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Abstract and Applied Analysis
VolumeΒ 2008, Article IDΒ 295817, 14 pages
http://dx.doi.org/10.1155/2008/295817
Research Article

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

Department of Mathematics, Anadolu University, Eskisehir 26470, , Turkey

Received 11 June 2007; Revised 30 August 2007; Accepted 6 November 2007

Academic Editor: AgacikΒ Zafer

Copyright Β© 2008 Khalik G. Guseinov and Ali S. Nazlipinar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 πœ‡0 in 𝐿𝑝([𝑑0,πœƒ];β„π‘š)(π‘βˆˆ(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 πœ‡0 in 𝐿𝑝𝑑0ξ‚„,πœƒ;β„π‘šξ‚.ξ‚ξ‚€π‘βˆˆ(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 HΓΆlder 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 β‹…π‘₯(𝑑)=𝑓𝑑,π‘₯(𝑑),𝑒(𝑑),π‘₯(𝑑0)βˆˆπ‘‹0,(1.1) where π‘₯βˆˆβ„π‘› is the phase-state vector of the system, π‘’βˆˆβ„π‘š is the control vector, π‘‘βˆˆ[𝑑0,πœƒ] is the time, and 𝑋0βŠ‚β„π‘› is a compact set.

For π‘βˆˆ(1,∞) and πœ‡0>0, we set π‘ˆπœ‡0𝑝=𝑒(β‹…)βˆˆπΏπ‘π‘‘ξ‚€ξ‚ƒ0ξ‚„,πœƒ,β„π‘šξ‚βˆΆβ€–β€–β€–β€–π‘’(β‹…)π‘β‰€πœ‡0,(1.2) where ‖𝑒(β‹…)β€–π‘βˆ«=(πœƒπ‘‘0‖𝑒(𝑑)‖𝑝𝑑𝑑)1/𝑝 and β€–β‹…β€– denotes the Euclidian norm.

A function 𝑒(β‹…)βˆˆπ‘ˆπœ‡0𝑝 is said to be an admissible control function. It is obvious that the set of all admissible control functions π‘ˆπœ‡0𝑝 is the closed ball centered at the origin with the radius πœ‡0 in 𝐿𝑝([𝑑0,πœƒ];β„π‘š).

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

(a)The function 𝑓(β‹…)∢[𝑑0,πœƒ]Γ—β„π‘›Γ—β„π‘šβ†’β„π‘› is continuous.(b)For any bounded set π·βŠ‚[𝑑0,πœƒ]×ℝ𝑛, there exist constants 𝐿1=𝐿1(𝐷)>0, 𝐿2=𝐿2(𝐷)>0, and 𝐿3=𝐿3(𝐷)>0 such that ‖‖𝑓𝑑,π‘₯1,𝑒1ξ‚ξ‚€βˆ’π‘“π‘‘,π‘₯2,𝑒2‖‖≀𝐿1+𝐿2‖‖𝑒2‖‖‖‖π‘₯1βˆ’π‘₯2β€–β€–+𝐿3‖‖𝑒1βˆ’π‘’2β€–β€–(1.3) for any (𝑑,π‘₯1)∈𝐷, (𝑑,π‘₯2)∈𝐷, 𝑒1βˆˆβ„π‘š, and 𝑒2βˆˆβ„π‘š.(c) There exists a constant 𝑐>0 such that β€–β€–β€–β€–ξ‚€β€–β€–π‘₯‖‖‖‖𝑒‖‖𝑓(𝑑,π‘₯,𝑒)≀𝑐1+1+(1.4) for every (𝑑,π‘₯,𝑒)∈[𝑑0,πœƒ]Γ—β„π‘›Γ—β„π‘š.

If the right-hand side of the system (1.1) is affine, that is, if 𝑓(𝑑,π‘₯,𝑒)=πœ‘(𝑑,π‘₯)+𝐡(𝑑,π‘₯)𝑒 and the functions πœ‘(β‹…)∢[𝑑0,πœƒ]×ℝ𝑛→ℝ𝑛,𝐡(β‹…)∢[𝑑0,πœƒ]Γ—β„π‘›β†’β„π‘š satisfy the assumptions given in [11–14], then, under these assumptions, the conditions (a), (b), and (c) are also fulfilled.

Let π‘’βˆ—(β‹…)βˆˆπ‘ˆπœ‡0𝑝. The absolutely continuous function π‘₯βˆ—(β‹…)∢[𝑑0,πœƒ]→ℝ𝑛, which satisfies the equation Μ‡π‘₯βˆ—(𝑑)=𝑓(𝑑,π‘₯βˆ—(𝑑),π‘’βˆ—(𝑑)) a.e. in [𝑑0,πœƒ], and the initial condition π‘₯βˆ—(𝑑0)=π‘₯0βˆˆπ‘‹0 is said to be a solution of the system (1.1) with initial condition π‘₯βˆ—(𝑑0)=π‘₯0, generated by the admissible control function π‘’βˆ—(β‹…). By the symbol π‘₯(β‹…;𝑑0,π‘₯0,𝑒(β‹…)), we denote the solution of the system (1.1) with initial condition π‘₯(𝑑0)=π‘₯0, which is generated by the admissible control function 𝑒(β‹…). Note that the conditions (a)–(c) guarantee the existence, uniqueness, and extendability of the solutions up to the instant of time πœƒ for every given π‘’βˆ—(β‹…)βˆˆπ‘ˆπœ‡0𝑝 and π‘₯0βˆˆπ‘‹0.

