Chapter 19
Ordinary Dierential
Equations
19.1
Classes of ODE
In this chapter we will deal with the class of functions satisfying the
following ordinary dierential equation
{˙ (w) = i (w> { (w)) for almost all w 5 [w0 > w0 + ]
{ (w0 ) = {0
i :R×X $X
(19.1)
where i is a nonlinear function and X is a Banach space (any concrete
space of functions). The Cauchy’s problem for (19.1) consists in resolving of (19.1), or, in other words, in finding a function { (w) which
satisfies (19.1).
For the simplicity we also will use the following abbreviation:
- ODE meaning an ordinary dierential equation,
- DRHS meaning the discontinuous right-hand side.
Usually the following three classes of ODE (19.1) are considered:
1) Regular ODE:
571
572
Chapter 19. Ordinary Dierential Equations
i (w> {) is continuous in both variables. In this case { (w), satisfying (19.1), should be continuous dierentiable, i.e.,
{ (w) 5 F 1 [w0 > w0 + ]
(19.2)
2) ODE of the Caratheodory’s type:
i (w> {) in (19.1) is measurable in w and continuous in {.
3) ODE with discontinuous right-hand side:
i (w> {) in (19.1) is continuous in w and discontinuous in {. In
fact, this type of ODE equation is related to the dierential
inclusion:
{˙ (w) 5 I (w> { (w))
(19.3)
where I (w> {) is a set in R × X . If this set for some pair (w> {)
consists of one point, then I (w> {) = i (w> {).
19.2
Regular ODE
19.2.1
Theorems on existence
Theorem based on the contraction principle
Theorem 19.1 (on local existence and uniqueness) Let i (w> {)
be a continuous in w on [w0 > w0 + ], ( 0) and for any w 5 [w0 > w0 + ]
it satisfies the, so-called, local Lipschitz condition in {, that is,
there exists constant f> Oi A 0 such that
ki (w> {)k f
(19.4)
ki (w> {1 ) i (w> {2 )k Oi k{1 {2 k
for all w 5 [w0 > w0 + ] and all {> {1 > {2 5 Eu ({0 ) where
Eu ({0 ) := {{ 5 X | k{ {0 k u}
Then the Cauchy’s problem (19.1) has a unique solution on the timeinterval [w0 > w0 + 1 ], where
©
ª
1 ? min u@f> O31
(19.5)
i >
19.2. Regular ODE
573
Proof.
1) First, show that the Cauchy’s problem (19.1) is equivalent to
finding the continuos solution to the following integral equation
Zw
{ (w) = {0 +
i (v> { (v)) gv
(19.6)
v=w0
Indeed, if { (w) is a solution of (19.1), then, obviously, it is a dierentiable function on [w0 > w0 + 1 ]. By integration of (19.1) on [w0 > w0 + 1 ]
we obtain (19.6). Inverse, suppose { (w) is continuos function satisfying
(19.6). Then, by the assumption (19.4) of the theorem, it follows
ki (v> { (v)) i (v0 > { (v0 ))k =
k[i (v> { (v)) i (v> { (v0 ))] + [i (v> { (v0 )) i (v0 > { (v0 ))]k ki (v> { (v)) i (v> { (v0 ))k + ki (v> { (v0 )) i (v0 > { (v0 ))k Oi k{ (v) { (v0 )k + ki (v> { (v0 )) i (v0 > { (v0 ))k
This implies that if v> v0 5 [w0 > w0 + 1 ] and v $ v0 , then the right
hand-side of the last inequality tends to zero, and, hence, i (v> { (v))
is continuous at each point of the interval [w0 > w0 + 1 ]. And, moreover,
we also obtain that { (w) is dierentiable on this interval, satisfies (19.1)
and { (w0 ) = {0 .
2) Using this equivalence, let us introduce the Banach space
F [w0 > w0 + 1 ] of abstract continuous functions { (w) with values in X
and with the norm
k{ (w)kF :=
max
wM[w0 >w0 +1 ]
k{ (w)kX
(19.7)
Consider in F [w0 > w0 + 1 ] the ball Eu ({0 ) and notice that the nonlinear
operator x : F [w0 > w0 + 1 ] $ F [w0 > w0 + 1 ] defined by
Zw
x ({) = {0 +
v=w0
i (v> { (v)) gv
(19.8)
574
Chapter 19. Ordinary Dierential Equations
transforms Eu ({0 ) in to Eu ({0 ) since
°
° w
°
°Z
°
°
°
i
(v>
{
(v))
gv
kx ({) {0 k = max °
°
wM[w0 >w0 +1 ] °
°
°
v=w0
Zw
max
ki (v> { (v))k gv 1 f ? u
wM[w0 >w0 +1 ]
v=w0
Moreover, the operator x is a contraction (see Definition 14.20) on
Eu ({0 ). Indeed, by the local Lipschitz condition (19.4), it follows
°
° w
°
°Z
°
°
°
kx ({1 ) x ({2 )k = max °
[i (v> {1 (v)) i (v> {2 (v))] gv°
°
wM[w0 >w0 +1 ] °
°
v=w0
Zw
max
ki (v> {1 (v)) i (v> {2 (v))k gv wM[w0 >w0 +1 ]
v=w0
1 Oi k{1 {2 kF = t k{1 {2 kF
where t := 1 Oi ? 1 for small enough u. Then, by Theorem (the
contraction principle) 14.17, we conclude that (19.6) has a unique
solution { (w) 5 F [w0 > w0 + 1 ]. Theorem is proven.
Corollary 19.1 If in the conditions of Theorem 19.1 the Lipschitz
condition (19.4) is fulfilled not locally, but globally, that is for all
{1 > {2 5 X (that corresponds with the case u = 4), then the Cauchy’s
problem (19.1) has a unique solution for [w0 > w0 + ] for any big
enough.
Proof. It directly follows from Theorem 19.1 if take u $ 4. But
here we prefer to present also another proof based on another type of
norm dierent from (19.7). Again, let us use the integral equivalent
form (19.6). Introduce in the Banach space F [w0 > w0 + 1 ] the following
norm equivalent to (19.7):
k{ (w)kmax :=
°
°
max °h3Oi w { (w)°X
wM[w0 >w0 +]
(19.9)
19.2. Regular ODE
575
Then
Zw
kx ({1 ) x ({2 )k Oi
v=w0
Zw
¡
¢
hOi v h3Oi v k{1 (v) {2 (v)kF gv Oi
v=w0
Zw
hOi v k{1 (w) {2 (w)kmax gv =
Oi
Zw
Oi
h3Oi v hOi v k{1 (v) {2 (v)kF =
v=w0
¡
¢
hOi v gv k{1 (w) {2 (w)kmax = hOi w 1 k{1 (w) {2 (w)kmax
v=w0
Multiplying this inequality by h3Oi w and taking
max
wM[w0 >w0 +]
we get
¡
¢
kx ({1 ) x ({2 )kmax 1 h3Oi k{1 (w) {2 (w)kmax
Since t := 1 h3Oi ? 1 we conclude that x is a contraction. Taking
then big enough we obtain the result. Corollary is proven.
