impulsive control systems

8/2/2019 Impulsive Control Systems

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Introduction

Impulsive Control Systems and Their Trajectories

A New Sampling Method

Invariance Properties

Open Problems and Future Work

Impulsive Control Systems

Wei [email protected]

Department of MathematicsLouisiana State University

Dissertation DefenseApril 28, 2009

Wei Cai Impulsive Control Systems
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Introduction





Background

outline

Introduction

Impulsive control system is one kind of dynamics systems

whose states may change fast with respect to different time

scales. My major contributions achieved through differentialinclusion and graph completion contain

A new sampling method

Invariance propertiesExtension of Hamilton-Jacobi theorey

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Introduction





Background

outline

Background

Cauchy or initial-value problem:

x(t) = f

t, x(t)

a.e. t [a, b]

x(a) = x0.

(1)

Standard control system:

x(t) = f(t, x(t), u(t)) a.e. t [0, T)u(t)

U a.e. t

[0, T)

x(0) = x0.(2)

A solution(or trajectory) is an absolutely continuous function

x : [0, T) Rn which satisfies the systems.

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Introduction





Background

outline

Background


x(t) = f

t, x(t)

a.e. t [a, b]

x(a) = x0.

(1)


x(t) = f(t, x(t), u(t)) a.e. t [0, T)u(t)

U a.e. t

[0, T)

x(0) = x0.(2)



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Introduction





Background

outline

Background


x(t) = f

t, x(t)

a.e. t [a, b]

x(a) = x0.

(1)


x(t) = f(t, x(t), u(t)) a.e. t [0, T)u(t)

U a.e. t

[0, T)

x(0) = x0.(2)




I d i
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Introduction





Background

outline

Background

Control System described in Differential Inclusion Form:x(t) F(t, x(t)) a.e. t [0, T)x(0) = x0,

(3)

where F : RRn Rn is a multifunction (set-valued map) on[0, T].

Its relationship with (1) and (2):

If F is singleton-valued (that is, F(t, x) = f(t, x)), then (3)subsumes Cauchy Problem (1).

If F(t, x) = f(t, x, U), then (3) subsumes Standard ControlSystem (2).


I t d ti
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Introduction





Background

outline

Background


(3)






Introduction
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Introduction





Background

outline

Background


(3)






Introduction
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Introduction





Background

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Background


(3)






Introduction
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Introduction





Background

outline

Outline

1 Impulsive Control Systems and Their Trajectories


Insight Into Measure

Assumptions

Their TrajectoriesGraph Completions

2 A New Sampling Method

Time Discretization and Euler Solutions

A Sampling Method for Impulsive Systems

3 Invariance Properties

Weak Invariance

Strong Invariance

4 Open Problems and Future Work


Introduction Impulsive Control Systems
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Introduction







Assumptions

Their Trajectories

Graph Completions

Impulsive Control System

An Impulsive control system (or called measure-driven

dynamics systems) is formulated as

dx F(x(t))dt + G(x(t))(dt)x(0) = x0, (4)where F() and G() are multifunctions whose values,respectively, are subsets of Rn and Mnm, and is avector-valued measure with values in a close convex coneK Rm. Distribution function u(t) = ([0, t]) is the control.This idea integrates the effects of the slow movement and the

fast movement (or called jump).


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Introduction







Assumptions

Their Trajectories

Graph Completions

Impulsive Control System

An Impulsive control system (or called measure-driven

dynamics systems) is formulated as

dx F(x(t))dt + G(x(t))(dt)x(0) = x0, (4)where F() and G() are multifunctions whose values,respectively, are subsets of Rn and Mnm, and is avector-valued measure with values in a close convex coneK Rm. Distribution function u(t) = ([0, t]) is the control.This idea integrates the effects of the slow movement and the

fast movement (or called jump).


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p y


Assumptions

Their Trajectories

Graph Completions


Recall that an arc x() of bounded variation induces a measuredx that have decomposition of absolutely continuous,

continuous singular, and discrete (i.e. purely atomic) parts:

dx = x(t)dt + dx + dxD,

where dxD :=

iI xti

. (xti := x(ti+) x(ti), the point massjump of x at ti).

Correspondingly, the measure is decomposed into

= udt + + D, where D =iI

uti .