Let us define

the sets 𝑋𝑝𝑑0,𝑋0,πœ‡0=π‘₯ξ‚€β‹…;𝑑0,π‘₯0,𝑒(β‹…)∢[𝑑0,πœƒ]βŸΆβ„π‘›βˆ£π‘₯0βˆˆπ‘‹0,𝑒(β‹…)βˆˆπ‘ˆπœ‡0𝑝,𝑋𝑝𝑑;𝑑0,𝑋0,πœ‡0=π‘₯(𝑑)βˆˆβ„π‘›βˆΆπ‘₯(β‹…)βˆˆπ‘‹π‘ξ‚€π‘‘0,𝑋0,πœ‡0,(1.5) where π‘‘βˆˆ[𝑑0,πœƒ].

The set 𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0) is called the attainable set of the system (1.1) at the instant of time 𝑑. It is obvious that the set 𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0) consists of all π‘₯βˆˆβ„π‘› to which the system (1.1) can be steered at the instant of time π‘‘βˆˆ[𝑑0,πœƒ].

The Hausdorff distance between the sets π΄βŠ‚β„π‘› and πΈβŠ‚β„π‘› is denoted by β„Ž(𝐴,𝐸) and is defined as β„Žξ€·ξ€Έπ΄,𝐸=max{supπ‘₯βˆˆπ΄π‘‘(π‘₯,𝐸),supπ‘¦βˆˆπΈπ‘‘(𝑦,𝐴)},(1.6) where 𝑑(π‘₯,𝐸)=inf{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘¦βˆˆπΈ}.

By 𝐢([𝑑0,πœƒ];ℝ𝑛), we denote the space of continuous functions π‘₯(β‹…)∢[𝑑0,πœƒ]→ℝ𝑛 with norm β€–β€–β€–β€–π‘₯(β‹…)𝐢=maxπ‘‘βˆˆ[𝑑0,πœƒ]β€–β€–β€–β€–π‘₯(𝑑).(1.7)

Also, β„ŽπΆ(π‘ˆ,𝑉) denotes the Hausdorff distance between the sets π‘ˆβŠ‚πΆ([𝑑0,πœƒ];ℝ𝑛) and π‘‰βŠ‚πΆ([𝑑0,πœƒ];ℝ𝑛).

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 π‘βˆˆ(1,∞), πœ‡βˆ—βˆˆ(0,πœ‡0+1), β„Ž(𝑋0,π‘‹βˆ—)≀1. Then for any π‘₯βˆ—(β‹…)βˆˆπ‘‹π‘(𝑑0,π‘‹βˆ—,πœ‡βˆ—), the inequality β€–β€–π‘₯βˆ—β€–β€–(β‹…)πΆβ‰€π‘Ÿβˆ—(2.1) holds, where π‘Ÿβˆ—=𝑑1𝑑exp(π‘˜),(2.2)1=1+π‘‘βˆ—π‘‘+π‘˜,βˆ—ξ‚†β€–β€–π‘₯β€–β€–=max∢π‘₯βˆˆπ‘‹0,π‘˜=π‘ξ‚ƒξ‚€πœƒβˆ’π‘‘0+π‘™βˆ—ξ‚€πœ‡0,𝑙+1(2.3)βˆ—=maxξ‚†ξ‚€πœƒβˆ’π‘‘0,1,(2.4)𝑐>0𝛾>0
and 𝐷𝛾=𝑑(𝑑,π‘₯)∈0ξ‚„,πœƒΓ—β„π‘›βˆΆβ€–β€–π‘₯‖‖,𝐡≀𝛾𝑛(𝛾)=π‘₯βˆˆβ„π‘›βˆΆβ€–β€–π‘₯‖‖.≀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 (𝑑,π‘₯(𝑑))∈𝐷(π‘Ÿβˆ—), we set π‘βˆˆ(1,∞),

We get from Proposition 2.1 that π‘₯(β‹…)βˆˆπ‘‹π‘(𝑑0, for every π‘‹βˆ—,πœ‡βˆ—),π‘‘βˆˆ[𝑑0,πœƒ],πœ‡βˆ—βˆˆ(0,πœ‡0+1)π‘‹βˆ—βŠ‚β„π‘›β„Ž(𝑋0,π‘‹βˆ—)≀1., and compact 𝑋𝑝(𝑑0,𝑋0,πœ‡0) such that 𝑋𝑝(𝑑;𝑑0,𝑋0, So, we have the validity of the following corollary.

Corollary 2.2. The set πœ‡0)βŠ‚π΅π‘›(π‘Ÿβˆ—) is uniformly bounded, and consequently π‘‘βˆˆ[𝑑0,πœƒ]π‘Ÿβˆ— for every 𝐷(π‘Ÿβˆ—), where 𝐷 is defined by (2.2).

Here and henceforth, we will have in mind the cylinder π‘˜βˆ—ξ‚€=𝑐1+π‘Ÿβˆ—π‘™ξ‚ξ‚€βˆ—+πœ‡0,(2.6) as the set π‘Ÿβˆ— in condition (b). We set also π‘™βˆ— where 𝑋𝑝(𝑑0,𝑋0,πœ‡0) is defined by (2.2), πœ€>0 is defined by (2.4).

Proposition 2.3. The set π‘₯(β‹…)βˆˆπ‘‹π‘(𝑑0,𝑋0,πœ‡0) is equicontinuous.