Remark 19.1 Sure, the global Lipschitz condition (19.4) with u =
4 holds for very narrow class of functions which is known as the
class of "quasi-linear" functions, that’s why Corollary 19.1 is too
conservative. On the other hand, the conditions of Theorem 19.1 for
finite (small enough) u ? 4 is not so restrictive valid for any function
satisfying somewhat mild smoothness conditions.
Remark 19.2 The main disadvantage of Theorem 19.1 is that the
solution of the Cauchy’s problem (19.1) exists only on the interval
[w0 > w0 + 1 ] (where 1 satisfies (19.5)), but not at the complete interval
[w0 > w0 + ], that is very restrictive. For example, the Cauchy problem
{˙ (w) = {2 (w) > { (0) = 1
has the exact solution { (w) =
for all [0> 4).
1
that exists only on [0> 1) but not
1w
576
Chapter 19. Ordinary Dierential Equations
The theorem presented below gives a constructive (direct) method
of finding a unique solution of the problem (19.1). It has several
forms. Here we present the version of this result which does not use
any Lipschitz conditions: neither local, no global.
Theorem 19.2 (Picard-Lindelöf, 1890) Consider the Cauchy’s
problem (19.1) where the function i (w> {) is continuous on
V := {(w> {) 5 R1+q | |w w0 | w0 , k{ {0 kF u}
(19.10)
C
and the partial derivative
i : V $ Rq is also continuous on V.
C{
Define the sets
°
°
°C
°
°
i
(w>
{)
M := max ki (w> {)k, O := max °
(19.11)
°
(w>{)MV
(w>{)MV ° C{
and choose the real number such that
0 ? u, M u, t := O ? 1
(19.12)
Then
1) the Cauchy’s problem (19.1) has unique solution on V;
2) the sequence {{q (w)} of functions generated iteratively by
Zw
{q+1 (w) = {0 +
i (v> {q (v)) gv
(19.13)
v=w0
{0 (w) = {0 , q = 0> 1> ===; w0 w + w0
to { (w) in the Banach space X with the norm (19.7) converges
geometrically as
k{q+1 (w) { (w)kF t q+1 k{0 { (w)kF
(19.14)
Proof. Consider the integral equation (19.6) and the integral operator x (19.8) given on V. So, (19.6) can be represented as
x ({ (w)) = { (w) , { (w) 5 Eu ({0 )
19.2. Regular ODE
577
where x : P $ X . For all w 5 [w0 > w0 + ] we have
° w
°
°Z
°
°
°
i (v> {q (v)) gv°
kx ({ (w)) {0 k = max °
°
°
wM[w0 >w0 +] °
°
v=w0
max (w w0 ) max ki (w> {)k M u
wM[w0 >w0 +]
(w>{)MV
i.e., x (P) P. By the classical mean value theorem 16.5
°
°
°C
°
° O k{ |k
ki (w> {) i (w> |)k = °
({
|)
i
(w>
})
|
}M[{>|]
° C{
°
and, hence,
°
° w
°
°Z
°
°
°
kx ({ (w)) x (| (w))k = max °
[i
(v>
{
(v))
i
(v>
|
(v))]
gv
°
wM[w0 >w0 +] °
°
°
v=w0
O max k{ (w) | (w)k = t k{ (w) | (w)kF
wM[w0 >w0 +]
Applying now the contraction principle we obtain that (19.6) has a
unique solution { 5 Eu ({0 ). We also have
Zw
k{q+1 (w) { (w)kF =
max
wM[w0 >w0 +]
v=w0
[i (v> {q (v)) i (v> { (v))] gv
t k{q (w) { (w)kF tq+1 k{0 { (w)kF
Theorem is proven.
Theorem based on the Schauder fixed-point theorem
The next theorem to be proved drops the assumption of Lipschitz
continuity but, also, the assertion of uniqueness.
Theorem 19.3 (Peano, 1890) Consider the Cauchy´s problem (19.1)
where the function i (w> {) is continuous on V (19.10) where the real
parameter is selected in such a way that
0 ? u, M u
(19.15)
Then the Cauchy’s problem (19.1) has at least one solution on V.
578
Chapter 19. Ordinary Dierential Equations
Proof.
a) The Schauder fixed-point theorem use (Zeidler 1995). By the
same arguments as in the proof of Theorem 19.2 it follows that the
operator x : Eu ({0 ) $ Eu ({0 ) is compact (see Definition ). So, by
the Schauder fixed-point theorem 18.20 we conclude that the operator
equation x ({ (w)) = { (w), { (w) 5 Eu ({0 ) has at least one solution.
This completes the proof.
b) Direct proof (Hartman 2002). Let A 0 and {0 (w) 5 F 1 [w0 > w0 ]
satisfy on [w0 > w0 ] the following conditions: {0 (w) = {0 , k{0 (w) {0 k
u and k{00 (w)k g. For 0 ? % define a function {% (w) on
[w0 > w0 + %] by putting {% (w0 ) = {0 on [w0 > w0 ] and
Zw
{% (w) = {0 +
i (v> {% (v %)) gv
(19.16)
v=w0
on [w0 > w0 + %]. Note that {% (w) is F 0 -function on [w0 > w0 + %] satisfying
k{% (w) {0 k u and k{% (w) {% (v)k g |w v|
Thus, for the family of functions {{%q (w)}, %q $ 0 whereas q $
4 it follows that the limit {(w) = lim {%q (w) exists uniformly on
[w0 > w0 + ] > that implies that
q<"
ki (w> {%q (w %q )) i (w> { (w))k $ 0
uniformly as q $ 4. So, term-by-term integration of (19.16) with
% = %q , gives (19.6) and, hence, { (w) is a solution of (19.1).
The following corollary of the Peano’s existence theorem is often
used.
Corollary 19.2 ((Hartman 2002)) Let i (w> {) be a continuous on
an open (w> {) - set of E R1+q satisfying ki (w> {)k M. Let also
E0 be a compact subset of E. Then there exists a A 0, depending on
E, E0 and M, such that if (w0 > {0 ) 5 E0 , then (19.6) has a solution on
|w w0 | .
19.2. Regular ODE
19.2.2
579
Dierential inequalities, extension and
uniqueness
The most important technique in ODE-theory involves the "integration" of the, so-called, dierential inequalities. In this subsection we
present results dealing with this integration process and extensively
used throughout, and, then there will be presented its immediate application to the extension and uniqueness problems.
Bihari and Gronwall-Bellman inequalities
Lemma 19.1 ((Bihari 1956)) Let
1) y (w) and (w) be nonnegative continuous functions on [w0 > 4),
that is,
y (w) 0> (w) 0 ;w 5 [w0 > 4) > y (w) > (w) 5 F [w0 > 4)
(19.17)
2) for any w 5 [w0 > 4) the following inequality holds
Zw
y (w) f +
( ) x (y ( )) g
(19.18)
=w0
where f is a positive constant (f A 0) and x (y) is a positive
non-decreasing continuous function, that is,
0 ? x (y) 5 F [w0 > 4) ;y 5 (0> ȳ) > ȳ 4
Denote
Zy
[ (y) :=
v=f
gv
(0 ? y ? ȳ)
x (v)
(19.19)
(19.20)
If in addition
Zw
( ) g ? [ (ȳ 0) > w 5 [w0 > 4)
=w0
(19.21)
580
Chapter 19. Ordinary Dierential Equations
then for any w 5 [w0 > 4)
3
4
Zw
y (w) [31 C
( ) g D
(19.22)
=w0
where [31 (|) is the function inverse to [ (y), that is,
| = [ (y) > y = [31 (|)
(19.23)
In particular, if ȳ = 4 and [ (4) = 4, then the inequality (19.22)
is fulfilled without any constraints.
