑡∈[𝑡₀,𝜃].

The Hausdorff distance between the sets 𝐴⊂ℝ^𝑛 and 𝐸⊂ℝ^𝑛 is denoted by ℎ(𝐴,𝐸) and is defined as ℎ𝐴,𝐸=max{sup_𝑥∈𝐴𝑑(𝑥,𝐸),sup_𝑦∈𝐸𝑑(𝑦,𝐴)},(1.6) where 𝑑(𝑥,𝐸)=inf{‖𝑥−𝑦‖∶𝑦∈𝐸}.

By 𝐶([𝑡₀,𝜃];ℝ^𝑛), we denote the space of continuous functions 𝑥(⋅)∶[𝑡₀,𝜃]→ℝ^𝑛 with norm ‖‖𝑥(⋅)‖‖_𝐶=max_{𝑡∈[𝑡0,𝜃]}‖‖𝑥(𝑡)‖‖.(1.7)

Also, ℎ_𝐶(𝑈,𝑉) denotes the Hausdorff distance between the sets 𝑈⊂𝐶([𝑡₀,𝜃];ℝ^𝑛) and 𝑉⊂𝐶([𝑡₀,𝜃];ℝ^𝑛).

2. Precompactness of the Set of Solutions

The following proposition asserts that the set of solutions and the attainable sets of the control system (1.1) with constraint (1.2) are bounded.

Proposition 2.1. Let 𝑝∈(1,∞), 𝜇_∗∈(0,𝜇₀+1), ℎ(𝑋₀,𝑋_∗)≤1. Then for any 𝑥_∗(⋅)∈𝑋_𝑝(𝑡₀,𝑋_∗,𝜇_∗), the inequality ‖‖𝑥_∗(⋅)‖‖_𝐶≤𝑟_∗(2.1) holds, where 𝑑𝑟_∗=𝑑₁exp(𝑘),(2.2)₁𝑑=1+𝑑_∗+𝑘,_∗𝑘==𝑐max𝜃−‖‖𝑥𝑡₀‖‖∶𝑥∈𝑋₀,𝑙+𝑙^∗𝜇⁰+1,(2.3)_∗=max𝜃−𝑡₀,1,(2.4)𝑐>0𝛾>0
and 𝐵𝐷𝛾=(𝑡,𝑥)∈𝑡₀,𝜃×ℝ^𝑛∶‖‖𝑥‖‖≤𝛾_𝑛,(𝛾)=𝑥∈ℝ^𝑛∶‖‖𝑥‖‖≤𝛾.(2.5) is the constant given in condition (c).

The proof of the proposition follows from condition (c) and Gronwall's inequality.

For given (𝑡,𝑥(𝑡))∈𝐷(𝑟_∗), we set 𝑝∈(1,∞),

We get from Proposition 2.1 that 𝑥(⋅)∈𝑋_𝑝(𝑡₀, for every 𝑋_∗,𝜇_∗),𝑡∈[𝑡₀,𝜃],𝜇_∗∈(0,𝜇₀+1)𝑋_∗⊂ℝ^𝑛ℎ(𝑋₀,𝑋_∗)≤1., and compact 𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) such that 𝑋_𝑝(𝑡;𝑡₀,𝑋₀, So, we have the validity of the following corollary.

Corollary 2.2. The set 𝜇₀)⊂𝐵_𝑛(𝑟_∗) is uniformly bounded, and consequently 𝑡∈[𝑡₀,𝜃]𝑟_∗ for every 𝐷(𝑟_∗), where 𝐷 is defined by (2.2).

Here and henceforth, we will have in mind the cylinder 𝑘_∗=𝑐1+𝑟_∗𝑙^∗+𝜇₀,(2.6) as the set 𝑟_∗ in condition (b). We set also 𝑙_∗ where 𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) is defined by (2.2), 𝜀>0 is defined by (2.4).

Proposition 2.3. The set 𝑥(⋅)∈𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) is equicontinuous.