Proof. Let 𝑑1,𝑑2∈[𝑑0,πœƒ]. be an arbitrarily given number. Now, let us choose an arbitrary 𝑑1≀𝑑2 and β€–β€–π‘₯𝑑1ξ‚ξ‚€π‘‘βˆ’π‘₯2ξ‚β€–β€–β‰€βˆ«π‘‘2𝑑1𝑐‖‖‖‖‖‖‖‖1+π‘₯(𝜏)1+𝑒(𝜏)π‘‘πœ.(2.7) Without loss of generality, we assume that β€–π‘₯(β‹…)β€–πΆβ‰€π‘Ÿβˆ—. Then from condition (c), we have π‘Ÿβˆ—
According to Proposition 2.1, β€–β€–π‘₯(𝑑1ξ‚βˆ’π‘₯(𝑑2‖‖≀𝑐1+π‘Ÿβˆ—|||𝑑2βˆ’π‘‘1|||+πœ‡0|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝≀|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝𝑐1+π‘Ÿβˆ—ξ‚ξ‚€ξ‚€πœƒβˆ’π‘‘01/𝑝+πœ‡0≀|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝𝑐1+π‘Ÿβˆ—π‘™ξ‚ξ‚€βˆ—+πœ‡0=π‘˜βˆ—|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝.(2.8), where πœ€>0 is defined by (2.2). Then we get from (2.4), (2.6), (2.7), and HΓΆlder's inequality that 𝛿(πœ€)=(πœ€/π‘˜βˆ—)𝑝/(π‘βˆ’1)
Thus for given β€–π‘₯(𝑑1)βˆ’π‘₯(𝑑2)β€–<πœ€, setting |𝑑1βˆ’π‘‘2|<𝛿(πœ€)., we obtain π‘₯(β‹…)βˆˆπ‘‹π‘(𝑑0,𝑋0,πœ‡0) for 𝑋𝑝(𝑑0,𝑋0,πœ‡0) Since 𝑋𝑝(𝑑0,𝑋0,πœ‡0) is arbitrarily chosen, the equicontinuity of the set 𝐢([𝑑0,πœƒ],ℝ𝑛) follows.

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

Corollary 2.4. The set 𝑒, is a precompact subset of the space π‘ˆπœ‡0𝑝.