Proof. Since x (y) is a positive non decreasing continuous function
the inequality (19.18) implies that
3
4
Zw
x (y (w)) x Cf +
( ) x (y ( )) g D
=w0
and
(w) x (y (w))
4 (w)
Zw
x Cf +
( ) x (y ( )) g D
3
=w0
Integrating the last inequality, we obtain
Zw
v=w0
Zw
(v) x (y (v))
3
4 gv (v) gv
Zv
v=w0
x Cf +
( ) x (y ( )) g D
=w0
Denote
Zw
z (w) := f +
( ) x (y ( )) g
=w0
Then evidently
ż (w) = (w) x (y (w))
(19.24)
19.2. Regular ODE
581
Hence, in view of (19.20), the inequality (19.24) may be represented
as
z(w)
Zw
Z
ż (v)
gz
gv =
x (z (v))
x (z)
z=z(w0 )
v=w0
Zw
= [ (z (w)) [ (z (w0 )) (v) gv
v=w0
Taking into account that z (w0 ) = f and [ (z (w0 )) = 0, from the last
inequality it follows
Zw
[ (z (w)) (v) gv
(19.25)
v=w0
Since
[´(y) =
1
(0 ? y ? ȳ)
x (y)
the function [ (y) has the uniquely defined continuous monotonically
increasing inverse function [31 (|) defined within the open interval
([ (+0) > [ (ȳ 0)). Hence, (19.25) directly implies
3
Zw
Zw
(v) gvD
( ) x (y ( )) g [31 C
z (w) = f +
=w0
4
v=w0
that, in view of (19.18), leads to (19.22). Indeed,
3
Zw
Zw
( ) x (y ( )) g [31 C
y (w) f +
=w0
4
(v) gvD
v=w0
The case ȳ = 4 and [ (4) = 4 is evident. Lemma is proven.
Corollary 19.3 Taking in (19.22)
x (y) = y p (p A 0> p 6= 1)
582
Chapter 19. Ordinary Dierential Equations
it follows
5
6
Zw
y (w) 7f13p + (1 p)
( ) g 8
1
p1
for 0 ? p ? 1 (19.26)
=w0
and
5
63
Zw
y (w) f 71 (1 p) fp31
( ) g 8
1
p1
=w0
for
Zw
p A 1 and
( ) g ?
=w0
1
(p 1) fp31
Corollary 19.4 ((Gronwall 1919) ) If y (w) and (w) are nonnegative continuous functions on [w0 > 4) verifying
Zw
y (w) f +
( ) y ( ) g
(19.27)
=w0
then for any w 5 [w0 > 4) the following inequality holds:
3 w
4
Z
y (w) f exp C (v) gvD
v=w0
This results remains true if f = 0.
Proof. Taking in (19.18) and (19.20)
x (y) = y
we obtain (19.193) and, hence, for the case f A 0
Zy
[ (y) :=
v=f
³y ´
gv
= ln
v
f
(19.28)
19.2. Regular ODE
583
and
[31 (|) = f · exp (|)
that implies (19.28). The case f = 0 follows from (19.28) applying
f $ 0.
Dierential inequalities
Here we completely follow (Hartman 2002).
Definition 19.1 Let i (w> {) be a continuous function on a plane (w> {)set E. By a maximal solution {0 (w) of the Cauchy’s problem
{(w)
˙ = i (w> {), { (w0 ) = {0 5 R
(19.29)
is meant a solution of (19.29) on a maximal interval of existence such
that if { (w) is any solution of (19.29) then
{ (w) {0 (w)
(19.30)
holds (by component-wise) on the common interval of existence of { (w)
and {0 (w). The minimal solution is similarly defined.
Lemma 19.2 Let i (w> {) be a continuous function on a rectangle
©
ª
V + := (w> {) 5 R2 | w0 w w0 + w0 , k{ {0 kF u
(19.31)
and on V +
ki (w> {)k M and := min {; u@M}
Then the Cauchy’s problem (19.29) has a solution on [w0 > w0 + ) such
that every solution { = {(w) of {(w)
˙
= i (w> {), { (w0 ) {0 satisfies
(19.30) on [w0 > w0 + ).
Proof. Let 0 ? 0 ? . Then, by the Peano’s existence theorem
19.3, the Cauchy problem
1
{(w)
˙ = i (w> {) + > {(w0 ) = {0
q
(19.32)
has a solution { = {q (w) on [w0 > w0 + 0 ] if q is su!ciently large. Then
there exists a subsequence {qn }n=1>2>== such that the limit {0 (w) =
584
Chapter 19. Ordinary Dierential Equations
lim {qn (w) exists uniformly on [w0 > w0 + 0 ] and {0 (w) is a solution of
n<"
(19.29). To prove that (19.30) holds on [w0 > w0 + 0 ] it is su!cient to
verify
{ (w) {q (w) on [w0 > w0 + 0 ]
(19.33)
for large enough q. If this is not true, then there exists a w = w1 5
(w0 > w0 + 0 ) such that { (w1 ) A {q (w1 ). Hence there exists a largest w2
on [w0 > w1 ) such that { (w2 ) = {q (w2 ) and { (w) A {q (w). But by (19.32)
1
{0q (w2 ) = {0 (w2 ) + h, so that {q (w) A { (w) for w A w2 near w2 . This
q
contradiction proves (19.33). Since 0 ? is arbitrary, the lemma
follows.
Corollary 19.5 Let i (w> {) be a continuous function on an open set E
and (w0 > {0 ) 5 E R2 . Then the Cauchy’s problem (19.29) a maximal
and minimal solution near (w0 > {0 ).
Right derivatives
Lemma 19.3
1. If q = 1 and { 5 F 1 [d> e] then |{ (w)| has a right derivative
1
[|{ (w + k)| |{ (w)|]
0?k<0 k
(19.34)
{0 (w) sign ({ (w)) if { (w) 6= 0
|{0 (w)|
if { (w) = 0
(19.35)
GU |{ (w)| := lim
such that
½
GU |{ (w)| =
and
|GU |{ (w)|| = |{0 (w)|
(19.36)
2. If q A 1 and { 5 F 1 [d> e] then k{ (w)k has a right derivative
1
[k{ (w + k)k k{ (w)k]
0?k<0 k
GU k{ (w)k := lim
(19.37)
such that on w 5 [d> e)
kGU k{ (w)kk = max GU |{n (w)| n=1>===>q
k{0 (w)k := max {|{01 (w)| > ===> |{0q (w)|}
(19.38)
19.2. Regular ODE
585
Proof. The assertion 1) is clear when { (w) 6= 0 and the case
{ (w) = 0 follows from the identity
{ (w + k) = { (w) + k{0 (w) + r (k) = k [{0 (w) + r (1)]
so that, in general, when k $ 0
|{ (w + k)| = |{ (w)| + k [|{0 (w)| + r (1)]
The multidimensional case 2) follows from 1) if take into account that
|{n (w + k)| = |{n (w)| + k [|{0n (w)| + r (1)]
Taking the max of these identities, we obtain
n=1>===>q
k{ (w + k)k = k{ (w)k + k
max
n=1>===>q
|{0n
¸
(w)| + r (1)
whereas k $ 0. This proves (19.38).
