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

Authors: Khalik G. Guseinov; Ali S. Nazlipinar
Journal: Abstract and Applied Analysis
 (2008)

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## 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 radiusμ0 in Lp([t0,θ];ℝm)(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.


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## Body

## 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 onp 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 radiusμ0 in Lp([t0,θ];ℝm)⁢(p∈(1,+∞)).In Section2, 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(1.1)x⋅(t)=f(t,x(t),u(t)),x(t0)∈X0,
where x∈ℝn is the
phase-state vector of the system, u∈ℝm is the control
vector, t∈[t0,θ] is the time,
and X0⊂ℝn is a compact
set.Forp∈(1,∞) and μ0>0, we set
(1.2)Upμ0={u(⋅)∈Lp([t0,θ],ℝm):∥u(⋅)∥p≤μ0},
where ∥u(⋅)∥p=(∫t0θ∥u(t)∥pdt)1/p and ∥⋅∥ denotes the
Euclidian norm.A functionu(⋅)∈Upμ0 is said to be
an admissible control function. It is obvious that the set of all admissible
control functions Upμ0 is the closed
ball centered at the origin with the radius μ0 in Lp([t0,θ];ℝm).It is assumed that the right-hand side of the system
(1.1) satisfies the following conditions.(a)
The functionf(⋅):[t0,θ]×ℝn×ℝm→ℝn is continuous.(b)
For any bounded
setD⊂[t0,θ]×ℝn, there exist constants L1=L1(D)>0, L2=L2(D)>0, and L3=L3(D)>0 such that
(1.3)∥f(t,x1,u1)−f(t,x2,u2)∥≤(L1+L2∥u2∥)∥x1−x2∥+L3∥u1−u2∥
for any (t,x1)∈D, (t,x2)∈D, u1∈ℝm, and u2∈ℝm.(c)
There exists a
constantc>0 such that
(1.4)∥f(t,x,u)∥≤c(1+∥x∥)(1+∥u∥)
for every (t,x,u)∈[t0,θ]×ℝn×ℝm.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]×ℝn→ℝn,B(⋅):[t0,θ]×ℝn→ℝm satisfy the
assumptions given in [11–14], then, under these assumptions,
the conditions (a), (b), and (c) are also fulfilled.Letu*(⋅)∈Upμ0. The absolutely continuous function x*(⋅):[t0,θ]→ℝn, which satisfies the equation x˙*(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*(⋅)∈Upμ0 and x0∈X0.Let us definethe sets(1.5)Xp(t0,X0,μ0)={x(⋅;t0,x0,u(⋅)):[t0,θ]⟶ℝn∣x0∈X0,u(⋅)∈Upμ0},Xp(t;t0,X0,μ0)={x(t)∈ℝn:x(⋅)∈Xp(t0,X0,μ0)},
where t∈[t0,θ].The setXp(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 Xp(t;t0,X0,μ0) consists
of all x∈ℝn to which the
system (1.1) can be steered at the instant of time t∈[t0,θ].The Hausdorff distance between the setsA⊂ℝn and E⊂ℝn is denoted by h(A,E) and is defined
as
(1.6)h(A,E)=max{sup⁡x∈Ad(x,E),⁢sup⁡y∈Ed(y,A)},
where d(x,E)=inf{∥x−y∥:y∈E}.ByC([t0,θ];ℝn), we denote the space of continuous functions x(⋅):[t0,θ]→ℝn with norm
(1.7)∥x(⋅)∥C=max⁡t∈[t0,θ]∥x(t)∥.Also,hC(U,V) denotes the
Hausdorff distance between the sets U⊂C([t0,θ];ℝn) and V⊂C([t0,θ];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.