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p y


Assumptions

Their Trajectories

Graph Completions


Recall that an arc x() of bounded variation induces a measuredx that have decomposition of absolutely continuous,

continuous singular, and discrete (i.e. purely atomic) parts:

dx = x(t)dt + dx + dxD,

where dxD :=

iI xti

. (xti := x(ti+) x(ti), the point massjump of x at ti).

Correspondingly, the measure is decomposed into

= udt + + D, where D =iI

uti .


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Assumptions

Their Trajectories

Graph Completions

Assumptions

The following assumptions are in effect throughout this

research.

(H1) A closed convex pointed cone K Rm, (wherepointed" is defined as K K = {0} );(H2) A multifunction F : Rn Rn with closed graph andconvex values, and satisfying

f F(x) f c(1 + x) x Rn,

(where c > 0 is a given constant);(H3) A multifunction G : RnMnm with closed graphand convex values, and satisfying

g G(x) g c(1 + x) x Rn.


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Assumptions

Their Trajectories

Graph Completions

Assumptions


research.


f F(x) f c(1 + x) x Rn,


g G(x) g c(1 + x) x Rn.


Introduction

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I i h I M
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Assumptions

Their Trajectories

Graph Completions

Assumptions


research.


f F(x) f c(1 + x) x Rn,


g G(x) g c(1 + x) x Rn.


Introduction

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Assumptions

Their Trajectories

Graph Completions

Graph Completion

Definition

A graph completion of the distribution function

u() : [0, T] Rm of , given by u(t) = ([0, t]), is a Lipschitzcontinuous map (

0, ) : [0, S]

[0, T]

R

m so that

(GC1) 0() is non-decreasing;(GC2) for every t [0, T], there exists s [0, S] so that(0(s), (s)) = (t, u(t));

(GC3) for almost all s [0, S],(s) K.Use graph completion to define a three-tuple solution:

X := (x(), (0(), ()), {yi()}iI)


Introduction



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Assumptions

Their Trajectories

Graph Completions

Graph Completion

Definition

A graph completion of the distribution function

u() : [0, T] Rm of , given by u(t) = ([0, t]), is a Lipschitzcontinuous map (

0, ) : [0, S]

[0, T]

R

m so that

(GC1) 0() is non-decreasing;(GC2) for every t [0, T], there exists s [0, S] so that(0(s), (s)) = (t, u(t));

(GC3) for almost all s [0, S],(s) K.Use graph completion to define a three-tuple solution:

X := (x(), (0(), ()), {yi()}iI)


Introduction



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Assumptions

Their Trajectories

Graph Completions

Trajectories (Direct Solutions)

Definition (Direct Solution)

The three-tuple X is a direct solution provided

for almost all t

[0, T],

x(t) F(x(t)) + G(x(t))u(t),x(0) = x0;

there exists a bounded -measurable selection

(t) G(x(t)) with dx = (t); andthe set of atoms of dx is T = {ti}iI, and for each i I,yi(s

i ) = x(ti), yi(s+i ) = x(ti+), andyi(s) G(yi(s))(s) a.e. s Ii.


IntroductionImpulsive Control Systems and Their Trajectories

Impulsive Control SystemsInsight Into Measure
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Assumptions

Their Trajectories

Graph Completions

Trajectories (Reparemerized Solutions)

Definition (Reparameterized Solution)

Consider a three-tuple X, and let

y(s) = x(t) if s / iIIi, t = 0(s),

yi(s) if s Ii.

Then X is a reparameterized solution provided y() is Lipschitzon [0, S] and satisfies

y(s) F(y(s))0(s) + G(y(s))(s), a.e. s [0, S],y(0) = x0.



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Assumptions

Their Trajectories

Graph Completions

Equivalence of Solutions

In Wolenski and Zabics paper, the two type of solutions are

proved to be exactly equivalent, which inspires switches

between graph completion and measure in many cases forconvenience of illustration.

Theorem

Suppose

BK([0, T]). Then X is a reparameterized solution

of (4) if and only if X is a direct solution (4).