Note that if the right-hand side of the system (1.1) is affine with respect to the control vector Μ‡π‘₯=βˆ’π‘¦2+𝑒2,π‘₯(0)=0,̇𝑦=𝑒,𝑦(0)=0,(2.9) then the weak compactness of the set of admissible control functions (π‘₯,𝑦)βˆˆβ„2 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 π‘’βˆˆβ„ where π‘‘βˆˆ[0,1]. is the phase-state vector of the system, πœ‡0=1 is the control vector, 𝑒(β‹…)∈𝐿2([0,1];ℝ) It is assumed that ∫10𝑒2(𝑑)𝑑𝑑≀1,(2.10) and the control function 𝑒(β‹…)βˆˆπ‘ˆ12. of the system (2.9) satisfies the integral constraint 𝑋2=π‘₯0,(0,0),1⋅;0,(0,0),𝑒(β‹…),𝑦⋅;0,(0,0),𝑒(β‹…)ξ‚ξ‚βˆΆπ‘’(β‹…)βˆˆπ‘ˆ12,𝑋2=𝑑;0,(0,0),1π‘₯(𝑑),𝑦(𝑑)βˆˆβ„2βˆΆξ‚€ξ‚π‘₯(β‹…),𝑦(β‹…)βˆˆπ‘‹2ξ‚€.0,(0,0),1(2.11) that is, 𝑋2(0,(0,0),1) Let us denote 𝑋2(𝑑;0,(0,0),1)
Thus π‘‘βˆˆ[0,1], is the set of solutions, 𝑒(β‹…)βˆˆπ‘ˆ12. is the attainable set of the control system (2.9) at the instant of time 𝑋2(0,(0,0),1) generated by control functions (π‘₯(β‹…),𝑦(β‹…))βˆˆπ‘‹2(0,(0,0),1)
Now, let us prove that the solution set 𝑒(β‹…)βˆˆπ‘ˆ12 is bounded. Let ξ€œπ‘₯(𝑑)=𝑑0βˆ’π‘¦2ξ€œ(𝜏)π‘‘πœ+𝑑0𝑒2ξ€œ(𝜏)π‘‘πœ,(2.12)𝑦(𝑑)=𝑑0𝑒(𝜏)π‘‘πœ(2.13) be an arbitrarily chosen solution of the system (2.9) with integral constraint (2.10). Then there exists π‘‘βˆˆ[0,1] such that ||||||β‰€βˆ«π‘¦(𝑑)𝑑0||||||ξ‚€βˆ«π‘’(𝜏)π‘‘πœβ‰€π‘‘012ξ‚π‘‘πœ1/2ξ‚€βˆ«π‘‘0||||||𝑒(𝜏)2ξ‚π‘‘πœ1/2β‰€βˆšπ‘‘β‰€1(2.14)π‘‘βˆˆ[0,1].for any ||||||β‰€βˆ«π‘₯(𝑑)𝑑0||||||𝑦(𝜏)2βˆ«π‘‘πœ+𝑑0||||||𝑒(𝜏)2βˆ«π‘‘πœβ‰€π‘‘0π‘‘πœπ‘‘πœ+1=1+22≀32(2.15). From (2.10), (2.13), and HΓΆlder's inequality, the inequality π‘‘βˆˆ[0,1] holds for all β€–β€–β€–β€–(π‘₯(β‹…),𝑦(β‹…))𝐢=maxπ‘‘βˆˆ[0,1]β€–β€–β€–β€–(π‘₯(𝑑),𝑦(𝑑))=maxπ‘‘βˆˆ[0,1]√π‘₯2(𝑑)+𝑦2ξ‚™(𝑑)≀91+4<2.(2.16) Then we get from (2.10), (2.12), and (2.14) that (π‘₯(β‹…),𝑦(β‹…))βˆˆπ‘‹2(0,(0,0),1) for all 𝑋2(0,(0,0),1).
However, (2.14) and (2.15) imply that (1,0)βˆ‰π‘‹2(1;0,(0,0),1) Since (1,0)βˆˆπ‘‹2(1;0,(0,0),1) is arbitrarily chosen, we get that the set (π‘₯βˆ—(β‹…),π‘¦βˆ—(β‹…))βˆˆπ‘‹2(0,(0,0),1) is bounded.
Now we prove that π‘₯βˆ—(1)=1,π‘¦βˆ—(1)=0.(2.17). Let us assume the contrary, that is, let (π‘₯βˆ—(β‹…),π‘¦βˆ—(β‹…))βˆˆπ‘‹2(0,(0,0),1). Then there exists π‘’βˆ—(β‹…)βˆˆπ‘ˆ12 such that π‘₯βˆ—ξ€œ(𝑑)=𝑑0βˆ’π‘¦2βˆ—ξ€œ(𝜏)π‘‘πœ+𝑑0𝑒2βˆ—π‘¦(𝜏)π‘‘πœ,(2.18)βˆ—ξ€œ(𝑑)=𝑑0π‘’βˆ—(𝜏)π‘‘πœ(2.19)
Since π‘‘βˆˆ[0,1]., then there exists π‘₯βˆ—βˆ«(1)=10𝑒2βˆ—βˆ«(𝜏)π‘‘πœβˆ’10ξ‚€βˆ«πœ0π‘’βˆ—ξ‚(𝑠)𝑑𝑠2π‘‘πœ=1.(2.20) such that π‘’βˆ—(β‹…)βˆˆπ‘ˆ12π‘’βˆ—(𝑑)=0for all π‘‘βˆˆ[0,1].
From (2.17), (2.18), and (2.19), it follows that π‘₯βˆ—(𝑑)=0,
Since π‘¦βˆ—(𝑑)=0, then it follows from (2.20) that π‘‘βˆˆ[0,1], for almost all ξ€·ξ€Έ1,0βˆ‰π‘‹2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)Ξ“π‘˜={0,1/2π‘˜,2/2π‘˜,…,(2π‘˜βˆ’1)/2π‘˜,1} for every [0,1] which contradicts (2.17). Thus π‘˜=1,2,…
Let us show that {π‘’π‘˜(β‹…)}βˆžπ‘˜=1,.
Let π‘’π‘˜βŽ§βŽͺ⎨βŽͺβŽ©ξ‚ƒ(𝑑)=1,π‘‘βˆˆ2𝑖,2π‘˜2𝑖+1,2π‘˜βˆ’1,π‘‘βˆˆ2𝑖+1,2π‘˜2𝑖+2,2π‘˜(2.22) be a uniform partition of the closed interval 𝑖=0,1,…,π‘˜βˆ’1, where π‘’π‘˜(β‹…)βˆˆπ‘ˆ12. Now we define a sequence of functions π‘˜=1,2,… setting (π‘₯π‘˜(β‹…),π‘¦π‘˜(β‹…))βˆˆπ‘‹2(0,(0,0),1) where π‘’π‘˜(β‹…)βˆˆπ‘ˆ12.
It is obvious that π‘₯π‘˜ξ€œ(𝑑)=𝑑0βˆ’π‘¦2π‘˜ξ€œ(𝜏)π‘‘πœ+𝑑0𝑒2π‘˜π‘¦(𝜏)π‘‘πœ,(2.23)π‘˜ξ€œ(𝑑)=𝑑0π‘’π‘˜(𝜏)π‘‘πœ(2.24) for all π‘‘βˆˆ[0,1].. Let π‘¦π‘˜βŽ§βŽͺ⎨βŽͺ⎩(𝑑)=π‘‘βˆ’2𝑖2π‘˜,π‘‘βˆˆ2𝑖,2π‘˜2𝑖+12π‘˜βˆ’π‘‘+2𝑖+22π‘˜,π‘‘βˆˆ2𝑖+1,2π‘˜2𝑖+22π‘˜(2.25) be the solution of the system (2.9) generated by the admissible control function π‘‘βˆˆ[0,1]. Then it follows from (2.9) that 𝑖=0,1,…,π‘˜βˆ’10β‰€π‘¦π‘˜1(𝑑)≀2π‘˜(2.26)for every π‘‘βˆˆ[0,1]
We get from (2.22) and (2.24) that 0≀𝑦2π‘˜1(𝑑)≀4π‘˜2(2.27) for every π‘‘βˆˆ[0,1]. where 𝑒2π‘˜(𝑑)=1.
Then, from (2.25) we have π‘‘βˆˆ[0,1]. for every 11βˆ’4π‘˜2≀⋅π‘₯π‘˜(𝑑)≀1(2.28), and consequently π‘‘βˆˆ[0,1] for every ξ‚€11βˆ’4π‘˜2𝑑≀π‘₯π‘˜(𝑑)≀𝑑(2.29)
According to (2.22), π‘‘βˆˆ[0,1] for all π‘˜=1,2,… Then, from (2.23) and (2.27) we obtain that ξ‚€11βˆ’4π‘˜2≀π‘₯π‘˜(1)≀1(2.30) for almost all π‘˜=1,2,…, and consequently ξ‚€π‘₯π‘˜(1),π‘¦π‘˜ξ‚(1)⟢(1,0)asπ‘˜βŸΆβˆž.(2.31) for every (π‘₯π‘˜(1),π‘¦π‘˜(1))βˆˆπ‘‹2(1;0,(0,0),1), where π‘˜=1,2,….
We conclude from the last inequality that 𝑋(1,0)∈cl2ξ‚€ξ€·ξ€Έ1;0,0,0,1.(2.32) for every 𝑋2(1;0,(0,0),1).
It follows from (2.26) and (2.30) that 𝑑
Since 𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0) for every 𝑑→𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0), from (2.31) we obtain that 𝑑
However, (2.21) and (2.32) imply that π΄βŠ‚β„π‘› 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=supπ‘₯,π‘¦βˆˆπ΄β€–β€–β€–β€–π‘₯βˆ’π‘¦.(3.1) and will show that the set-valued map 𝑋𝑝(𝑑;𝑑0,𝑋0, is HΓΆlder continuous with respect to πœ‡0)..