Proof. Let 𝑡₁,𝑡₂∈[𝑡₀,𝜃]. be an arbitrarily given number. Now, let us choose an arbitrary 𝑡₁≤𝑡₂ and ‖‖𝑥𝑡₁−𝑥𝑡²‖‖≤∫^𝑡2𝑡₁𝑐1+‖‖𝑥(𝜏)‖‖1+‖‖𝑢(𝜏)‖‖𝑑𝜏.(2.7) Without loss of generality, we assume that ‖𝑥(⋅)‖_𝐶≤𝑟_∗. Then from condition (c), we have 𝑟_∗
According to Proposition 2.1, ‖‖𝑥(𝑡₁−𝑥(𝑡²‖‖≤𝑐1+𝑟^∗|||𝑡²−𝑡₁|||+𝜇₀|||𝑡₂−𝑡₁|||^{(𝑝−1)/𝑝}≤|||𝑡₂−𝑡₁|||^{(𝑝−1)/𝑝}≤|||𝑡₂𝑐1+𝑟_∗𝜃−𝑡⁰^1/𝑝+𝜇₀−𝑡₁|||^{(𝑝−1)/𝑝}𝑐1+𝑟_∗𝑙^∗+𝜇₀=𝑘^∗|||𝑡₂−𝑡₁|||^{(𝑝−1)/𝑝}.(2.8), where 𝜀>0 is defined by (2.2). Then we get from (2.4), (2.6), (2.7), and Hölder's inequality that 𝛿(𝜀)=(𝜀/𝑘_∗)^{𝑝/(𝑝−1)}
Thus for given ‖𝑥(𝑡₁)−𝑥(𝑡₂)‖<𝜀, setting |𝑡₁−𝑡₂|<𝛿(𝜀)., we obtain 𝑥(⋅)∈𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) for 𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) Since 𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) is arbitrarily chosen, the equicontinuity of the set 𝐶([𝑡₀,𝜃],ℝ^𝑛) follows.

From Corollary 2.2 and Proposition 2.3, we get the validity of the following corollary.

Corollary 2.4. The set 𝑢, is a precompact subset of the space 𝑈^𝜇0_𝑝.

Note that if the right-hand side of the system (1.1) is affine with respect to the control vector 𝑥𝑦̇̇==𝑢−,𝑦²+𝑦𝑢(²0,)=0𝑥,(0)=0,(2.9) then the weak compactness of the set of admissible control functions (𝑥,𝑦)∈ℝ² guaranties the closeness of the attainable sets; but the attainable sets of the control system (1.1) with constraint (1.2), in general, are not closed. In [19, 20], the example is given which illustrates that the attainable set of nonlinear control system with geometric constraint on control is not closed. We use that example to show that the attainable set of nonlinear control system with integral constraint on control is not also closed.

