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
Department of MathematicsLouisiana State University
Dissertation DefenseApril 28, 2009
Wei Cai 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
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
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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.
Wei Cai Impulsive Control Systems
I d i
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Introduction
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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).
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I t d ti
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Introduction
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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).
Wei Cai Impulsive Control Systems
Introduction
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Introduction
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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).
Wei Cai Impulsive Control Systems
Introduction
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Introduction
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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).
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Introduction
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Introduction
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Background
outline
Outline
1 Impulsive Control Systems and Their Trajectories
Impulsive Control Systems
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
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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
Insight Into Measure
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
Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Impulsive Control Systems
Insight Into Measure
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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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
p y
Insight Into Measure
Assumptions
Their Trajectories
Graph Completions
Insight Into Measure
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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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
p y
Insight Into Measure
Assumptions
Their Trajectories
Graph Completions
Insight Into Measure
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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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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.
Wei Cai Impulsive Control Systems
Introduction
I l i C l S d Th i T j i
Impulsive Control Systems
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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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.
Wei Cai Impulsive Control Systems
Introduction
I l i C t l S t d Th i T j t i
Impulsive Control Systems
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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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)
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Impulsive Control Systems and Their Trajectories
Impulsive Control Systems
Insight Into Measure
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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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)
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Introduction
Impulsive Control Systems and Their Trajectories
Impulsive Control Systems
Insight Into Measure
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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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.
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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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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Impulsive Control Systems and Their Trajectories
A New Sampling Method
Invariance Properties
Open Problems and Future Work
Insight Into Measure
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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p y j
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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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p y j
A New Sampling Method
Invariance Properties
Open Problems and Future Work
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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A New Sampling Method
Invariance Properties
Open Problems and Future Work
Some Useful Theorems
A Sampling Method for Impulsive Systems
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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Invariance Properties
Open Problems and Future Work
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A Sampling Method for Impulsive Systems
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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IntroductionImpulsive Control Systems and Their Trajectories
A N S li M h d
Time Discretization and Euler Solutions
S U f l Th
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A New Sampling Method
Invariance Properties
Open Problems and Future Work
Some Useful Theorems
A Sampling Method for Impulsive Systems
Some Useful Theorems
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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A Sampling Method for Impulsive Systems
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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A Sampling Method for Impulsive Systems
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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A Sampling Method for Impulsive Systems
Graph of Sampled Trajectories
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 New Sampling Method
Invariance Properties
Open Problems and Future Work
Some Useful Theorems
A Sampling Method for Impulsive Systems
Graph of Sampled Trajectories
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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IntroductionImpulsive Control Systems and Their Trajectories
A New Sampling Method
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A New Sampling Method
Invariance Properties
Open Problems and Future Work
Some Useful Theorems
A Sampling Method for Impulsive Systems
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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A Sampling Method for Impulsive Systems
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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IntroductionImpulsive Control Systems and Their Trajectories
A New Sampling Method
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A Sampling Method for Impulsive Systems
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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A Sampling Method for Impulsive Systems
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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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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A Sampling Method for Impulsive Systems
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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S S
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A Sampling Method for Impulsive Systems
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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I i P ti
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A Sampling Method for Impulsive Systems
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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A Sampling Method for Impulsive Systems
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
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A Sampling Method for Impulsive Systems
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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A Sampling Method for Impulsive Systems
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
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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.
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Invariance Properties
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.
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Strong Invariance
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p
Open Problems and Future Work
Invariance Properties
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.
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Weak Invariance
Strong Invariance
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p
Open Problems and Future Work
Invariance Properties
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.
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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
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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].
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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.
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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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Proof of Strong Invariance (=)
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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Proof of Strong Invariance (=)
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).
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Proof of Strong Invariance (=)
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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Proof of Strong Invariance (=)
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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Proof of Strong Invariance (=)
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.
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Open Problems and Future Work
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)}.
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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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