We denote the diameter of a set π‘βˆˆ(1,+∞) by diam𝑋𝑝𝑑;𝑑0,𝑋0,πœ‡0≀𝑑0+π‘Ÿ1ξ‚„ξ‚€π‘Ÿ(𝑑,𝑝)exp0(𝑑,𝑝)(3.2) and define it as π‘‘βˆˆ[𝑑0,πœƒ]

The following proposition characterizes the diameter of the attainable set 𝑑0=diam𝑋0,π‘Ÿ(3.3)0(𝑑,𝑝)=𝐿1ξ‚€π‘‘βˆ’π‘‘0+𝐿2ξ‚€π‘‘βˆ’π‘‘0(π‘βˆ’1)/𝑝,π‘Ÿ(3.4)1(𝑑,𝑝)=2𝐿3πœ‡0ξ‚€π‘‘βˆ’π‘‘0(π‘βˆ’1)/𝑝.(3.5)π‘‘βˆˆ[𝑑0,πœƒ]

Proposition 3.1. For every π‘₯1(𝑑)βˆˆπ‘‹π‘(𝑑;𝑑0,𝑋0,πœ‡0),, the inequality π‘₯2(𝑑)βˆˆπ‘‹π‘(𝑑;𝑑0,𝑋0,πœ‡0) holds for any π‘₯1βˆˆπ‘‹0, where π‘₯1(β‹…)βˆˆπ‘‹π‘(𝑑0,π‘₯0,πœ‡0)𝑒1(β‹…)βˆˆπ‘ˆπœ‡0𝑝,π‘₯2βˆˆπ‘‹0,π‘₯2(β‹…)∈

Proof. Let 𝑋𝑝(𝑑0,π‘₯0,πœ‡0) and 𝑒2(β‹…)βˆˆπ‘ˆπœ‡0𝑝π‘₯1(𝑑)=π‘₯1+ξ€œπ‘‘π‘‘0π‘“ξ‚€πœ,π‘₯1(𝜏),𝑒1π‘₯(𝜏)π‘‘πœ,2(𝑑)=π‘₯2+ξ€œπ‘‘π‘‘0π‘“ξ‚€πœ,π‘₯2(𝜏),𝑒2(𝜏)π‘‘πœ.(3.6) be arbitrarily chosen. Then there exist β€–π‘₯1βˆ’π‘₯2‖≀𝑑0, β€–β€–π‘₯1(𝑑)βˆ’π‘₯2β€–β€–(𝑑)≀𝑑0+ξ€œπ‘‘π‘‘0𝐿1+𝐿2‖‖𝑒2‖‖‖‖π‘₯(𝜏)1(𝜏)βˆ’π‘₯2β€–β€–+ξ€œ(𝜏)π‘‘πœπ‘‘π‘‘0𝐿3‖‖𝑒1(𝜏)βˆ’π‘’2β€–β€–(𝜏)π‘‘πœ.(3.7), 𝑒1(β‹…),𝑒2(β‹…)βˆˆπ‘ˆπœ‡0𝑝𝐿3βˆ«π‘‘π‘‘0‖‖𝑒1(𝜏)βˆ’π‘’2β€–β€–(𝜏)π‘‘πœβ‰€2𝐿3πœ‡0ξ‚€π‘‘βˆ’π‘‘0(π‘βˆ’1)/𝑝=π‘Ÿ1(𝑑,𝑝),(3.8)π‘Ÿ1(𝑑,𝑝), π‘‘βˆˆ[𝑑0,πœƒ] such that β€–β€–π‘₯1(𝑑)βˆ’π‘₯2‖‖≀𝑑(𝑑)0+π‘Ÿ1ξ‚„ξ‚€ξ€œ(𝑑,𝑝)exp𝑑𝑑0𝐿1+𝐿2‖‖𝑒2‖‖≀𝑑(𝜏)π‘‘πœ0+π‘Ÿ1𝐿(𝑑,𝑝)exp1ξ‚€π‘‘βˆ’π‘‘0+𝐿2ξ‚€π‘‘βˆ’π‘‘0(π‘βˆ’1)/π‘πœ‡0=𝑑0+π‘Ÿ1ξ‚„ξ‚€π‘Ÿ(𝑑,𝑝)exp0,(𝑑,𝑝)(3.9)
Since π‘Ÿ0(𝑑,𝑝), then from (3.6), and the condition (b), we get diam𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0)β†’diam𝑋0
Since 𝑑→𝑑0., then the HΓΆlder and Minkowski inequalities imply that 𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0) where 𝑑. is defined by (3.5). Since 𝑑1∈[𝑑0,πœƒ], is arbitrarily chosen, we obtain from (3.7), (3.8), and Gronwall's inequality that 𝑑2∈[𝑑0,πœƒ]. where β„Žξ‚€π‘‹π‘ξ‚€π‘‘1;𝑑0,𝑋0,πœ‡0,𝑋𝑝𝑑2;𝑑0,𝑋0,πœ‡0ξ‚ξ‚β‰€π‘˜βˆ—|||𝑑1βˆ’π‘‘2|||(π‘βˆ’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. π‘˜βˆ—>0 as 𝑑1<𝑑2.

The following proposition asserts that the attainable set 𝑦1βˆˆπ‘‹π‘(𝑑1;𝑑0,𝑋0,πœ‡0) is HΓΆlder continuous with respect to π‘₯0βˆˆπ‘‹0

Proposition 3.3. Let π‘₯0(β‹…)βˆˆπ‘‹π‘(𝑑0,𝑋0,πœ‡0)𝑒0(β‹…)βˆˆπ‘ˆπœ‡0𝑝 Then 𝑦1=π‘₯0𝑑1=π‘₯0+βˆ«π‘‘1𝑑0π‘“ξ‚€πœ,π‘₯0(𝜏),𝑒0(𝜏)π‘‘πœ.(3.11) where 𝑦2=π‘₯0𝑑2=π‘₯0+βˆ«π‘‘2𝑑0π‘“ξ‚€πœ,π‘₯0(𝜏),𝑒0(𝜏)π‘‘πœ.(3.12) is defined by (2.6).