Example 19.1 Let { (w) := (w w0 )2 . Then {0 (w) := 2 (w w0 ) is continuous and, hence, { (w) 5 F 1 . By Lemma 19.3 it follows GU |{ (w)| =
2 |w w0 |.
Dierential inequalities
The next theorem deals with the integration of dierential inequalities
and is most used in the ODE-theory.
Theorem 19.4 ((Hartman 2002)) Let i (w> {) be continuous on an
open (w> {)-set E R and {0 (w) be the maximal solution of (19.6). Let
y (w) be a continuous on [w0 > w0 + ] function such that
y (w0 ) {0 > (w> y) 5 E
GU y (w) i (w> y (w))
¾
(19.39)
Then, on the common interval of existence of {0 (w) and y (w)
y (w) {0 (w)
(19.40)
586
Chapter 19. Ordinary Dierential Equations
Remark 19.3 If the inequalities (19.39) are reversed with the leftderivative GO y (w) instead of GU y (w), then the conclusion (19.40)
must be replaced by y (w) {0 (w) where {0 (w) is the minimal solution
of (19.6).
Proof. It is su!cient to show that there exists a A 0 such that
(19.40) holds for [w0 > w0 + ]. Indeed, if this is the case and if y (w) and
{0 (w) are defined on [w0 > w0 + ], then it follows that the set of w-values,
where (19.40) holds, can not have an upper bound dierent from .
Let in Lemma 19.2 q A 0 be large enough and be chosen independent
of q such that (19.32) has a solution { = {q (w) on [w0 > w0 + ]. In view
of Lemma 19.2 it is su!cient to verify that y (w) {q (w) on [w0 > w0 + ].
But the proof of this fact is absolutely identical to the proof of (19.33).
Theorem is proven.
In fact, the following several consequences of this theorem are
widely used in the ODE-theory.
Corollary 19.6 If y (w) is continuous on [w0 > wi ] and GU y (w) 0 when
w 5 [w0 > wi ], then
y (w) y (w0 ) for any w 5 [w0 > wi ]
(19.41)
Corollary 19.7 (Lemma on dierential inequalities) Let i (w> {),
{0 (w) be as in Theorem 19.4 and j (w> {) be continuous on an open
(w> {)-set E R2 satisfying
j (w> {) i (w> {)
(19.42)
Let also y (w) be a solution of the following ODE:
ẏ (w) = j (w> y (w)), y (w0 ) := y0 {0
(19.43)
on [w0 > w0 + ]. Then
y (w) {0 (w)
(19.44)
holds on any common interval of existence of y (w) and {0 (w) to the
right of w0 .
19.2. Regular ODE
587
Corollary 19.8 Let {0 (w) be the maximal solution of
{˙ (w) = i (w> { (w)) > { (w0 ) := {0 5 R
and {0 (w) be the minimal solution of
{˙ (w) = i (w> { (w)) > { (w0 ) := {0 0
Let also | = | (w) be a F 1 vector valued function on [w0 > w0 + ] such
that
{0 k| (w0 )k {0
(w> |) 5 E R2
(19.45)
g
(k| (w)k) i (w> k| (w)k)
gw
Then the first (second) of two inequalities
{0 (w) k| (w)k {0 (w)
(19.46)
holds on any common interval of existence of {0 (w) and | (w) (or {0 (w)
and | (w)).
Corollary 19.9 Let i (w> {) be continuous and non-decreasing on {
when w 5 [w0 > w0 + ]. Let {0 (w) be a maximal solution of (19.6) which
exists on [w0 > w0 + ]. Let another continuous function y (w) satisfies on
[w0 > w0 + ] the integral inequality
Zw
y (w) y0 +
i (v> y (v)) gv
(19.47)
v=w0
where y0 {0 . Then
y (w) {0 (w)
(19.48)
holds on [w0 > w0 + ]. This results is false if we omit: i (w> {) is nondecreasing on {.
Proof. Denote by Y (w) the right-hand side of (19.47), so that
y (w) Y (w), and, by the monotonicity property with respect the
second argument, we have
Ẏ (w) = i (w> y (w)) i (w> Y (w))
By Theorem 19.4 we have Y (w) {0 (w) on [w0 > w0 + ]. Thus y (w) {0 (w) that completes the proof.
588
Chapter 19. Ordinary Dierential Equations
Existence of solutions on the complete axis [w0 > 4)
Here we show that the condition ki (w> {)k n k{k guaranties the
existence of the solutions of ODE {˙ (w) = i (w> { (w)), { (w0 ) = {0 5 Uq
for any w w0 . In fact, the following more general result holds.
Theorem 19.5 ((Wintner 1945)) Let for any w w0 and { 5 Rq
¡
¢
({> i (w> {)) [ k{k2
(19.49)
where the function [ satisfies the condition
Z"
v=v0
gv
= 4, [ (v) A 0 as v v0 0
[ (v)
(19.50)
Then the Cauchy’s problem
{˙ (w) = i (w> { (w)) > { (w0 ) = {0 5 Uq
has a solution on the complete semi-axis [w0 > 4) for any {0 5 Rq .
Proof. Notice that for the function z (w) := k{ (w)k2 in view of
(19.49) we have
g
z (w) = 2 ({ (w) > {˙ (w)) = 2 ({ (w) > i (w> { (w))) gw
¢
¡
2[ k{ (w)k2 = 2[ (z (w))
Then by Theorem 19.4 (see (19.44)) it follows that z (w0 ) v0 implies
z (w) v (w), where v (w) satisfies
v̇ (w) = 2[ (v (w)) > v (w0 ) = v0 := k{0 k2
But the solution of the last ODE is always bounded for any finite
w w0 . Indeed,
Zv(w)
gv
(19.51)
= 2 (w w0 )
[ (v)
v=v0
and [ (v) A 0 as v v0 implies that v̇ (w) A 0, and, hence, v (w) A 0 for
all w A w0 . But the solution v (w) can fail to exist on a bounded interval
19.2. Regular ODE
589
[w0 > w0 + d] only if it exists on [w0 > w0 + ] with ? d and v (w) $ 4
if w $ w0 + d. But this gives the contradiction to (19.50) since the
left-hand side of (19.51) tend to infinity and the right-hand side of
(19.51) remains finite and equal to 2d.
Remark 19.4 The admissible choices of [ (v) may be, for example,
F> Fv> Fv ln v>... for large enough v and F as a positive constant.
Remark 19.5 Some generalizations of this theorem can be found in
(Hartman 2002).