Letp∈(1,∞),  μ*∈(0,μ0+1),  h(X0,X*)≤1. Then for any  x*(⋅)∈Xp(t0,X*,μ*), the inequality
(2.1)∥x*(⋅)∥C≤r*
holds, where
(2.2)r*=d1exp⁡(k),(2.3)d1=1+d*+k,d*=max{∥x∥:x∈X0},k=c[(θ−t0)+l*(μ0+1)],(2.4)l*=max{(θ−t0),1},
andc>0 is the constant
given in condition (c).The proof of the proposition follows from condition
(c) and Gronwall's inequality.For givenγ>0, we set
(2.5)D(γ)={(t,x)∈[t0,θ]×ℝn:∥x∥≤γ},Bn(γ)={x∈ℝn:∥x∥≤γ}.We get from Proposition2.1 that (t,x(t))∈D(r*) for every p∈(1,∞),x(⋅)∈Xp(t0,X*,μ*),t∈[t0,θ],μ*∈(0,μ0+1), and compact X*⊂ℝn such that h(X0,X*)≤1. So, we have the
validity of the following corollary.Corollary 2.2.
The
setXp(t0,X0,μ0)
is uniformly
bounded, and consequently  Xp(t;t0,X0,μ0)⊂Bn(r*)  for every  t∈[t0,θ], where  r*  is defined by
(2.2).Here and henceforth, we will have in mind the cylinderD(r*) as the set D in condition (b). We set also
(2.6)k*=c(1+r*)(l*+μ0),
where r* is defined by
(2.2), l* is defined by
(2.4).Proposition 2.3.
The
setXp(t0,X0,μ0)
is
equicontinuous.Proof.
Letε>0 be an
arbitrarily given number. Now, let us choose an arbitrary x(⋅)∈Xp(t0,X0,μ0) and t1,t2∈[t0,θ]. Without loss of
generality, we assume that t1≤t2. Then from condition (c), we have
(2.7)∥x(t1)−x(t2)∥≤∫t1t2c(1+∥x(τ)∥)⁢(1+∥u(τ)∥)dτ.
According to Proposition2.1, ∥x(⋅)∥C≤r*, where r* is defined by
(2.2). Then we get from (2.4), (2.6), (2.7), and Hölder's inequality that
(2.8)∥x(t1)−x(t2)∥≤c(1+r*)(|t2−t1|+μ0|t2−t1|(p−1)/p)≤|t2−t1|(p−1)/pc(1+r*)((θ−t0)1/p+μ0)≤|t2−t1|(p−1)/pc(1+r*)(l*+μ0)=k*|t2−t1|(p−1)/p.
Thus for givenε>0, setting δ(ε)=(ε/k*)p/(p−1), we obtain ∥x(t1)−x(t2)∥<ε for |t1−t2|<δ(ε). Since x(⋅)∈Xp(t0,X0,μ0) is arbitrarily
chosen, the equicontinuity of the set Xp(t0,X0,μ0) follows.From Corollary2.2 and Proposition 2.3, we get the
validity of the following corollary.Corollary 2.4.
The setXp(t0,X0,μ0)
is a precompact
subset of the space  C([t0,θ],ℝn).Note that if the right-hand side of the system (1.1)
is affine with respect to the control vector u, then the weak
compactness of the set of admissible control functions Upμ0 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(2.9)x˙=−y2+u2,x(0)=0,y˙=u,y(0)=0,
where (x,y)∈ℝ2 is the
phase-state vector of the system, u∈ℝ is the control
vector, t∈[0,1]. It is assumed
that μ0=1 and the control
function u(⋅)∈L2([0,1];ℝ) of the system
(2.9) satisfies the integral constraint
(2.10)∫01u2(t)dt≤1,
that is, u(⋅)∈U21. Let us denote
(2.11)X2(0,(0,0),1)={(x(⋅;0,(0,0),u(⋅)),y(⋅;0,(0,0),u(⋅))):u(⋅)∈U21},X2(t;0,(0,0),1)={(x(t),y(t))∈ℝ2:(x(⋅),y(⋅))∈X2(0,(0,0),1)}.
ThusX2(0,(0,0),1) is the set of
solutions, X2(t;0,(0,0),1) is the
attainable set of the control system (2.9) at the instant of time t∈[0,1], generated by
control functions u(⋅)∈U21.