Example 2.5. Let us consider the control system 𝑢∈ℝ where 𝑡∈[0,1]. is the phase-state vector of the system, 𝜇₀=1 is the control vector, 𝑢(⋅)∈𝐿₂([0,1];ℝ) It is assumed that ∫¹⁰𝑢²(𝑡)𝑑𝑡≤1,(2.10) and the control function 𝑢(⋅)∈𝑈¹². of the system (2.9) satisfies the integral constraint 𝑋𝑋₂₂0,(0,0),1=𝑥⋅;0,(0,0),𝑢(⋅),𝑦⋅;0,(0,0),𝑢(⋅)∶𝑢(⋅)∈𝑈¹²,𝑡;0,(0,0),1=𝑥(𝑡),𝑦(𝑡)∈ℝ²∶𝑥(⋅),𝑦(⋅)∈𝑋₂0,(0,0),1.(2.11) that is, 𝑋₂(0,(0,0),1) Let us denote 𝑋₂(𝑡;0,(0,0),1)
Thus 𝑡∈[0,1], is the set of solutions, 𝑢(⋅)∈𝑈¹². is the attainable set of the control system (2.9) at the instant of time 𝑋₂(0,(0,0),1) generated by control functions (𝑥(⋅),𝑦(⋅))∈𝑋₂(0,(0,0),1)
Now, let us prove that the solution set 𝑢(⋅)∈𝑈¹² is bounded. Let 𝑥(𝑡)=^𝑡0𝑦−(𝑡)𝑦²=(𝜏)_𝑡0𝑑𝜏+^𝑡0𝑢²(𝜏)𝑑𝜏,(2.12)𝑢(𝜏)𝑑𝜏(2.13) be an arbitrarily chosen solution of the system (2.9) with integral constraint (2.10). Then there exists 𝑡∈[0,1] such that |||𝑦(𝑡)|||≤∫^𝑡0|||𝑢(𝜏)|||𝑑𝜏≤∫^𝑡01²𝑑𝜏^1/2∫^𝑡0|||𝑢(𝜏)|||²𝑑𝜏^1/2≤√𝑡≤1(2.14)𝑡∈[0,1].for any |||𝑥(𝑡)|||≤∫^𝑡0|||𝑦(𝜏)|||²𝑑𝜏+∫^𝑡0|||𝑢(𝜏)|||²𝑑𝜏≤∫^𝑡0𝜏𝑑𝜏+1=1+𝑡²2≤32(2.15). From (2.10), (2.13), and Hölder's inequality, the inequality 𝑡∈[0,1] holds for all ‖‖(𝑥(⋅),𝑦(⋅))‖‖_𝐶=max_𝑡∈[0,1]‖‖(𝑥(𝑡),𝑦(𝑡))‖‖=max_𝑡∈[0,1]√𝑥²(𝑡)+𝑦²(𝑡)≤1+94<2.(2.16) Then we get from (2.10), (2.12), and (2.14) that (𝑥(⋅),𝑦(⋅))∈𝑋₂(0,(0,0),1) for all 𝑋₂(0,(0,0),1).
However, (2.14) and (2.15) imply that (1,0)∉𝑋₂(1;0,(0,0),1) Since (1,0)∈𝑋₂(1;0,(0,0),1) is arbitrarily chosen, we get that the set (𝑥_∗(⋅),𝑦_∗(⋅))∈𝑋₂(0,(0,0),1) is bounded.
Now we prove that 𝑥_∗(1)=1,𝑦_∗(1)=0.(2.17). Let us assume the contrary, that is, let (𝑥_∗(⋅),𝑦_∗(⋅))∈𝑋₂(0,(0,0),1). Then there exists 𝑢_∗(⋅)∈𝑈¹² such that 𝑥_∗(𝑡)=𝑦^𝑡0−𝑦^2∗(𝜏)𝑑𝜏+^𝑡0𝑢^2∗(𝜏)𝑑𝜏,(2.18)_∗(𝑡)=^𝑡0𝑢_∗(𝜏)𝑑𝜏(2.19)
Since 𝑡∈[0,1]., then there exists 𝑥_∗(1)=∫¹⁰𝑢^2∗(𝜏)𝑑𝜏−∫¹⁰∫^𝜏0𝑢_∗(𝑠)𝑑𝑠²𝑑𝜏=1.(2.20) such that 𝑢_∗(⋅)∈𝑈¹²𝑢_∗(𝑡)=0for all 𝑡∈[0,1].
From (2.17), (2.18), and (2.19), it follows that 𝑥_∗(𝑡)=0,
Since 𝑦_∗(𝑡)=0, then it follows from (2.20) that 𝑡∈[0,1], for almost all 1,0∉𝑋₂1;0,0,0,1.(2.21) Then we have from (2.18) and (2.19) that (1,0)∈cl(𝑋₂(1;0,(0,0)),1)Γ^𝑘={0,1/2𝑘,2/2𝑘,…,(2𝑘−1)/2𝑘,1} for every [0,1] which contradicts (2.17). Thus 𝑘=1,2,…
Let us show that {𝑢_𝑘(⋅)}^∞𝑘=1,.
Let 𝑢_𝑘(𝑡)=⎧⎪⎨⎪⎩1,𝑡∈2𝑖−1,𝑡∈22𝑖𝑘+,21𝑖2+𝑘1,2𝑘,2𝑖2+𝑘2,(2.22) be a uniform partition of the closed interval 𝑖=0,1,…,𝑘−1, where 𝑢_𝑘(⋅)∈𝑈¹². Now we define a sequence of functions 𝑘=1,2,… setting (𝑥_𝑘(⋅),𝑦_𝑘(⋅))∈𝑋₂(0,(0,0),1) where 𝑢_𝑘(⋅)∈𝑈¹².
It is obvious that 𝑥_𝑘(𝑡)=𝑦^𝑡0−𝑦^2𝑘(𝜏)𝑑𝜏+^𝑡0𝑢^2𝑘(𝜏)𝑑𝜏,(2.23)_𝑘(𝑡)=^𝑡0𝑢_𝑘(𝜏)𝑑𝜏(2.24) for all 𝑡∈[0,1].. Let 𝑦_𝑘(𝑡)=⎧⎪⎨⎪⎩𝑡−2𝑖−𝑡+2𝑘2,𝑖+2𝑡∈22𝑘𝑖,2𝑖2+𝑘12𝑘,𝑡∈2𝑖2+𝑘1,2𝑖2+𝑘2(2.25) be the solution of the system (2.9) generated by the admissible control function 𝑡∈[0,1]. Then it follows from (2.9) that 𝑖=0,1,…,𝑘−10≤𝑦_𝑘(𝑡)≤12𝑘(2.26)for every 𝑡∈[0,1]
We get from (2.22) and (2.24) that 0≤𝑦^2𝑘(𝑡)≤14𝑘²(2.27) for every 𝑡∈[0,1]. where 𝑢^2𝑘(𝑡)=1.
Then, from (2.25) we have 𝑡∈[0,1]. for every 1−14𝑘²≤𝑥^⋅_𝑘(𝑡)≤1(2.28), and consequently 𝑡∈[0,1] for every 1−41𝑘²𝑡≤𝑥^𝑘(𝑡)≤𝑡(2.29)
According to (2.22), 𝑡∈[0,1] for all 𝑘=1,2,… Then, from (2.23) and (2.27) we obtain that 1−41𝑘²≤𝑥^𝑘(1)≤1(2.30) for almost all 𝑘=1,2,…, and consequently 𝑥^𝑘(1),𝑦_𝑘(1)⟶(1,0)as𝑘⟶∞.(2.31) for every (𝑥_𝑘(1),𝑦_𝑘(1))∈𝑋₂(1;0,(0,0),1), where 𝑘=1,2,….
We conclude from the last inequality that (1,0)∈cl𝑋₂1;0,0,0,1.(2.32) for every 𝑋₂(1;0,(0,0),1).
It follows from (2.26) and (2.30) that 𝑡
Since 𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀) for every 𝑡→𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀), from (2.31) we obtain that 𝑡
However, (2.21) and (2.32) imply that 𝐴⊂ℝ^𝑛 is not a closed set.