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Assumptions

Their Trajectories

Graph Completions

Canonical Graph Completions

For further investigation, two special graph completions are

worth mention:

Definition

A canonical graph completionof a distribution functionu() : [0, T] Rm of , given by u(t) = ([0, t]), is a Lipschitzcontinuous pair (0, ) : [0, S] [0, T] Rm so that

(CG1) 0() is the filled-in inverse of (t) := t + ([0, t]),which means that

0(s) = t for (t

)

s

(t+);

(CG2) For every t [0, T], there exists s [0, S] so that(0(s), (s)) = (t, u(t)); and

(CG3) For almost all s [0, S], (s) K.



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g

Assumptions

Their Trajectories

Graph Completions

Normalized Graph Completions

Definition

Normalized graph completion of distribution function

u() : [0, T] Rm of , given by u(t) = ([0, t]), is a Lipschitzcontinuous pair map (0, ) : [0, S] [0, T] R

m

so that(NG1) 0 0 1 almost everywhere on [0, S],(NG2) for every t [0, T] there exists s [0, S] so that(0(s), (s)) = (t, u(t)) and

(NG3) (s) = (1 0(s))k(s), for almost all s [0, S],where k(s) K1 = K S1.

Any X represented originally by graph completion can be also

represented by canonical and normalized ones, by rescaling s.



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g

Assumptions

Their Trajectories

Graph Completions

Normalized Graph Completions

Definition

Normalized graph completion of distribution function

u() : [0, T] Rm of , given by u(t) = ([0, t]), is a Lipschitzcontinuous pair map (0, ) : [0, S] [0, T] R

m

so that(NG1) 0 0 1 almost everywhere on [0, S],(NG2) for every t [0, T] there exists s [0, S] so that(0(s), (s)) = (t, u(t)) and

(NG3) (s) = (1 0(s))k(s), for almost all s [0, S],where k(s) K1 = K S1.

Any X represented originally by graph completion can be also

represented by canonical and normalized ones, by rescaling s.


IntroductionImpulsive Control Systems and Their Trajectories Time Discretization and Euler Solutions
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Some Useful Theorems


Time Discretization

Recall the sampling method on the initial problem (1). Let

= {t0, t1,..., tN1, tN} be a partition of [a, b] with equal lengthh = (b a)/N, where t0 = aand tN = b. The nodes areobtained as follows,

v0 = f(t0, x0) x1 = x0 + hv0...

...

vi = f(ti, xi) xi+1 = xi + hvi... ...

vN1 = f(tN1, xN1) xN = xN1 + hvN1


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Euler Solution

The Euler polygonal arc is given by

xN(t) = xi + (t ti)vi whenever t [ti, ti+1].

An Euler solutionto the initial-value problem (1) is any uniform

limit x(

) of Euler polygonal arcs xN(

) as N

.



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Theorem

For system (1), suppose that for positive constants and c , wehave the linear growth condition:

f(t, x)

x

+ c.

At least one Euler solution x to the initial-value problem

exists, and any Euler solution is Lipschitz.

Any Euler arc x for f on [a,b] satisfies

x(t)

x(a)

(t

a)e(ta)(

x(a)

+ c), a

t

b.

If f is continuous, then any Euler arc x of f on (a, b) iscontinuously differentiable on(a, b) and satisfiesx(t) = f(t, x(t)), t (a, b).



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Compactness of Trajectories Theorem

Theorem (Compactness of Trajectories Theorem)

For system (3), let{xi} be a sequence of arcs on[a, b] suchthat the set x

i(a) is bounded, and satisfying

xi(t) F(i(t), xi(t) + yi(t)) + ri(t)B a.e.,where{yi}, {ri} and{i} are sequences of measurablefunctions on[a, b] such that yi(

) converges to0 in L2, ri

0

converges to0 in L2 andi converges a.e. to t. Then there is asubsequence of{xi} which converges uniformly to an arc xwhich is trajectory of F.





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Graph of Sampled Trajectories

The impulsive control systems are represented by normalized

graph completions: y F(y) 0(s) + (1 0(s))G(y)k(s), for son [0, S].

With S > 0 and x0 C fixed, let h := SN be the mesh size forN N. Let sN0 = 0 and sNj = jh for j = 1, 2,..., N. The samplednodes {xNj }Nj=0 are defined as follows, for the initial pieces,

xN

0 := x0 and xN

1 := xN

0 + N

0 hfN

0 + (1 N

0 )hgN

0 kN

0 with

N0 [0, 1] fN0 F(xN0 ) kN0 K1 gN0 G(xN0 ).