Now, let us prove that the solution setX2(0,(0,0),1) is bounded. Let (x(⋅),y(⋅))∈X2(0,(0,0),1) be an
arbitrarily chosen solution of the system (2.9) with integral constraint (2.10).
Then there exists u(⋅)∈U21 such that
(2.12)x(t)=∫0t−y2(τ)dτ+∫0tu2(τ)dτ,(2.13)y(t)=∫0tu(τ)dτfor any t∈[0,1]. From (2.10), (2.13), and Hölder's inequality, the
inequality
(2.14)|y(t)|≤∫0t|u(τ)|dτ≤(∫0t12dτ)1/2(∫0t|u(τ)|2dτ)1/2≤t≤1
holds for all t∈[0,1]. Then we get
from (2.10), (2.12), and (2.14) that
(2.15)|x(t)|≤∫0t|y(τ)|2dτ+∫0t|u(τ)|2dτ≤∫0tτdτ+1=1+t22≤32
for all t∈[0,1].
However, (2.14) and (2.15) imply that
(2.16)∥(x(⋅),y(⋅))∥C=max⁡t∈[0,1]∥(x(t),y(t))∥=max⁡t∈[0,1]x2(t)+y2(t)≤1+94<2.
Since (x(⋅),y(⋅))∈X2(0,(0,0),1) is arbitrarily
chosen, we get that the set X2(0,(0,0),1) is bounded.
Now we prove that(1,0)∉X2(1;0,(0,0),1). Let us assume
the contrary, that is, let (1,0)∈X2(1;0,(0,0),1). Then there
exists (x*(⋅),y*(⋅))∈X2(0,(0,0),1) such that
(2.17)x*(1)=1,y*(1)=0.
Since(x*(⋅),y*(⋅))∈X2(0,(0,0),1), then there exists u*(⋅)∈U21 such that
(2.18)x*(t)=∫0t−y*2(τ)dτ+∫0tu*2(τ)dτ,(2.19)y*(t)=∫0tu*(τ)dτfor all t∈[0,1].
From (2.17), (2.18), and (2.19), it follows that
(2.20)x*(1)=∫01u*2(τ)dτ−∫01(∫0τu*(s)ds)2dτ=1.
Sinceu*(⋅)∈U21, then it follows from (2.20) that u*(t)=0 for almost all t∈[0,1]. Then we have
from (2.18) and (2.19) that x*(t)=0,y*(t)=0 for every t∈[0,1], which
contradicts (2.17). Thus
(2.21)(1,0)∉X2(1;0,(0,0),1).
Let us show that(1,0)∈cl(X2(1;0,(0,0)),1).
LetΓk={0,1/2k,2/2k,…,(2k−1)/2k,1} be a uniform
partition of the closed interval [0,1], where k=1,2,…. Now we define a sequence of functions {uk(⋅)}k=1∞, setting
(2.22)uk(t)={1,t∈[2i2k,2i+12k),−1,t∈[2i+12k,2i+22k),
where i=0,1,…,k−1.
It is obvious thatuk(⋅)∈U21 for all k=1,2,…. Let (xk(⋅),yk(⋅))∈X2(0,(0,0),1) be the solution
of the system (2.9) generated by the admissible control function uk(⋅)∈U21. Then it follows from (2.9) that
(2.23)xk(t)=∫0t−yk2(τ)dτ+∫0tuk2(τ)dτ,(2.24)yk(t)=∫0tuk(τ)dτfor every t∈[0,1].
We get from (2.22) and (2.24) that
(2.25)yk(t)={t−2i2k,t∈[2i2k,2i+12k)−t+2i+22k,t∈[2i+12k,2i+22k)
for every t∈[0,1] where i=0,1,…,k−1.
Then, from (2.25) we have
(2.26)0≤yk(t)≤12k
for every t∈[0,1], and consequently
(2.27)0≤yk2(t)≤14k2
for every t∈[0,1].
According to (2.22), uk2(t)=1 for all t∈[0,1]. Then, from
(2.23) and (2.27) we obtain that
(2.28)1−14k2≤x⋅k(t)≤1
for almost all t∈[0,1], and consequently
(2.29)(1−14k2)t≤xk(t)≤t
for every t∈[0,1], where k=1,2,….
We conclude from the last inequality that(2.30)(1−14k2)≤xk(1)≤1
for every k=1,2,….
It follows from (2.26) and (2.30) that
(2.31)(xk(1),yk(1))⟶(1,0)ask⟶∞.
Since(xk(1),yk(1))∈X2(1;0,(0,0),1) for every k=1,2,…, from (2.31) we obtain that
(2.32)(1,0)∈cl(X2(1;0,(0,0),1)).
However, (2.21) and (2.32) imply that X2(1;0,(0,0),1) is not a closed
set.