Example 19.2 If D (w) is a continuous q × q matrix and j (w) is continuous on [w0 > w0 + d] vector function, then the Cauchy’s problem
{˙ (w) = D (w) { (w) + j (w) > { (w0 ) = {0 5 Uq
(19.52)
has a unique solution { (w) on [w0 > w0 + d]. It follows from the Wintner
theorem 19.5 if take [ (v) := F (1 + v) with F A 0.
The continuous dependence of the solution on a parameter
and on the initial conditions
Theorem 19.6 If the right-hand side of ODE
{˙ (w) = i (w> { (w) > ) > { (w0 ) = {0 5 Uq
(19.53)
is continuous with respect to on [3 > + ] and satisfies the condition
of Theorem 19.1 with the Lipschitz constant Oi which is independent
of , then the solution { (w> ) of (19.53) depends continuously on
5 [3 > + ] 5 Rp as well as on {0 in some neighborhood.
Proof. The proof of this assertion repeats word by word the proof
of Theorem 19.1. Indeed, by the same reasons as in Theorem 19.1,
the solution { (w> ) is a continuous function of both w and if Oi is
independent of . As for the proof of the continuous dependence of the
solution on the initial conditions, it can be transform to the proof of
the continuous dependence of the solution on the parameter. Indeed,
putting
:= w w0 > } := { (w> ) {0
590
Chapter 19. Ordinary Dierential Equations
we obtain that (19.53) is converted to
g
} = i ( + w0 > } + {0 > ) > } (0) = 0
g
where {0 may be considered as a new parameter so that i is continuous
on {0 by the assumption. This proves the theorem.
19.2.3
Linear ODE
Linear vector ODE
Lemma 19.4 The solution { (w) of the linear ODE (or, the corresponding Cauchy’s problem)
{˙ (w) = D (w) { (w), { (w0 ) = {0 5 Rq×q , w w0
(19.54)
where D (w) is a continuous q × q-matrix function, may be presented
as
{ (w) = x (w> w0 ) {0
(19.55)
where the matrix x (w> w0 ) is, so-called, the fundamental matrix of
the system (19.54) and satisfies the following matrix ODE
g
x (w> w0 ) = D (w) x (w> w0 ), x (w0 > w0 ) = L
gw
(19.56)
and fulfills the group property
x (w> w0 ) = x (w> v) x (v> w0 ) ;v 5 (w0 > w)
(19.57)
Proof. Assuming (19.55), the direct dierentiation of (19.55) implies
g
{˙ (w) = x (w> w0 ) {0 = D (w) x (w> w0 ) {0 = D (w) { (w)
gw
So, (19.55) verifies (19.54). Uniqueness of such presentation follows
from Example 19.2. The property (19.57) results from the fact that
{ (w) = x (w> v) {v = x (w> v) x (v> w0 ) { (w0 ) = x (w> w0 ) { (w0 )
Lemma is proven.
19.2. Regular ODE
591
Liouville’s theorem
The next results serves for the demonstration that the transformation
x (w> w0 ) is non-singular (or, has its inverse) on any finite time interval.
Theorem 19.7 (Liouville, 1836) If x (w> w0 ) is the solution to
(19.56), then
; w
?Z
det x (w> w0 ) = exp
=
<
@
trD (v) gv
>
(19.58)
v=w0
Proof. The usual expansion for the determinant det x (w> w0 ) and
the rule for dierentiating the product of scalar functions show that
X
g
det x (w> w0 ) =
det x̃m (w> w0 )
gw
m=1
q
where x̃m (w> w0 ) is the matrix obtained by replacing the m-th row
xm>1 (w> w0 ) > ===> xm>q (w> w0 ) of x (w> w0 ) by its derivatives ẋm>1 (w> w0 ) > ===>
ẋm>q (w> w0 ). But since
ẋm>n (w> w0 ) =
q
X
dm>l (w) xl>n (w> w0 ) > D (w) = kdm>l (w)km=l=1>===>q
l=1
it follows
det x̃m (w> w0 ) = dm>m (w) det x (w> w0 )
that gives
X g
g
det x (w> w0 ) =
det x̃m (w> w0 )
gw
gw
m=1
q
=
q
X
dm>m (w) det x (w> w0 ) = wu {D (w)} det x (w> w0 )
m=1
and, as a result, we obtain (19.58) that completes the proof.
592
Chapter 19. Ordinary Dierential Equations
Corollary 19.10 If for the system (19.54)
ZW
trD (v) gv A 4
(19.59)
v=w0
then for any w 5 [w0 > W ]
det x (w> w0 ) A 0
(19.60)
Proof. It is the direct consequence of (19.58).
ZW
trD (v) gv A 4,
Lemma 19.5 If (19.59) is fulfilled, namely,
v=w0
then the solution { (w) on [0> W ] of the linear non autonomous ODE
{˙ (w) = D (w) { (w) + j (w), { (w0 ) = {0 5 Rq×q , w w0
(19.61)
where D (w) and i (w) are assumed to be continuous matrix and vector
functions, may be presented by the Cauchy formula
5
6
Zw
{ (w) = x (w> w0 ) 7{0 +
x31 (v> w0 ) j (v) gv8
(19.62)
v=w0
where x31 (w> w0 ) exists for all w 5 [w0 > W ] and satisfies
g 31
x (w> w0 ) = x31 (w> w0 ) D (w), x31 (w0 > w0 ) = L
gw
(19.63)
Proof. By the previous corollary, x31 (w> w0 ) exists within the interval [w0 > W ]. The direct derivation of (19.62) implies
5
6
Zw
x31 (v> w0 ) j (v) gv8
{˙ (w) = ẋ (w> w0 ) 7{0 +
v=w0
+5x (w> w0 ) x31 (w> w0 ) j (w) = 6
Zw
D(w)x (w> w0 ) 7{0 +
x31 (v> w0 ) j (v) gv8 + j (w)
v=w0
= D(w){ (w) + j (w)
19.2. Regular ODE
593
that coincides with (19.61). Notice that the integral in (19.62) is well
defined in view of the continuity property of the participating functions
to be integrated. By identities
x (w> w0 ) x31 (w> w0 ) = L
g
[x (w> w0 ) x31 (w> w0 )] = ẋ (w> w0 ) x31 (w> w0 )
gw
g
+ x (w> w0 ) x31 (w> w0 ) = 0
gw
it follows
h
i
g 31
x (w> w0 ) = x31 (w> w0 ) ẋ (w> w0 ) x31 (w> w0 ) =
gw
x31 (w> w0 ) [D (w) x (w> w0 )] x31 (w> w0 ) = x31 (w> w0 ) D (w)
Lemma is proven.