Proof. Without loss of generality, let us assume that 𝑦2βˆˆπ‘‹π‘(𝑑2;𝑑0,𝑋0,πœ‡0). Let ‖‖𝑦1βˆ’π‘¦2β€–β€–β‰€ξ€œπ‘‘2𝑑1𝑐‖‖π‘₯1+0‖‖‖‖𝑒(𝜏)1+0‖‖(𝜏)π‘‘πœβ‰€π‘1+π‘Ÿβˆ—ξ‚ξ€œπ‘‘2𝑑1‖‖𝑒1+0‖‖(𝜏)π‘‘πœβ‰€π‘1+π‘Ÿβˆ—|||𝑑2βˆ’π‘‘1|||+πœ‡0|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝≀𝑐1+π‘Ÿβˆ—π‘™ξ‚ξ‚€βˆ—+πœ‡0|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝=π‘˜βˆ—||𝑑2βˆ’π‘‘1||(π‘βˆ’1)/𝑝,(3.13) be arbitrarily chosen. Then there exist 𝑐>0, 𝑦1βˆˆπ‘‹π‘(𝑑1,𝑑0,𝑋0,πœ‡0) and 𝑋𝑝𝑑1;𝑑0,𝑋0,πœ‡0ξ‚βŠ‚π‘‹π‘ξ‚€π‘‘2;𝑑0,𝑋0,πœ‡0+π‘˜βˆ—|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝𝐡𝑛1ξ€Έ.(3.14) such that 𝑋𝑝𝑑2;𝑑0,𝑋0,πœ‡0ξ‚βŠ‚π‘‹π‘ξ‚€π‘‘1;𝑑0,𝑋0,πœ‡0+π‘˜βˆ—|||𝑑2βˆ’π‘‘1|||(π‘βˆ’1)/𝑝𝐡𝑛1ξ€Έ.(3.15) Let 𝑑→𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0)
It is obvious that π‘‘βˆˆ[𝑑0,πœƒ], From Proposition 2.1, relations (2.4), (2.6), (3.11), (3.12), and the condition (c), we have (π‘βˆ’1)/𝑝 where 𝑑0 is the constant given in condition (c).
Since 𝑋0 is arbitrarily chosen, then (3.13) implies that (𝑑0,𝑋0)β†’
Analogously, it is possible to show that 𝑋𝑝(𝑑;𝑑0,𝑋0,πœ‡0)
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 πœ”βˆ—=𝐿1+𝐿2πœ‡0ξ‚π‘™βˆ—,(4.1), π‘™βˆ— is 𝐿1-HΓΆlder continuous.

4. Dependence of the Attainable Sets on Parameters 𝐿2 and 𝑑1β‰₯𝑑0

The following proposition characterizes the continuity of the set-valued map 𝑋0,𝑋1βŠ‚β„π‘›β„Žξ‚€π‘‹π‘ξ‚€π‘‘;𝑑0,𝑋0,πœ‡0,𝑋𝑝𝑑;𝑑1,𝑋1,πœ‡0β‰€ξ‚ƒβ„Žξ‚€π‘‹ξ‚ξ‚0,𝑋1+𝑑1βˆ’π‘‘0(π‘βˆ’1)/π‘π‘˜βˆ—ξ‚„ξ‚€πœ”expβˆ—ξ‚(4.2) in the Hausdorff metric.

Let us denote π‘‘βˆˆ[𝑑1,πœƒ] where π‘˜βˆ— is defined by (2.4), πœ”βˆ— and π‘‘βˆˆ[𝑑1,πœƒ] are the constants given in condition (b).

Proposition 4.1. Let π‘₯0(𝑑)βˆˆπ‘‹π‘(𝑑;𝑑0,𝑋0,πœ‡0) and π‘₯0(β‹…)∈ be compact sets. Then the inequality 𝑋𝑝(𝑑0,𝑋0,πœ‡0) holds for all π‘₯0βˆˆπ‘‹0, where 𝑒0(β‹…)βˆˆπ‘ˆπœ‡0𝑝 is defined by (2.6), π‘₯0(𝑑)=π‘₯0+βˆ«π‘‘π‘‘0π‘“ξ‚€πœ,π‘₯0(𝜏),𝑒0(𝜏)π‘‘πœ(4.3) is defined by (4.1).