## 3. Diameter of the Attainable Set and Continuity with Respect tot

In this section
we will give an upper estimation for the diameter of the attainable setXp(t;t0,X0,μ0) and will show
that the set-valued map t→Xp(t;t0,X0,μ0) is Hölder
continuous with respect to t.We denote the diameter of a setA⊂ℝn by diam(A) and define
it as
(3.1)diam⁢A=sup⁡x,y∈A∥x−y∥.The following proposition characterizes the diameter
of the attainable setXp(t;t0,X0,μ0).Proposition 3.1.
For
everyp∈(1,+∞), the inequality
(3.2)diam⁢Xp(t;t0,X0,μ0)≤[d0+r1(t,p)]exp⁡(r0(t,p))
holds for any t∈[t0,θ], where
(3.3)d0=diam⁢X0,(3.4)r0(t,p)=L1(t−t0)+L2(t−t0)(p−1)/p,(3.5)r1(t,p)=2L3μ0(t−t0)(p−1)/p.Proof.
Lett∈[t0,θ] and x1(t)∈Xp(t;t0,X0,μ0),x2(t)∈Xp(t;t0,X0,μ0) be arbitrarily
chosen. Then there exist x1∈X0, x1(⋅)∈Xp(t0,x0,μ0), u1(⋅)∈Upμ0,x2∈X0,x2(⋅)∈Xp(t0,x0,μ0), u2(⋅)∈Upμ0 such that
(3.6)x1(t)=x1+∫t0tf(τ,x1(τ),u1(τ))dτ,x2(t)=x2+∫t0tf(τ,x2(τ),u2(τ))dτ.
Since∥x1−x2∥≤d0, then from (3.6), and the condition (b), we
get
(3.7)∥x1(t)−x2(t)∥≤d0+∫t0t(L1+L2∥u2(τ)∥)∥x1(τ)−x2(τ)∥dτ+∫t0tL3∥u1(τ)−u2(τ)∥dτ.
Sinceu1(⋅),u2(⋅)∈Upμ0, then the
Hölder and Minkowski inequalities imply that
(3.8)L3∫t0t∥u1(τ)−u2(τ)∥dτ≤2L3μ0(t−t0)(p−1)/p=r1(t,p),
where r1(t,p) is defined by
(3.5). Since t∈[t0,θ] is arbitrarily
chosen, we obtain from (3.7), (3.8), and Gronwall's inequality that
(3.9)∥x1(t)−x2(t)∥≤[d0+r1(t,p)]exp⁡(∫t0t(L1+L2∥u2(τ)∥)dτ)≤[d0+r1(t,p)]exp⁡(L1(t−t0)+L2(t−t0)(p−1)/pμ0)=[d0+r1(t,p)]exp⁡(r0(t,p)),
where r0(t,p) is defined by
(3.4).Note that an estimation for diameter of the attainable
set can be obtained from Propo-sition2.1; but the estimation given by Proposition 3.1 is more precise.Corollary 3.2.
diam⁢Xp(t;t0,X0,μ0)→diam⁢X0  as  t→t0.The following proposition asserts that the attainable
setXp(t;t0,X0,μ0) is Hölder
continuous with respect to t.Proposition 3.3.
Lett1∈[t0,θ],t2∈[t0,θ].  Then
(3.10)h(Xp(t1;t0,X0,μ0),⁢Xp(t2;t0,X0,μ0))≤k*|t1−t2|(p−1)/p,
where k*>0 is defined by
(2.6).Proof.
Without loss of generality, let us
assume thatt1<t2. Let y1∈Xp(t1;t0,X0,μ0) be arbitrarily
chosen. Then there exist x0∈X0, x0(⋅)∈Xp(t0,X0,μ0) and u0(⋅)∈Upμ0 such that
(3.11)y1=x0(t1)=x0+∫t0t1f(τ,x0(τ),u0(τ))dτ.
Let
(3.12)y2=x0(t2)=x0+∫t0t2f(τ,x0(τ),u0(τ))dτ.
It is obvious thaty2∈Xp(t2;t0,X0,μ0). From
Proposition 2.1, relations (2.4), (2.6), (3.11), (3.12), and the condition (c),
we have
(3.13)∥y1−y2∥≤∫t1t2c(1+∥x0(τ)∥)(1+∥u0(τ)∥)dτ≤c(1+r*)∫t1t2(1+∥u0(τ)∥)dτ≤c(1+r*)(|t2−t1|+μ0|t2−t1|(p−1)/p)≤c(1+r*)(l*+μ0)|t2−t1|(p−1)/p)=k*|t2−t1|(p−1)/p,
where c>0 is the constant
given in condition (c).
Sincey1∈Xp(t1,t0,X0,μ0) is arbitrarily
chosen, then (3.13) implies that
(3.14)Xp(t1;t0,X0,μ0)⊂Xp(t2;t0,X0,μ0)+k*|t2−t1|(p−1)/pBn(1).
Analogously, it is possible to show that(3.15)Xp(t2;t0,X0,μ0)⊂Xp(t1;t0,X0,μ0)+k*|t2−t1|(p−1)/pBn(1).
In fact, (3.14) and (3.15)
yield the proof.From Proposition3.3, we obtain the following
corollary.Corollary 3.4.
The
set-valued mapt→Xp(t;t0,X0,μ0),  t∈[t0,θ],  is  (p−1)/p-Hölder
continuous.