Remark 19.6 The solution (19.62) can be rewritten as
Zw
{ (w) = x (w> w0 ) {0 +
x (w> v) j (v) gv
(19.64)
v=w0
since by (19.57)
x (w> v) = x (w> w0 ) x31 (v> w0 )
(19.65)
Bounds for norm of ODE solutions
Let kDk := sup kD{k where k{k is Euclidean or Chebishev’s type.
k{k=1
Lemma 19.6 Let { (w) be a solution of (19.61). Then
Zw
k{ (w)k (k{ (w0 )k+
v=w0
Zw
kj (v)k gv) exp(
v=w0
kD (v)k gv)
(19.66)
594
Chapter 19. Ordinary Dierential Equations
Proof. By (19.61) it follows
k{˙ (w)k kD (w)k k{ (w)k + kj (w)k
Let y (w) be the unique solution of the following ODE:
ẏ (w) = kD (w)k y (w) + kj (w)k > y (w0 ) = k{ (w0 )k
which solution is
Zw
Zv
Zw
ki (v)k exp(kD (u)k gu)] exp(
kD (v)k gv)
y (w) =[y (w0 ) +
v=w0
u=w0
v=w0
Then, by Lemma 19.7, it follows that k{ (w)k y (w) for any w w0
that gives (19.66).
Corollary 19.11 Similarly, if z (w) is the solution of
ẏ (w) = kD (w)k y (w) kj (w)k > z (w0 ) = k{ (w0 )k
then k{ (w)k y (w) for any w w0 that gives
Zw
k{ (w)k (k{ (w0 )k Zw
kj (v)k gv) exp(
v=w0
kD (v)k gv)
(19.67)
v=w0
Stationary linear ODE
If in (19.1) D (w) = D is a constant matrix, then it is easily to check
that
"
X
1 n n
D(w3w0 )
Dw
D w
x (w> w0 ) := h
where h =
(19.68)
n!
n=0
and (19.62), (19.64) become
5
6
Zw
{ (w) = hD(w3w0 ) 7{0 +
h3D(v3w0 ) j (v) gv8
v=w0
Zw
= hD(w3w0 ) {0 +
v=w0
hD(w3v) j (v) gv
(19.69)
19.2. Regular ODE
595
Linear ODE with periodic matrices
In this subsection we show that the case of variable, but periodic,
coe!cients can be reduced to the case of constant coe!cients.
Theorem 19.8 (Floquet 1883) Let in ODE
{˙ (w) = D (w) { (w)
(19.70)
the matrix D (w) 5 Rq×q (4 ? w ? 4) be a continuous and periodic
of period W , that is, for any w
D (w + W ) = D (w)
(19.71)
Then the fundamental x (w> w0 ) of (19.70) has a representation of the
form
x (w> w0 ) = x̃ (w w0 ) = ] (w w0 ) hU(w3w0 )
] ( ) = ] ( + W )
(19.72)
and U is a constant q × q matrix.
Proof. Since x̃ ( ) is fundamental matrix of (19.70), then x̃ ( + W )
is fundamental too. By the group property (19.57) it follows x̃ ( + W )
= x̃ ( ) x̃ (W ). Since det x̃ (W ) 6= 0 one can represent x̃ (W ) as x̃ (W ) =
hUW and hence
x̃ ( + W ) = x̃ ( ) hUW
(19.73)
So, defining ] ( ) := x̃ ( ) h3U > we get
] ( + W ) =i x̃ ( + W ) h3U( +W ) =
h
x̃ ( + W ) h3UW h3U = x̃ ( ) h3U = ] ( )
that completes the proof.
First integrals and related adjoint linear ODE
Definition 19.2 A function I = I (w> {) : R × Cq $ C, belonging to
F 1 [R × Cq ], is called the first integral to ODE (19.1) if it is constant
596
Chapter 19. Ordinary Dierential Equations
over trajectories of { (w) generated by (19.1), that is, if for any w w0
and any {0 5 Cq
μ
¶
g
C
C
I (w> { (w)) = I (w> { (w)) +
I (w> { (w)) > {˙ (w)
gw
Cw μ
C{
¶
(19.74)
C
C
= I (w> {) +
I (w> { (w)) > i (w> { (w)) = 0
Cw
C{
In the case of linear ODE (19.54) the condition (19.74) is converted
into the following:
μ
¶
C
C
I (w> {) +
I (w> { (w)) > D (w) = 0
(19.75)
Cw
C{
Let us try to find a first integral for (19.54) as a linear form of { (w),
i.e., let us try to satisfy (19.75) selecting I as
I (w> {) = (} (w) > { (w)) := } W (w) { (w)
(19.76)
where } W (w) 5 Cq is from F 1 [Cq ].
The existence of the first integral for ODE (19.1) permits to decrease the order of the system to be integrated since if the equation
I (w> { (w)) = f can be resolved with respect one of components, say,
{ (w) = * (w> {1 (w) > ===> {31 (w) > {+1 (w) > ===> {q (w))
then the order of ODE (19.1) becomes to be equal to (q 1). If one
can find all q first integrals I (w> { (w)) = f ( = 1> ===> q) which are
linearly independent, then the ODE system (19.1) can be considered
to be solved.
Lemma 19.7 A first integral I (w> {) for (19.54) is linear on { (w) as
in (19.76) if and only if
}˙ (w) = DW (w) } (w) > } (w0 ) = }0 5 Rq×q > w w0
(19.77)
Proof. a) Necessity. If a linear I (w> {) = (} (w) > { (w)) is a first
integral, then
g
I (w> { (w)) = (}˙ (w) > { (w)) + (} (w) > {˙ (w)) =
gw
(}˙ (w) > { (w)) + (} (w) > D (w) { (w)) = (}˙ (w) > { (w)) +
(DW (w) } (w) > { (w)) = (}˙ (w) + DW (w) } (w) > { (w))
(19.78)
19.2. Regular ODE
597
Suppose that }˙ (w0 ) + DW (w0 ) } (w0 ) 6= 0 for some w0 w0 . Put
{ (w0 ) := }˙ (w0 ) + DW (w0 ) } (w0 )
Since { (w0 ) = x (w0 > w0 ) {0 and x31 (w0 > w0 ) always exists, then for {0 =
x31 (w0 > w0 ) { (w0 ) we obtain
g
I (w0 > { (w0 )) = (}˙ (w0 ) + DW (w0 ) } (w0 ) > { (w0 ))
gw
= k}˙ (w0 ) + DW (w0 ) } (w0 )k2 6= 0
that is in the contradiction with the assumption that I (w> { (w)) is a
first integral.
b) Su!ciency. It directly results from (19.78) Lemma is proven.
Definition 19.3 The system (19.77) is called the ODE system adjoint to (19.54). For the corresponding inhomogeneous system (19.61)
the adjoint system is
}˙ (w) = DW (w) } (w) j̃ (w) > } (w0 ) = }0 5 Rq×q > w w0
(19.79)
There are several results concerning the joint behavior of (19.54)
and (19.77).
Lemma 19.8 A matrix x (w> w0 ) is a fundamental matrix for the linear
W
ODE (19.54) if and only if (xW (w> w0 ))31 = (x31 (w> w0 )) is a fundamental matrix for the adjoint system (19.77).
Proof. Since x (w> w0 ) x31 (w> w0 ) = L by dierentiation it follows
that
g 31
g
x (w> w0 ) = x31 (w> w0 ) x (w> w0 ) x31 (w> w0 ) = x31 (w> w0 ) D (w)
gw
gw
and, taking the complex conjugate transpose of the last identity gives
¢W
¡
¢W
g ¡ 31
x (w> w0 ) = DW (w) x31 (w> w0 )
gw
The converse is proved similarly.