3. Diameter of the Attainable Set and Continuity with Respect to diam(𝐴)

In this section we will give an upper estimation for the diameter of the attainable set diam𝐴=sup_{𝑥,𝑦∈𝐴}‖‖𝑥−𝑦‖‖.(3.1) and will show that the set-valued map 𝑋_𝑝(𝑡;𝑡₀,𝑋₀, is Hölder continuous with respect to 𝜇₀)..

We denote the diameter of a set 𝑝∈(1,+∞) by diam𝑋_𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀≤𝑑⁰+𝑟₁(𝑡,𝑝)exp𝑟₀(𝑡,𝑝)(3.2) and define it as 𝑡∈[𝑡₀,𝜃]

The following proposition characterizes the diameter of the attainable set 𝑟𝑑₀=diam𝑋₀,(3.3)₀(𝑡,𝑝𝑟)=𝐿₁𝑡−𝑡⁰+𝐿²𝑡−𝑡⁰^{(𝑝−1)/𝑝},(3.4)₁(𝑡,𝑝)=2𝐿₃𝜇₀𝑡−𝑡⁰^{(𝑝−1)/𝑝}.(3.5)𝑡∈[𝑡₀,𝜃]

Proposition 3.1. For every 𝑥₁(𝑡)∈𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀),, the inequality 𝑥₂(𝑡)∈𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀) holds for any 𝑥₁∈𝑋₀, where 𝑥₁(⋅)∈𝑋_𝑝(𝑡₀,𝑥₀,𝜇₀)𝑢₁(⋅)∈𝑈^𝜇0_𝑝,𝑥₂∈𝑋₀,𝑥₂(⋅)∈