## 4. Dependence of the Attainable Sets on Parameterst0 and X0

The following
proposition characterizes the continuity of the set-valued map(t0,X0)→Xp(t;t0,X0,μ0) in the
Hausdorff metric.Let us denote(4.1)ω*=(L1+L2μ0)l*,
where l* is defined by
(2.4), L1 and L2 are the
constants given in condition (b).Proposition 4.1.
Lett1≥t0  and  X0,X1⊂ℝn
be compact
sets. Then the inequality
(4.2)h(Xp(t;t0,X0,μ0),Xp(t;t1,X1,μ0))≤[h(X0,X1)+(t1−t0)(p−1)/pk*]exp⁡(ω*)
holds for all t∈[t1,θ], where k* is defined by
(2.6), ω* is defined by
(4.1).Proof.
Let us choose arbitraryt∈[t1,θ] and x0(t)∈Xp(t;t0,X0,μ0), where x0(⋅)∈Xp(t0,X0,μ0). Then there
exist x0∈X0 and u0(⋅)∈Upμ0 such that
(4.3)x0(t)=x0+∫t0tf(τ,x0(τ),u0(τ))dτ
holds.
According to the definition of Hausdorff distance, there exists x1∈X1 such that ∥x1−x0∥≤h(X0,X1). Let x1(⋅) be a solution
of the control system (1.1), generated by the admissible control function u0(⋅) with initial
condition x1(t1)=x1∈X1. Then
(4.4)x1(t)=x1+∫t1tf(τ,x1(τ),u0(τ))dτ
and x1(t)∈Xp(t;t1,X1,μ0).
From (4.3), (4.4), and conditions (b) and (c), we have
(4.5)∥x0(t)−x1(t)∥≤h(X0,X1)+c∫t0t1(1+∥u0(τ)∥)(1+∥x0(τ)∥)dτ+∫t1t(L1+L2∥u0(τ)∥)(∥x0(τ)−x1(τ)∥)dτ.
Proposition2.1 implies that
(4.6)c∫t0t1(1+∥u0(τ)∥)(1+∥x0(τ)∥)dτ≤(t1−t0)(p−1)/pk*,
where k* is defined by
(2.6). Since t∈[t1,θ] is arbitrarily
chosen, from (4.5), (4.6), and Gronwall's inequality, we get
(4.7)∥x0(t)−x1(t)∥≤[h(X0,X1)+(t1−t0)(p−1)/pk*]exp⁡(∫t1t(L1+L2∥u0(τ)∥)dτ)≤[h(X0,X1)+(t1−t0)(p−1)/pk*]exp⁡(ω*).
Hence, we obtain from (4.7) that
(4.8)Xp(t;t0,X0,μ0)⊂Xp(t;t1,X1,μ0)+[h(X0,X1)+(t1−t0)(p−1)/pk*]exp⁡(ω*)Bn(1).
Similarly, one can prove that(4.9)Xp(t;t1,X1,μ0)⊂Xp(t;t0,X0,μ0)+[h(X0,X1)+(t1−t0)(p−1)/pk*]exp⁡(ω*)Bn(1).
Finally, (4.8) and (4.9) complete the proof.From Proposition4.1, the validity of the following
corollaries follow.Corollary 4.2.
The
inequality(4.10)h(Xp(t;t0,X0,μ0),⁢Xp(t;t0,X1,μ0))≤h(X0,X1)exp⁡(ω*)
holds for all t∈[t0,θ], where ω*>0 is defined by
(4.1).Corollary 4.3.
The
inequality(4.11)hC(Xp(t0,X0,μ0),⁢Xp(t0,X1,μ0))≤h(X0,X1)exp⁡(ω*)
holds.Corollary 4.4.
LetX0⊂ℝn  and  Xn⊂ℝn
be compact sets
for all  n=1,2,….  Assume that  h(Xn,X0)→0  and  tn→t0+0  as  n→∞. Then for all  t∈(t0,θ],(4.12)h(Xp(t;tn,Xn,μ0),⁢Xp(t;t0,X0,μ0))⟶0asn⟶∞.