598
Chapter 19. Ordinary Dierential Equations
Lemma 19.9 The direct (19.61) and the corresponding adjoint (19.79)
linear systems can be presented in the Hamiltonian form, i.e.,
{˙ (w) =
C
C
K (}> {) > }˙ (w) = K (}> {)
C}
C{
(19.80)
where
K (w> }> {) := (}> i (w> {)) = (}> D (w) { + j (w))
(19.81)
is called the Hamiltonian function for the system (19.61). In the
stationary homogeneous case when
{˙ (w) = D{ (w) > { (w0 ) = {0 5 Rq×q > w w0
(19.82)
the Hamiltonian function is a first integral for (19.82).
Proof. The representation (19.80) follows directly from (19.81).
C
In the stationary, when K (w> }> {) = 0, we have
Cw
μ
¶ μ
¶
g
C
C
C
K (w> }> {) = K (w> }> {) +
K (}> {) > }˙ +
K (}> {) > {˙
gw μ
Cw
C{
¶ C} μ
¶
C
C
C
C
=
K (}> {) > K (}> {) +
K (}> {) > K (}> {) = 0
C}
C{
C{
C}
So, K (w> }> {) is a constant.
Lemma 19.10 If D (w) = DW (w) is skew Hermitian, then
k{ (w)k = const
w
(19.83)
Proof. One has directly
g
k{ (w)k2 = ({˙ (w) > { (w)) + ({ (w) > {˙ (w)) = (D (w) { (w) > { (w)) +
gw
({ (w) > D (w) { (w)) = (D (w) { (w) > { (w)) + (DW (w) { (w) > { (w)) =
([D (w) + DW (w)] { (w) > { (w)) = 0
that proves the result.
19.2. Regular ODE
599
Lemma 19.11 (Green’s formula) Let D (w), j (w) and j̃ (w) be continuous for w 5 [d> e]; { (w) be a solution of (19.61) and } (w) be a
solution of (19.79). Then for all w 5 [d> e]
Zw
[j | (v) } (v) -{ (v)| j̃ (v)] gv = {| (w) } (w)-{| (d) > } (d)
(19.84)
v=d
Proof. The relation (19.84) is proved by showing that both sides
have the same derivatives, since (D|> }) = (|> DW }).
19.2.4
Index of increment for ODE solutions
Definition 19.4 A number is called a Lyapunov order number
(or the index of the increment) for a vector function { (w) defined for w w0 , if for every % A 0 there exists positive constants F%0
and F% such that
k{ (w)k F% h( +%)w for all large w
(19.85)
k{ (w)k F%0 h( 3%)w
for some arbitrary large w
that equivalently can be formulated as
= lim sup w31 ln k{ (w)k
(19.86)
w<"
Lemma 19.12 If { (w) is the solution of (19.61), then it has the Lyapunov order number
3
lim sup w31 ln Ck{ (w0 )k +
w<"
Zw
+ lim sup w31
w<"
4
Zw
ki (v)k gvD
v=w0
kD (v)k gv
v=w0
Proof. It follows directly from (19.66).
(19.87)
600
19.2.5
Chapter 19. Ordinary Dierential Equations
Riccati dierential equation
Let us introduce the symmetric q × q matrix function S (w) = S | (w)
5 F 1 [0> W ] which satisfies the following ODE:
<
Ṡ (w) = S (w) D (w) + D (w)| S (w) A
A
@
S (w) U (w) S (w) + T (w)
A
A
>
S (W ) = J 0
(19.88)
D (w) > T (w) 5 Rq×q > U (w) 5 Rp×p
(19.89)
with
Definition 19.5 We call ODE (19.88) the matrix Riccati dierential equation.
Theorem
symmetric
there exist
isfying the
19.9 (on the structure of the solution) Let S (w) be a
nonnegative solution of (19.88) defined on [0> W ]. Then
two functional q × q matrices [ (w) > \ (w) 5 F 1 [0> W ] satfollowing linear ODE
μ
[˙ (w)
\˙ (w)
¶
μ
= K (w)
[ (w)
\ (w)
¶
(19.90)
[ (W ) = L> \ (W ) = S (W ) = J
with
K (w) =
D (w) U (w)
T (w) D| (w)
¸
(19.91)
where D (w) and T (w) are as in (19.88) and such that S (w) may be
uniquely represented as
S (w) = \ (w) [ 31 (w)
(19.92)
for any finite w 5 [0> W ].
Proof.
a) Notice that the matrices [ (w) and \ (w) exist since they are
defined by the solution to the ODE (19.90).
19.2. Regular ODE
601
b) Show that they satisfy the relation (19.92). Firstly, remark that
[ (W ) = L, so det [ (W ) = 1 A 0. From (19.90) it follows that [ (w)
is a continuous matrix function and, hence, there exists a time such
that for all w 5 (W > W ] det [ (w) A 0. As a result, [ 31 (w) exists
within the small semi-open interval (W > W ]. Then, directly using
(19.90) and in view of the identities
[ 31 (w) [ (w) = L>
g £ 31 ¤
[ (w) [ (w) + [ 31 (w) [˙ (w) = 0
gw
it follows
g
[[ 31 (w)] = [ 31 (w) [˙ (w) [ 31 (w) =
gw
[ 31 (w) [D (w) [ (w) U (w) \ (w)] [ 31 (w) =
(19.93)
[ 31 (w) D (w) + [ 31 (w) U (w) \ (w) [ 31 (w)
and, hence, for all w 5 (W > W ] in view of (19.88)
g
g
[\ (w) [ 31 (w)] = \˙ (w) [ 31 (w) + \ (w) [[ 31 (w)] =
gw
gw
[T (w) [ (w) D| (w) \ (w)] [ 31 (w) +
\ (w) [[ 31 (w) D (w) + [ 31 (w) U (w) \ (w) [ 31 (w)] =
T (w) D| (w) S (w) S (w) D (w) + S (w) U (w) S (w) = Ṡ (w)
that implies
g
[\ (w) [ 31 (w) S (w)] = 0, or,
gw
\ (w) [ 31 (w) S (w) = const
wM(W 3 >W ]
But for w = W we have
const
wM(W 3 >W ]
= \ (W ) [ 31 (W ) S (W ) = \ (W ) S (W ) = 0
So, for all w 5 (W > W ] it follows S (w) = \ (w) [ 31 (w).