Proof. Let 𝑋_𝑝(𝑡₀,𝑥₀,𝜇₀) and 𝑢₂(⋅)∈𝑈^𝜇0_𝑝𝑥𝑥₁(𝑡)=𝑥₁+^𝑡𝑡₀𝑓𝜏,𝑥₁(𝜏),𝑢₁(𝜏)𝑑𝜏,₂(𝑡)=𝑥₂+^𝑡𝑡₀𝑓𝜏,𝑥₂(𝜏),𝑢₂(𝜏)𝑑𝜏.(3.6) be arbitrarily chosen. Then there exist ‖𝑥₁−𝑥₂‖≤𝑑₀, ‖‖𝑥₁(𝑡)−𝑥₂(𝑡)‖‖≤𝑑₀+^𝑡𝑡₀𝐿¹+𝐿₂+‖‖𝑢₂(𝜏)‖‖‖‖𝑥₁(𝜏)−𝑥₂(𝜏)‖‖𝑑𝜏_𝑡𝑡0𝐿₃‖‖𝑢₁(𝜏)−𝑢₂(𝜏)‖‖𝑑𝜏.(3.7), 𝑢₁(⋅),𝑢₂(⋅)∈𝑈^𝜇0_𝑝𝐿₃∫^𝑡𝑡₀‖‖𝑢₁(𝜏)−𝑢₂(𝜏)‖‖𝑑𝜏≤2𝐿₃𝜇₀𝑡−𝑡⁰^{(𝑝−1)/𝑝}=𝑟₁(𝑡,𝑝),(3.8)𝑟₁(𝑡,𝑝), 𝑡∈[𝑡₀,𝜃] such that ‖‖𝑥₁(𝑡)−𝑥₂(𝑡)‖‖≤𝑑₀+𝑟₁(𝑡,𝑝)exp^𝑡𝑡₀𝐿¹+𝐿₂≤𝑑‖‖𝑢₂(𝜏)‖‖𝑑𝜏₀=𝑑₀+𝑟₁(𝑡,𝑝)exp𝐿₁𝑡−𝑡⁰+𝐿²𝑡−𝑡⁰^{(𝑝−1)/𝑝}𝜇₀+𝑟₁(𝑡,𝑝)exp𝑟₀(𝑡,𝑝),(3.9)
Since 𝑟₀(𝑡,𝑝), then from (3.6), and the condition (b), we get diam𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀)→diam𝑋₀
Since 𝑡→𝑡₀., then the Hölder and Minkowski inequalities imply that 𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀) where 𝑡. is defined by (3.5). Since 𝑡₁∈[𝑡₀,𝜃], is arbitrarily chosen, we obtain from (3.7), (3.8), and Gronwall's inequality that 𝑡₂∈[𝑡₀,𝜃]. where ℎ𝑋_𝑝𝑡¹;𝑡₀,𝑋₀,𝜇₀,𝑋^𝑝𝑡²;𝑡₀,𝑋₀,𝜇₀≤𝑘^∗|||𝑡₁−𝑡₂|||^{(𝑝−1)/𝑝},(3.10) is defined by (3.4).

Note that an estimation for diameter of the attainable set can be obtained from Propo-sition 2.1; but the estimation given by Proposition 3.1 is more precise.

Corollary 3.2. 𝑘_∗>0 as 𝑡₁<𝑡₂.

The following proposition asserts that the attainable set 𝑦₁∈𝑋_𝑝(𝑡₁;𝑡₀,𝑋₀,𝜇₀) is Hölder continuous with respect to 𝑥₀∈𝑋₀

Proposition 3.3. Let 𝑥₀(⋅)∈𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀)𝑢₀(⋅)∈𝑈^𝜇0_𝑝 Then 𝑦₁=𝑥₀𝑡¹=𝑥⁰+∫^𝑡1_𝑡0𝑓𝜏,𝑥₀(𝜏),𝑢₀(𝜏)𝑑𝜏.(3.11) where 𝑦₂=𝑥₀𝑡²=𝑥⁰+∫^𝑡2_𝑡0𝑓𝜏,𝑥₀(𝜏),𝑢₀(𝜏)𝑑𝜏.(3.12) is defined by (2.6).

Proof. Without loss of generality, let us assume that 𝑦₂∈𝑋_𝑝(𝑡₂;𝑡₀,𝑋₀,𝜇₀). Let ‖‖𝑦₁−𝑦₂‖‖≤^𝑡2_𝑡1𝑐1+‖‖𝑥₀(𝜏)‖‖1+‖‖𝑢₀(𝜏)‖‖𝑑𝜏≤𝑐1+𝑟_∗≤𝑐1+𝑟_∗^𝑡2_𝑡11+‖‖𝑢⁰(𝜏)‖‖𝑑𝜏|||𝑡²−𝑡₁|||+𝜇₀|||𝑡₂−𝑡₁|||^{(𝑝−1)/𝑝}≤𝑐1+𝑟_∗𝑙^∗+𝜇₀|||𝑡²−𝑡₁|||^{(𝑝−1)/𝑝}=𝑘^∗||𝑡₂−𝑡₁||^{(𝑝−1)/𝑝},(3.13) be arbitrarily chosen. Then there exist 𝑐>0, 𝑦₁∈𝑋_𝑝(𝑡₁,𝑡₀,𝑋₀,𝜇₀) and 𝑋_𝑝𝑡¹;𝑡₀,𝑋₀,𝜇₀⊂𝑋^𝑝𝑡²;𝑡₀,𝑋₀,𝜇₀+𝑘^∗|||𝑡₂−𝑡₁|||^{(𝑝−1)/𝑝}𝐵_𝑛1.(3.14) such that 𝑋_𝑝𝑡²;𝑡₀,𝑋₀,𝜇₀⊂𝑋^𝑝𝑡¹;𝑡₀,𝑋₀,𝜇₀+𝑘^∗|||𝑡₂−𝑡₁|||^{(𝑝−1)/𝑝}𝐵_𝑛1.(3.15) Let 𝑡→𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀)
It is obvious that 𝑡∈[𝑡₀,𝜃], From Proposition 2.1, relations (2.4), (2.6), (3.11), (3.12), and the condition (c), we have (𝑝−1)/𝑝 where 𝑡₀ is the constant given in condition (c).
Since 𝑋₀ is arbitrarily chosen, then (3.13) implies that (𝑡₀,𝑋₀)→
Analogously, it is possible to show that 𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀)
In fact, (3.14) and (3.15) yield the proof.