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Generally for j, xNj := xN

j1 + Nj1hf

Nj1 + (1 Nj1)hgNj1kNj1 with

Nj1 [0, 1] fNj1 F(xNj1) kNj1 K1 gNj1 G(xNj1).

We denote the graph of a sampled trajectory by Nas

N := {(sNj , xNj ) : j = 0, 1, ..., N}.One fact needed to mention is that there exists a constant c1

independent of N and j so that

maxj

{xj, fj, gj} c1





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Generally for j, xNj := xN

j1 + Nj1hf

Nj1 + (1 Nj1)hgNj1kNj1 with

Nj1 [0, 1] fNj1 F(xNj1) kNj1 K1 gNj1 G(xNj1).

We denote the graph of a sampled trajectory by Nas

N := {(sNj , xNj ) : j = 0, 1, ..., N}.One fact needed to mention is that there exists a constant c1

independent of N and j so that

maxj

{xj, fj, gj} c1





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A Sampling Method for impulsive systems

Theorem (Sampling Method Theorem)

Suppose that S > 0 and x0

Care given. For every sequence

{N}N of graphs of sampled trajectories, there exist a timelength T, a solution X with some measure BK([0, T]) anda sequence{Nk}Nk constructed from{N}N so that

distH(Nk, gr y)

0 as k

,

where y() is defined as in three-tuple solution X of (4).





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Step1: Construct T

Define N() on [0, S] so that for every N NN(s) := Nj on [sj1, sj], j = 1, 2,..., N.

Define T as T := lim supNNj=1 Nj SN.Without loss of generality, there exists a selection of integer

sequence {Nk}k such that

Nkj=1 Nkj SNk T as Nk .Notice we also can rewrite the summation expression in

integral form: T = limNkS

0 Nk(s)ds.





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Step1: Construct T






0 Nk(s)ds.





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Step1: Construct T






0 Nk(s)ds.





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Step2: Construct Temporal Component 0(

)

Consider a dense subset of [0, S]: D := {Sq : q is a rational in[0, 1]}. For s1 = q1S = S (q1 = 1), let 0(s1) = T.

For s2 = q2S, consider s20 Nk(s)dsk. Without loss ofgenerality, by passing to a subsequence of {Nk}k, we supposethe integral sequence above converges to the supremum value.

Then we set 0(s2) = limNks2

0Nk(s)ds.

Similarly, 0(si) = limNk si0 Nk(s)ds, for i N.Generally, for any s, 0(s) is defined with 0 0(s) 1.





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Step2: Construct Temporal Component 0()




0Nk(s)ds.







S S
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Step2: Construct Temporal Component 0()




0Nk(s)ds.





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Step3: Consider The Inclusion Transformed by 0()

y(s) (s)F(y(s)) + (1 (s))G(y(s))K1, where(s) := 0(s).

Through the same method introduced previously, we construct

a new graph of sampled trajectory for each N,

N := {(sNj , xNj ) : j = 0, 1,..., N}, and its related Euler polygonal

arc, yN

(s) := xN

j1 +

ssj1

h (xN

j xN

j1), whenever s [sN

j1, sN

j ],







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Step4: Claim The Approaching

Easily, distH(N, gr yN()) 2max{h, 2c1h}Consider the multifunction

M(s, y) := (s)F(y) + (1

(s))G(y)coK1

We claim, by Compactness of Trajectories Theorem, there

exists a trajectory y() of M and a subsequence {yNk()}k of{yN()}N so that yNk() y() uniformly on [0, S]; that isdistH(gr y

Nk(), gr y(

))

0, as k

.

Finally, by the triangle inequality, we have

distH(Nk, gr y) 0 as k .Wei Cai Impulsive Control Systems






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M(s, y) := (s)F(y) + (1

(s))G(y)coK1



Nk(), gr y(

))

0, as k

.








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M(s, y) := (s)F(y) + (1

(s))G(y)coK1



Nk(), gr y(

))

0, as k

.