Proof. Let us choose arbitrary π‘₯1βˆˆπ‘‹1 and β€–π‘₯1βˆ’π‘₯0β€–β‰€β„Ž(𝑋0,𝑋1), where π‘₯1(β‹…)𝑒0(β‹…). Then there exist π‘₯1(𝑑1)=π‘₯1βˆˆπ‘‹1. and π‘₯1(𝑑)=π‘₯1+βˆ«π‘‘π‘‘1π‘“ξ‚€πœ,π‘₯1(𝜏),𝑒0(𝜏)π‘‘πœ(4.4) such that π‘₯1(𝑑)βˆˆπ‘‹π‘(𝑑;𝑑1,𝑋1,πœ‡0). holds. According to the definition of Hausdorff distance, there exists β€–β€–π‘₯0(𝑑)βˆ’π‘₯1‖‖𝑋(𝑑)β‰€β„Ž0,𝑋1ξ‚ξ€œ+𝑐𝑑1𝑑0‖‖𝑒1+0β€–β€–β€–β€–π‘₯(𝜏)1+0‖‖+ξ€œ(𝜏)π‘‘πœπ‘‘π‘‘1𝐿1+𝐿2‖‖𝑒0β€–β€–β€–β€–π‘₯(𝜏)0(𝜏)βˆ’π‘₯1‖‖(𝜏)π‘‘πœ.(4.5) such that π‘βˆ«π‘‘1𝑑0‖‖𝑒1+0β€–β€–β€–β€–π‘₯(𝜏)1+0‖‖𝑑(𝜏)π‘‘πœβ‰€1βˆ’π‘‘0(π‘βˆ’1)/π‘π‘˜βˆ—,(4.6). Let π‘˜βˆ— be a solution of the control system (1.1), generated by the admissible control function π‘‘βˆˆ[𝑑1,πœƒ] with initial condition β€–β€–π‘₯0(𝑑)βˆ’π‘₯1β€–β€–β‰€ξ‚ƒβ„Žξ‚€π‘‹(𝑑)0,𝑋1+𝑑1βˆ’π‘‘0(π‘βˆ’1)/π‘π‘˜βˆ—ξ‚„ξ‚€ξ€œexp𝑑𝑑1𝐿1+𝐿2‖‖𝑒0β€–β€–ξ‚ξ‚β‰€ξ‚ƒβ„Žξ‚€π‘‹(𝜏)π‘‘πœ0,𝑋1+𝑑1βˆ’π‘‘0(π‘βˆ’1)/π‘π‘˜βˆ—ξ‚„exp(πœ”βˆ—).(4.7) Then 𝑋𝑝𝑑;𝑑0,𝑋0,πœ‡0ξ‚βŠ‚π‘‹π‘ξ‚€π‘‘;𝑑1,𝑋1,πœ‡0+ξ‚ƒβ„Žξ‚€π‘‹0,𝑋1+𝑑1βˆ’π‘‘0(π‘βˆ’1)/π‘π‘˜βˆ—ξ‚„ξ‚€πœ”expβˆ—ξ‚π΅π‘›ξ€·1ξ€Έ.(4.8) and 𝑋𝑝𝑑;𝑑1,𝑋1,πœ‡0ξ‚βŠ‚π‘‹π‘ξ‚€π‘‘;𝑑0,𝑋0,πœ‡0+ξ‚ƒβ„Žξ‚€π‘‹0,𝑋1+𝑑1βˆ’π‘‘0(π‘βˆ’1)/π‘π‘˜βˆ—ξ‚„ξ‚€πœ”expβˆ—ξ‚π΅π‘›ξ€·1ξ€Έ.(4.9)
From (4.3), (4.4), and conditions (b) and (c), we have β„Žξ‚€π‘‹π‘ξ‚€π‘‘;𝑑0,𝑋0,πœ‡0,𝑋𝑝𝑑;𝑑0,𝑋1,πœ‡0ξ‚€π‘‹ξ‚ξ‚β‰€β„Ž0,𝑋1ξ‚ξ‚€πœ”expβˆ—ξ‚(4.10)
Proposition 2.1 implies that π‘‘βˆˆ[𝑑0,πœƒ] where πœ”βˆ—>0 is defined by (2.6). Since β„ŽπΆξ‚€π‘‹π‘ξ‚€π‘‘0,𝑋0,πœ‡0,𝑋𝑝𝑑0,𝑋1,πœ‡0ξ‚€π‘‹ξ‚ξ‚β‰€β„Ž0,𝑋1ξ‚ξ‚€πœ”expβˆ—ξ‚(4.11) is arbitrarily chosen, from (4.5), (4.6), and Gronwall's inequality, we get 𝑋0βŠ‚β„π‘›
Hence, we obtain from (4.7) that π‘‹π‘›βŠ‚β„π‘›
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 β„Ž(𝑋𝑛,𝑋0)β†’0 holds for all 𝑑𝑛→𝑑0+0, where π‘›β†’βˆž is defined by (4.1).

Corollary 4.3. The inequality π‘‘βˆˆ(𝑑0,πœƒ], holds.

Corollary 4.4. Let β„Žξ‚€π‘‹π‘ξ‚€π‘‘;𝑑𝑛,𝑋𝑛,πœ‡0,𝑋𝑝𝑑;𝑑0,𝑋0,πœ‡0ξ‚ξ‚βŸΆ0asπ‘›βŸΆβˆž.(4.12) and πœ‡0 be compact sets for all 𝑋𝑝(𝑑0,𝑋0,πœ‡0). Assume that πœ‡0. and π‘Ÿ1=𝐿3π‘™βˆ—ξ‚€πœ”expβˆ—ξ‚,(5.1) as πœ”βˆ—. Then for all 𝑋𝑝(𝑑0,𝑋0,πœ‡βˆ—)

5. Dependence of the Attainable Sets on 𝑋𝑝(𝑑0,𝑋0,πœ‡0)

In this section we specify dependence of the set β„ŽπΆ(𝑋𝑝(𝑑0,𝑋0,πœ‡βˆ—),𝑋𝑝(𝑑0,𝑋0,πœ‡0))β‰€π‘Ÿ1||πœ‡βˆ—βˆ’πœ‡0||(5.2) on the constraint parameter π‘Ÿ1 Let π‘₯0(β‹…)βˆˆπ‘‹π‘(𝑑0,𝑋0,πœ‡0) where π‘₯0βˆˆπ‘‹0 is defined by (4.1).

The following proposition characterizes the relation between the solutions sets 𝑒0(β‹…)βˆˆπ‘ˆπœ‡0𝑝π‘₯0(𝑑)=π‘₯0+βˆ«π‘‘π‘‘0π‘“ξ‚€πœ,π‘₯0(𝜏),𝑒0(𝜏)π‘‘πœ(5.3) and π‘‘βˆˆ[𝑑0,πœƒ]..