From Proposition 3.3, we obtain the following corollary.

Corollary 3.4. The set-valued map 𝜔_∗=𝐿₁+𝐿₂𝜇₀𝑙^∗,(4.1), 𝑙_∗ is 𝐿₁-Hölder continuous.

4. Dependence of the Attainable Sets on Parameters 𝐿₂ and 𝑡₁≥𝑡₀

The following proposition characterizes the continuity of the set-valued map 𝑋₀,𝑋₁⊂ℝ^𝑛ℎ𝑋_𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀,𝑋^𝑝𝑡;𝑡¹,𝑋₁,𝜇₀≤ℎ𝑋⁰,𝑋₁+𝑡¹−𝑡₀^{(𝑝−1)/𝑝}𝑘_∗exp𝜔^∗(4.2) in the Hausdorff metric.

Let us denote 𝑡∈[𝑡₁,𝜃] where 𝑘_∗ is defined by (2.4), 𝜔_∗ and 𝑡∈[𝑡₁,𝜃] are the constants given in condition (b).

Proposition 4.1. Let 𝑥₀(𝑡)∈𝑋_𝑝(𝑡;𝑡₀,𝑋₀,𝜇₀) and 𝑥₀(⋅)∈ be compact sets. Then the inequality 𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) holds for all 𝑥₀∈𝑋₀, where 𝑢₀(⋅)∈𝑈^𝜇0_𝑝 is defined by (2.6), 𝑥₀(𝑡)=𝑥₀+∫^𝑡𝑡₀𝑓𝜏,𝑥₀(𝜏),𝑢₀(𝜏)𝑑𝜏(4.3) is defined by (4.1).