## 5. Dependence of the Attainable Sets onμ0

In this section we specify dependence of the setXp(t0,X0,μ0) on the
constraint parameter μ0. Let
(5.1)r1=L3l*exp⁡(ω*),
where ω* is defined by
(4.1).The following proposition characterizes the relation
between the solutions setsXp(t0,X0,μ*) and Xp(t0,X0,μ0).Proposition 5.1.
The
inequality(5.2)hC(Xp(t0,X0,μ*),⁢Xp(t0,X0,μ0))≤r1|μ*−μ0|
is satisfied, where r1 is defined by
(5.1).Proof.
Letx0(⋅)∈Xp(t0,X0,μ0) be an
arbitrarily chosen solution. Then there exist x0∈X0 and u0(⋅)∈Upμ0 such that
(5.3)x0(t)=x0+∫t0tf(τ,x0(τ),u0(τ))dτ
for every t∈[t0,θ].
We define a new control functionu*(⋅):[t0,θ]→ℝm, setting
(5.4)u*(t)=μ*μ0u0(t),t∈[t0,θ].
It is not difficult to verify thatu*(⋅)∈Upμ*. Let x*(⋅) be a solution
of the control system (1.1), generated by u*(⋅)∈Upμ* from the
initial point (t0,x0). Then x*(⋅)∈Xp(t0,X0,μ*) and
(5.5)x*(t)=x0+∫t0tf(τ,x*(τ),u*(τ))dτ
for every t∈[t0,θ]. From (5.3), (5.4), (5.5), and condition (b), we get
(5.6)∥x*(t)−x0(t)∥≤L3|μ*μ0−1|∫t0t∥u0(τ)∥dτ+∫t0t(L1+L2∥u0(τ)∥)∥x*(τ)−x0(τ)∥dτ≤L3l*|μ*−μ0|+∫t0t(L1+L2∥u0(τ)∥)∥x*(τ)−x0(τ)∥dτ
for every t∈[t0,θ], where l* is defined by
(2.4).
The Gronwall inequality, (5.1), and (5.6)
yield that
(5.7)∥x*(t)−x0(t)∥≤L3l*|μ*−μ0|exp⁡(L1(θ−t0)+L2μ0(θ−t0)(p−1)/p)≤L3l*exp⁡(ω*)|μ*−μ0|=r1|μ*−μ0|
for all t∈[t0,θ].
Thus from (5.7) we get that for any fixed x0(⋅)∈Xp(t0,X0,μ0) there exists x1(⋅)∈Xp(t0,X0,μ*) such that
(5.8)∥x0(⋅)−x*(⋅)∥C≤r1|μ*−μ0|,
and consequently
(5.9)Xp(t0,X0,μ0)⊂Xp(t0,X0,μ*)+r1|μ*−μ0|BC,
where BC is the closed
unit ball centered at the origin in the space C([t0,θ],ℝn).
Analogously, it is possible to prove that(5.10)Xp(t0,X0,μ*)⊂Xp(t0,X0,μ0)+r1|μ*−μ0|BC.
Hence, from
(5.9) and (5.10), we obtain the proof of the proposition.From Proposition5.1, it follows that the following
corollaries are satisfied.Corollary 5.2.
The
inequality(5.11)h(Xp(t;t0,X0,μ*),⁢Xp(t;t0,X0,μ0))≤r1|μ*−μ0|
is satisfied for any t∈[t0,θ], where r1>0 is defined by
(5.1).Corollary 5.3.
Letμn→μ0  as  n→∞. Then
(5.12)hC(Xp(t0,X0,μn),⁢Xp(t0,X0,μ0))⟶0asn⟶∞,h(Xp(t;t0,X0,μn),⁢Xp(t;t0,X0,μ0))⟶0asn⟶∞
for every t∈[t0,θ].



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*Source: 295817-2008-03-09.xml*