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Final Step: Construct Trajectory and Control Measure

Selections f and g, and a function k() : [0, S] coK1 are alsoimplied so that y(s) = (s)f(y(s)) + (1 (s))g(y(s))k(s).We get a graph completion pair (0, )() : [0, S] [0, T] Rm,where (s) is defined as (s) := s0 (1 (s))k(s)ds.And get functions : [0, T] [0, S] and u : [0, T] Rm as(t) := 10 (t+), u(t) := ((t)). Let measure BK[0, T]such that u() is its distribution.Define other components of a solution X as follows. Letx() : [0, T] Rn be given by x(t) = y((t)), and the functionsyi() for i Ibe as the restriction of y() to each atom Ii.





Weak Invariance

Strong Invariance
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Variance properties of trajectories satisfying the system

represent its stability somehow by testing if trajectories remain

in a target set.We will show the weak invariance of impulsive control system

(4): the existence of a characterized trajectory lying in a closed

set C over all slow and fast times.

With an additional assumption, we also figure out the stronginvariance of the system: all trajectories lie in C.





Weak Invariance

Strong Invariance
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Weak Invariance

Strong Invariance
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Weak Invariance

Strong Invariance
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Weak Invariance

Definition

Given C Rn a closed set, the system is weak invariant on C ifand only if for any S > 0 and x0 C, there exist a timeT

[0, S], a measure

BK[0, T] and a three-tuple solution

X of the system such that x(t) C for all t [0, T] and foreach fast time arc {yi()}, yi(s) C for all s Ii.

Theorem

The system(3.1) is weak invariant on a closed set C if and onlyif for each x0 C and NpC(x0), there exist [0, 1] andv F(x0) + (1 )G(x0)K1 so that

, v 0.Wei Cai Impulsive Control Systems




Weak Invariance

Strong Invariance
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Proof of Weak Invariance (=)

Supposing for each x0 C and NpC(x0), there exist [0, 1] and v F(x0) + (1 )G(x0)K1 so that , v 0,we need to show it implies the weak invariance on closed set C.

Let S > 0, x0 C and N N. A sampled trajectory{sj, xj} : j = 0, 1,..., N, satisfies

for a c(xj) projC(xj), jhfj + (1 j)hgjkj, xj c(xj) 0.

By Sampling Method Theorem, there exists a solution X sothat the graphes of sampled trajectories converge to the graph

of y(). We claim y(s) C for all s [0, S].





Weak Invariance

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Proof of Weak Invariance (=)

In fact, x0 C,dC(x1) x1 x0 0hf0 + (1 0)g0k0 2hc1.

d2C(x2) x2 c(x1)2 8h2c21 ,

Generally, d2C(xj) 4jh2c21 4Shc21 .





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Proof of Weak Invariance (=)Suppose the weak invariance holds. Let x0 C, and consider asolution X with x(0) = x0 and the normalized graphcompletion (0, )(). For NPC(x0), a well known fact is thatthere is a > 0 satisfying

, x x0 x x02, for all x C.

Since 0 0(s) 1, we have 0(s) s. So there exists asequence {sj} decreasing and converging to 0 and such thatthe following limit exists:

:= limj+

0(sj)

sj= 0(0).






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Proof of Weak Invariance (=)If time t = 0 is an atom with (0+) = a> 0, then = 0 and fora large j, sj 10 (0) = [0, a]. By weak variance, a trajectoryy() satisfies y(s) G(y(s)) (s) and (s) K1 a.e on [0, a].Moreover,

y(sj) x0sj

=1

sj

sj0

y(s)ds G(x0)K1 + o(j),

where o(j) 0. Thus, y(sj)x0sj has at least one cluster pointdenoted by v G(x0)K1. By the well known fact,

, v = limj

, y(sj) x0sj

limj

sjy(sj) x02 = 0.






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Proof of Weak Invariance (=)If time t = 0 is not an atom and let tj := 0(sj). There istrajectory y() corresponding to solution X satisfies

y(sj) x0sj

=1

sj sj

0

f(s) 0(s)ds+1

sj sj

0

g(s) (s)ds.

We also can prove, by pass onto subsequence,

v := lim

j

y(sj) x0sj

F(x0) + (1

)G(x0)K1.

, v = limj

, y(sj) x0sj

limj

sjy(sj) x02 = 0.


Introduction





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Strong Invariance

Both F and G are required to be locally Lipschitz.

Definition

The system (4) is called strong invariance on a closed set

CR

n if for every x0

C and any T > 0, all measure BK[0, T] and all corresponding three-tuple solution X ofsystem with x(0) = x0 satisfy that x(t) C for all t [0, T]and yi(s) C as s Ii for each fast time arc {yi()}i.