Proof. Let us choose arbitrary 𝑥₁∈𝑋₁ and ‖𝑥₁−𝑥₀‖≤ℎ(𝑋₀,𝑋₁), where 𝑥₁(⋅)𝑢₀(⋅). Then there exist 𝑥₁(𝑡₁)=𝑥₁∈𝑋₁. and 𝑥₁(𝑡)=𝑥₁+∫^𝑡𝑡₁𝑓𝜏,𝑥₁(𝜏),𝑢₀(𝜏)𝑑𝜏(4.4) such that 𝑥₁(𝑡)∈𝑋_𝑝(𝑡;𝑡₁,𝑋₁,𝜇₀). holds. According to the definition of Hausdorff distance, there exists ‖‖𝑥₀(𝑡)−𝑥₁(𝑡)‖‖≤ℎ𝑋₀,𝑋₁++𝑐^𝑡1_𝑡01+‖‖𝑢⁰(𝜏)‖‖1+‖‖𝑥₀(𝜏)‖‖𝑑𝜏_𝑡𝑡1𝐿¹+𝐿₂‖‖𝑢₀(𝜏)‖‖‖‖𝑥₀(𝜏)−𝑥₁(𝜏)‖‖𝑑𝜏.(4.5) such that 𝑐∫^𝑡1_𝑡01+‖‖𝑢⁰(𝜏)‖‖1+‖‖𝑥₀(𝜏)‖‖𝑑𝜏≤𝑡₁−𝑡₀^{(𝑝−1)/𝑝}𝑘_∗,(4.6). Let 𝑘_∗ be a solution of the control system (1.1), generated by the admissible control function 𝑡∈[𝑡₁,𝜃] with initial condition ‖‖𝑥₀(𝑡)−𝑥₁(𝑡)‖‖≤ℎ𝑋₀,𝑋₁+𝑡¹−𝑡₀^{(𝑝−1)/𝑝}𝑘_∗exp^𝑡𝑡₁𝐿¹+𝐿₂≤ℎ𝑋‖‖𝑢₀(𝜏)‖‖𝑑𝜏₀,𝑋₁+𝑡¹−𝑡₀^{(𝑝−1)/𝑝}𝑘_∗exp(𝜔^∗).(4.7) Then 𝑋_𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀⊂𝑋^𝑝𝑡;𝑡¹,𝑋₁,𝜇₀+ℎ𝑋⁰,𝑋₁+𝑡¹−𝑡₀^{(𝑝−1)/𝑝}𝑘_∗exp𝜔^∗𝐵^𝑛1.(4.8) and 𝑋_𝑝𝑡;𝑡¹,𝑋₁,𝜇₀⊂𝑋^𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀+ℎ𝑋⁰,𝑋₁+𝑡¹−𝑡₀^{(𝑝−1)/𝑝}𝑘_∗exp𝜔^∗𝐵^𝑛1.(4.9)
From (4.3), (4.4), and conditions (b) and (c), we have ℎ𝑋_𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀,𝑋^𝑝𝑡;𝑡⁰,𝑋₁,𝜇₀≤ℎ𝑋⁰,𝑋₁exp𝜔^∗(4.10)
Proposition 2.1 implies that 𝑡∈[𝑡₀,𝜃] where 𝜔_∗>0 is defined by (2.6). Since ℎ_𝐶𝑋^𝑝𝑡⁰,𝑋₀,𝜇₀,𝑋^𝑝𝑡⁰,𝑋₁,𝜇₀≤ℎ𝑋⁰,𝑋₁exp𝜔^∗(4.11) is arbitrarily chosen, from (4.5), (4.6), and Gronwall's inequality, we get 𝑋₀⊂ℝ^𝑛
Hence, we obtain from (4.7) that 𝑋_𝑛⊂ℝ^𝑛
Similarly, one can prove that 𝑛=1,2,…
Finally, (4.8) and (4.9) complete the proof.

From Proposition 4.1, the validity of the following corollaries follow.

Corollary 4.2. The inequality ℎ(𝑋_𝑛,𝑋₀)→0 holds for all 𝑡_𝑛→𝑡₀+0, where 𝑛→∞ is defined by (4.1).

Corollary 4.3. The inequality 𝑡∈(𝑡₀,𝜃], holds.

Corollary 4.4. Let ℎ𝑋_𝑝𝑡;𝑡^𝑛,𝑋_𝑛,𝜇₀,𝑋^𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀⟶0as𝑛⟶∞.(4.12) and 𝜇₀ be compact sets for all 𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀). Assume that 𝜇₀. and 𝑟₁=𝐿₃𝑙_∗exp𝜔_∗,(5.1) as 𝜔_∗. Then for all 𝑋_𝑝(𝑡₀,𝑋₀,𝜇_∗)

5. Dependence of the Attainable Sets on 𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀)

In this section we specify dependence of the set ℎ_𝐶(𝑋_𝑝(𝑡₀,𝑋₀,𝜇_∗),𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀))≤𝑟₁||𝜇_∗−𝜇₀||(5.2) on the constraint parameter 𝑟₁ Let 𝑥₀(⋅)∈𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) where 𝑥₀∈𝑋₀ is defined by (4.1).

The following proposition characterizes the relation between the solutions sets 𝑢₀(⋅)∈𝑈^𝜇0_𝑝𝑥₀(𝑡)=𝑥₀+∫^𝑡𝑡₀𝑓𝜏,𝑥₀(𝜏),𝑢₀(𝜏)𝑑𝜏(5.3) and 𝑡∈[𝑡₀,𝜃]..

Proposition 5.1. The inequality 𝑢_∗(⋅)∶[𝑡₀,𝜃]→ℝ^𝑚 is satisfied, where 𝑢_∗(𝑡)=𝜇^∗𝜇₀𝑢₀(𝑡),𝑡∈𝑡₀,𝜃.(5.4) is defined by (5.1).