Proposition 5.1. The inequality π‘’βˆ—(β‹…)∢[𝑑0,πœƒ]β†’β„π‘š is satisfied, where π‘’βˆ—πœ‡(𝑑)=βˆ—πœ‡0𝑒0𝑑(𝑑),π‘‘βˆˆ0ξ‚„.,πœƒ(5.4) is defined by (5.1).

Proof. Let π‘’βˆ—(β‹…)βˆˆπ‘ˆπœ‡βˆ—π‘ be an arbitrarily chosen solution. Then there exist π‘₯βˆ—(β‹…) and π‘’βˆ—(β‹…)βˆˆπ‘ˆπœ‡βˆ—π‘ such that (𝑑0,π‘₯0) for every π‘₯βˆ—(β‹…)βˆˆπ‘‹π‘(𝑑0,𝑋0,
We define a new control function πœ‡βˆ—), setting π‘₯βˆ—(𝑑)=π‘₯0+βˆ«π‘‘π‘‘0π‘“ξ‚€πœ,π‘₯βˆ—(𝜏),π‘’βˆ—ξ‚(𝜏)π‘‘πœ(5.5)
It is not difficult to verify that π‘‘βˆˆ[𝑑0,πœƒ]. Let β€–β€–π‘₯βˆ—(𝑑)βˆ’π‘₯0β€–β€–(𝑑)≀𝐿3|||πœ‡βˆ—πœ‡0|||ξ€œβˆ’1𝑑𝑑0‖‖𝑒0β€–β€–ξ€œ(𝜏)π‘‘πœ+𝑑𝑑0𝐿1+𝐿2‖‖𝑒0‖‖‖‖π‘₯(𝜏)βˆ—(𝜏)βˆ’π‘₯0β€–β€–(𝜏)π‘‘πœβ‰€πΏ3π‘™βˆ—|||πœ‡βˆ—βˆ’πœ‡0|||+ξ€œπ‘‘π‘‘0𝐿1+𝐿2‖‖𝑒0‖‖‖‖π‘₯(𝜏)βˆ—(𝜏)βˆ’π‘₯0β€–β€–(𝜏)π‘‘πœ(5.6) be a solution of the control system (1.1), generated by π‘‘βˆˆ[𝑑0,πœƒ], from the initial point π‘™βˆ—. Then β€–β€–π‘₯βˆ—(𝑑)βˆ’π‘₯0β€–β€–(𝑑)≀𝐿3π‘™βˆ—|||πœ‡βˆ—βˆ’πœ‡0|||𝐿exp1(πœƒβˆ’π‘‘0+𝐿2πœ‡0ξ‚€πœƒβˆ’π‘‘0(π‘βˆ’1)/𝑝≀𝐿3π‘™βˆ—ξ‚€πœ”expβˆ—ξ‚|||πœ‡βˆ—βˆ’πœ‡0|||=π‘Ÿ1|||πœ‡βˆ—βˆ’πœ‡0|||(5.7)π‘‘βˆˆ[𝑑0,πœƒ] and π‘₯0(β‹…)βˆˆπ‘‹π‘(𝑑0,𝑋0,πœ‡0) for every π‘₯1(β‹…)βˆˆπ‘‹π‘(𝑑0,𝑋0,πœ‡βˆ—). From (5.3), (5.4), (5.5), and condition (b), we get β€–β€–π‘₯0(β‹…)βˆ’π‘₯βˆ—β€–β€–(β‹…)πΆβ‰€π‘Ÿ1|||πœ‡βˆ—βˆ’πœ‡0|||,(5.8) for every 𝑋𝑝𝑑0,𝑋0,πœ‡0ξ‚βŠ‚π‘‹π‘ξ‚€π‘‘0,𝑋0,πœ‡βˆ—ξ‚+π‘Ÿ1|||πœ‡βˆ—βˆ’πœ‡0|||𝐡𝐢,(5.9) where 𝐡𝐢 is defined by (2.4).
The Gronwall inequality, (5.1), and (5.6) yield that 𝐢([𝑑0,πœƒ],ℝ𝑛) for all 𝑋𝑝𝑑0,𝑋0,πœ‡βˆ—ξ‚βŠ‚π‘‹π‘ξ‚€π‘‘0,𝑋0,πœ‡0+π‘Ÿ1|||πœ‡βˆ—βˆ’πœ‡0|||𝐡𝐢.(5.10).
Thus from (5.7) we get that for any fixed β„Žξ‚€π‘‹π‘ξ‚€π‘‘;𝑑0,𝑋0,πœ‡βˆ—ξ‚,𝑋𝑝𝑑;𝑑0,𝑋0,πœ‡0ξ‚ξ‚β‰€π‘Ÿ1|||πœ‡βˆ—βˆ’πœ‡0|||(5.11) there exists π‘‘βˆˆ[𝑑0,πœƒ], such that π‘Ÿ1>0 and consequently πœ‡π‘›β†’πœ‡0 where π‘›β†’βˆž is the closed unit ball centered at the origin in the space β„ŽπΆξ‚€π‘‹π‘ξ‚€π‘‘0,𝑋0,πœ‡π‘›ξ‚,𝑋𝑝𝑑0,𝑋0,πœ‡0β„Žξ‚€π‘‹ξ‚ξ‚βŸΆ0asπ‘›βŸΆβˆž,𝑝𝑑;𝑑0,𝑋0,πœ‡π‘›ξ‚,𝑋𝑝𝑑;𝑑0,𝑋0,πœ‡0ξ‚ξ‚βŸΆ0asπ‘›βŸΆβˆž(5.12).
Analogously, it is possible to prove that π‘‘βˆˆ[𝑑0,πœƒ]. 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 𝐿𝑝 where 𝑝β‰₯1 is defined by (5.1).

Corollary 5.3. Let 𝐿𝑝 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.

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