TheoremThe system (4) is strong invariant on a closed set C if and only

if for each x C and NPC(x) we havev, 0 for all [0, 1] and every v F(x) + (1 )G(x)K1.


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Proof of Strong Invariance (=)

Suppose that system (4) is strongly invariant on a closed set C.

Any arc y() corresponding to solution X that satisfies

y(s) F(y(s)) 0(s) + G(y(s)) (s),remains within the set C.

For any fixed x C, let be any number in [0, 1] and letv F(x) + (1 )G(x)K1 arbitrarily, or v := f + (1 )gk.We need to show v, 0.


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For any y, we define v(y) to be the closest point to v inF(y) + (1 )G(y)k. Note that v(x) = v and the multifunctionF + (1

)Gk is locally Lipschitz.

V(y) :=

{v(y)

}.

For S = 1, consider BK([0, ]) so that 0(s) := s and(s) := (1 )ks represent a normalized graph completionwith this measure on [0, 1]. y() satisfies y F + (1 )Gk.We see the system y

V(y) is also weakly invariant, for the

point x C and v = v(x) V(x), we getv, 0, for all NPC(x).


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Now takeT 0 and BK([0, T]) arbitrarily. Supposedly, foreach x C and NPC(x), we have v, 0 for all [0, 1]and for every v F(x) + (1 )G(x)K1. We need to show thesystem (4) is strongly invariant on C. For any x0 C, let X bea solution of (4) with x(0) = x0. The given condition impliesthat for all y C,

maxv, 0, NP

C(y).


Introduction





Weak Invariance

Strong Invariance
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Consider y F(y) 0(s) + (1 0(s))G(y)K1, and M(s, y) :=F(y) 0(s) + (1 0(s))G(y)K1, where S := 10 (T+).

Given an arc y() of system (4), there exists a selectionf of Msuch that y = f(s, y), y(s) = x0.

Let m > 0 s.t. any y() satisfies y(t) x0 < m, for s [0, S].Consider any y x0 + mB and c projC(y). y c NPC(c).

Sincef(s, y) M(s, y), there exists v M(s, c) such thatv f(s, y) Lc y = LdC(y). By v, y c 0, we

deduce f(s, y), y c LdC(y)2.


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Weak Invariance

Strong Invariance
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Consider y F(y) 0(s) + (1 0(s))G(y)K1, and M(s, y) :=F(y) 0(s) + (1 0(s))G(y)K1, where S := 10 (T+).

Given an arc y() of system (4), there exists a selectionf of Msuch that y = f(s, y), y(s) = x0.

Let m > 0 s.t. any y() satisfies y(t) x0 < m, for s [0, S].Consider any y x0 + mB and c projC(y). y c NPC(c).

Sincef(s, y) M(s, y), there exists v M(s, c) such thatv f(s, y) Lc y = LdC(y). By v, y c 0, we

deduce f(s, y), y c LdC(y)2.


Introduction





Weak Invariance

Strong Invariance
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Then for any 0 < s S,

d

2

C(y(s)) d2

C(y()) 2s

f(r, y(r)), y() c() dr.

With both sides divided by s , and taking limit s 0,we get ddsd

2C(y(s)) 2Ld2C(y(s)).

So,d

dsdC(y(s)) LdC(y(s)), s [0, S], dC(y(0)) = 0,which implies dC(y(s)) = 0 by Gronwall inequality.


Introduction





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The transformed impulsive system, dx F(y)0(s) + G(y)(s),can be viewed an direct extension on autonomous system,

dx F(x(t))dt. We show the minimal time for x / C defined as

TC(x) := inf{S : there exists y() satisfying (4)with y(0) = x and y(S) C}.is the unique solution of Hamilton-Jacobi problem. However, we

need to investigate the complete HJ Theory by defining the

minimal time function on real time as

TC(x) := inf{T = 0(S) : (0(), ()) is defined as in Xand y() satisfies (4)}.


Introduction





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More problems coming with it are as follows,

Figure out a specific way to seek the minimal time by

solving HJ problemDevelop numerical methods to achieve this optimization

objective

General optimal control problem on impulsive control

system

Connect with hybrid systems

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impulsive control systems

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