Proof. Let 𝑢_∗(⋅)∈𝑈^𝜇∗_𝑝 be an arbitrarily chosen solution. Then there exist 𝑥_∗(⋅) and 𝑢_∗(⋅)∈𝑈^𝜇∗_𝑝 such that (𝑡₀,𝑥₀) for every 𝑥_∗(⋅)∈𝑋_𝑝(𝑡₀,𝑋₀,
We define a new control function 𝜇_∗), setting 𝑥_∗(𝑡)=𝑥₀+∫^𝑡𝑡₀𝑓𝜏,𝑥_∗(𝜏),𝑢_∗(𝜏)𝑑𝜏(5.5)
It is not difficult to verify that 𝑡∈[𝑡₀,𝜃]. Let ‖‖𝑥_∗(𝑡)−𝑥₀(𝑡)‖‖≤𝐿₃|||𝜇_∗𝜇₀≤𝐿₃−1|||^𝑡𝑡₀‖‖𝑢₀(𝜏)‖‖𝑑𝜏+^𝑡𝑡₀𝐿¹+𝐿₂‖‖𝑢₀(𝜏)‖‖‖‖𝑥_∗(𝜏)−𝑥₀(𝜏)‖‖𝑑𝜏𝑙_∗|||𝜇_∗−𝜇₀|||+^𝑡𝑡₀𝐿¹+𝐿₂‖‖𝑢₀(𝜏)‖‖‖‖𝑥_∗(𝜏)−𝑥₀(𝜏)‖‖𝑑𝜏(5.6) be a solution of the control system (1.1), generated by 𝑡∈[𝑡₀,𝜃], from the initial point 𝑙_∗. Then ‖‖𝑥_∗(𝑡)−𝑥₀(𝑡)‖‖≤𝐿₃𝑙_∗|||𝜇_∗−𝜇₀|||exp𝐿₁(𝜃−𝑡₀≤𝐿₃+𝐿²𝜇₀𝜃−𝑡⁰^{(𝑝−1)/𝑝}𝑙_∗exp𝜔_∗|||𝜇^∗−𝜇₀|||=𝑟₁|||𝜇_∗−𝜇₀|||(5.7)𝑡∈[𝑡₀,𝜃] and 𝑥₀(⋅)∈𝑋_𝑝(𝑡₀,𝑋₀,𝜇₀) for every 𝑥₁(⋅)∈𝑋_𝑝(𝑡₀,𝑋₀,𝜇_∗). From (5.3), (5.4), (5.5), and condition (b), we get ‖‖𝑥₀(⋅)−𝑥_∗(⋅)‖‖_𝐶≤𝑟₁|||𝜇_∗−𝜇₀|||,(5.8) for every 𝑋_𝑝𝑡⁰,𝑋₀,𝜇₀⊂𝑋^𝑝𝑡⁰,𝑋₀,𝜇_∗+𝑟¹|||𝜇_∗−𝜇₀|||𝐵_𝐶,(5.9) where 𝐵_𝐶 is defined by (2.4).
The Gronwall inequality, (5.1), and (5.6) yield that 𝐶([𝑡₀,𝜃],ℝ^𝑛) for all 𝑋_𝑝𝑡⁰,𝑋₀,𝜇_∗⊂𝑋^𝑝𝑡⁰,𝑋₀,𝜇₀+𝑟¹|||𝜇_∗−𝜇₀|||𝐵_𝐶.(5.10).
Thus from (5.7) we get that for any fixed ℎ𝑋_𝑝𝑡;𝑡⁰,𝑋₀,𝜇_∗,𝑋^𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀≤𝑟¹|||𝜇_∗−𝜇₀|||(5.11) there exists 𝑡∈[𝑡₀,𝜃], such that 𝑟₁>0 and consequently 𝜇_𝑛→𝜇₀ where 𝑛→∞ is the closed unit ball centered at the origin in the space ℎℎ^𝐶𝑋𝑋^𝑝𝑡⁰,𝑋₀,𝜇_𝑛,𝑋^𝑝𝑡⁰,𝑋₀,𝜇₀⟶0as𝑛⟶∞,_𝑝𝑡;𝑡⁰,𝑋₀,𝜇_𝑛,𝑋^𝑝𝑡;𝑡⁰,𝑋₀,𝜇₀⟶0as𝑛⟶∞(5.12).
Analogously, it is possible to prove that 𝑡∈[𝑡₀,𝜃]. Hence, from (5.9) and (5.10), we obtain the proof of the proposition.

From Proposition 5.1, it follows that the following corollaries are satisfied.

Corollary 5.2. The inequality 𝑝 is satisfied for any 𝐿^𝑝 where 𝑝≥1 is defined by (5.1).

Corollary 5.3. Let 𝐿_𝑝 as 𝐿₂. Then for every

Acknowledgments

This research was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) Project no. 106T012. The authors thank the referees for the careful reading of the manuscript and the helpful suggestions.