602
Chapter 19. Ordinary Dierential Equations
c) Show that det [ (W ) A 0. The relations (19.90) and (19.92)
lead to the following presentation within w 5 w 5 (W > W ]
[˙ (w) = D (w) [ (w) U (w) \ (w) = [D (w) U (w) S (w)] [ (w)
and, by the Liouville’s theorem 19.7, it follows
; W 3
<
?Z
@
det [ (W ) = det [ (0) exp
wu [D (w) U (w) S (w)] gw
=
>
w=0
; W
?Z
<
@
1 = det [ (W ) = det [ (0) exp
wu [D (w) U (w) S (w)] gw
=
>
w=0
;
<
? ZW
@
det [ (W ) = exp wu [D (w) U (w) S (w)] gw A 0
=
>
w=W 3
By continuity, again there exists a time 1 A that det [ (w) A 0 for
any w 5 [W > W 1 ]. Repeating the same considerations we may
conclude that det [ (w) A 0 for any w 5 [0> W ].
d) Show that the matrix J (w) := \ (w) [ 31 (w) is symmetric. One
has
g |
g
[\ (w) [ (w) [ | (w) \ (w)] = \˙ | (w) [ (w) + \ | (w) [[ (w)]
gw
gw
g |
[ (w) \ (w) [ | (w) \˙ (w) = [T (w) [ (w) D| (w) \ (w)]| [ (w) +
gw
\ | (w) [D (w) [ (w) U (w) \ (w)] [D (w) [ (w) U (w) \ (w)]| \ (w)
[ | (w) [T (w) [ (w) D| (w) \ (w)] = 0
and \ (W )| [ (W ) [[ (W )]| \ (W ) = \ | (W ) \ (W ) = J| J =
0 that implies \ | (w) [ (w) [ | (w) \ (w) = 0 for any w 5 [0> W ]. So,
\ | (w) = [ | (w) \ (w) [ 31 (w) = [ | (w) S (w) and, hence, by the transposition operation we get \ (w) = S | (w) [ (w) and S (w) = \ (w) [ 31 (w)
= S | (w). The symmetricity of S (w) is proven.
e) The Riccati dierential equation (19.88) is uniquely solvable
with S (w) = \ (w) [ 31 (w) 0 on [0> W ] since the matrices [ (w) and
\ (w) are uniquely defined by (19.92).
19.2. Regular ODE
19.2.6
603
Linear first order partial DE
Consider the following linear first order partial DE
q
X
l=1
[l ({> })
C}
= ] ({> })
C{l
(19.94)
where { 5 Rq is a vector of q independent real variables and } = } ({)
is a real-valued function of the class F 1 (X ), { 5 X Rq . Defining
¶|
μ
C}
C}
C}
|
[ ({> }) := ([1 ({> }) > ===> [q ({> })) >
> ===>
:=
C{
C{1
C{q
the equation (19.94) can be rewritten as follows
μ
¶
C}
[ ({> }) >
= ] ({> })
C{
(19.95)
Any function } = } ({) 5 F 1 (X ) satisfying (19.95) is its solution. If
so, then its full dierential g} is
¶ X
μ
q
C}
C}
> g{ =
g} =
g{l
(19.96)
C{
C{l
l=1
Consider also the following auxiliary system of ODE:
g{1
g{q
g}
=···=
=
[1 ({> })
[q ({> })
] ({> })
(19.97)
or, equivalently,
[131 ({> })
g{1
g{q
= · · · = [q31 ({> })
= ] 31 ({> })
g}
g}
or,
g{1
= [1 ({> }) ] 31 ({> })
g}
···
g{q
= [q ({> }) ] 31 ({> })
g}
(19.98)
<
A
A
A
@
A
A
A
>
(19.99)
which is called the system of characteristic ODE related to (19.95).
The following important result, describing the natural connection of
(19.95) and (19.98), is given below.
604
Chapter 19. Ordinary Dierential Equations
Lemma 19.13 If } = } ({) satisfies (19.97), then it satisfies (19.95)
too.
Proof. Indeed, by (19.99) and (19.96) we have
g{l = [l ({> }) ] 31 ({> }) g}
q
q
X
X
C}
C}
g{l =
[l ({> }) ] 31 ({> }) g}
g} =
C{
C{
l
l
l=1
l=1
that implies (19.94).
Cauchy’s method of characteristics
The method, presented here, permits to convert the solution of a linear
first order partial DE the solution of a system of nonlinear ODE.
Suppose that we can solve the system (19.98) of ODE and its
solution is
{l = {l (}> fl ) > l = 1> ===> q
(19.100)
where fl are some constants.
Definition 19.6 The solutions (19.100) are called the characteristics of (19.94).
Assume that this solution can be resolved with respect to the constants fl , namely, there exists functions
#l = #l ({> }) = fl (l = 1> ===> q)
(19.101)
Since these functions are constants on the solutions of (19.98) they are
the first integrals of (19.98). Evidently, any arbitrary function x : Rq
$ R of constants fl (l = 1> ===> q) is a constant too, that is,
x (f1 > ===> fq ) = const
(19.102)
Without the loss of a generality we can take const = 0, so the equation
(19.102) becomes
x (f1 > ===> fq ) = 0
(19.103)
19.2. Regular ODE
605
Theorem 19.10 (Cauchy’s method of characteristics) If the
first integrals (19.74) #l ({> }) of the system (19.98) are independent,
that is,
¸
C#l ({> })
det
6= 0
(19.104)
C{m
l>m=1>===>q
then the solution } = } ({) of (19.97) can be found from the algebraic
equation
x (#1 ({> }) > ===> # q ({> })) = 0
(19.105)
where x (#1 > ===> # q ) is an arbitrary function of its arguments.
Proof. By Theorem 16.8 on an implicit function, the systems
(19.74) can be uniquely resolved with respect to { if (19.104) is fulfilled. So, the obtained functions (19.100) satisfy (19.99) and, hence,
by Lemma 19.13 it follows that } = } ({) satisfies (19.95).
Example 19.3 Let us integrate the equation
q
X
l=1
{l
C}
= s} (s is a constant)
C{l
The system (19.97)
g{1
g{q
g}
=···=
=
{1
{q
s}
has the following first integrals
{sl } = fl (l = 1> ===> q)
So, } = } ({) can be found from the algebraic equation
x ({s1 }> ===> {sq }) = 0
where x (#1 > ===> # q ) is an arbitrary function, for example,
x (#1 > ===> # q ) :=
q
X
l #l ,
l=1
that gives
q
X
l=1
à q
!31 q
X
X
}=
l
l {sl
l=1
l=1
l 6= 0
(19.106)
606
Chapter 19. Ordinary Dierential Equations
19.3
Carathéodory’s Type ODE
19.3.1
Main definitions
The dierential equation
{˙ (w) = i (w> { (w)) > w w0
(19.107)
in the regular case (with continuous right-hand side in both variables)
is known to be equivalent to the integral equation
Zw
{ (w) = { (w0 ) +
i (v> { (v)) gv
(19.108)
v=w0
Definition 19.7 If the function i (w> {) is discontinuous in w and
continuous in { 5 Rq , then the functions { (w), satisfying the integral equation (19.108) where the integral is understood in the Lebesgue
sense, is called solutions ODE (19.107).
The material presented bellow follows (Filippov 1988).
Let us define more exactly the conditions which the function i (w> {)
should satisfy.
Condition 19.1 (Carathéodory’s conditions) Let in the domain
D of the (w> {)-space the following conditions be fulfilled:
1) the function i (w> {) be defined and continuous in { for almost
all w;
2) the function i (w> {) be measurable (see (15.97)) in w for each {;
3)
ki (w> {)k p (w)
(19.109)
where the function p (w) is summable (integrable in the Lebesgue
sense) on each finite interval (if w is unbounded in the domain
D).
19.3. Carathéodory’s Type ODE
607
Definition 19.8
a) The equation (19.107), where the function i (w> {) satisfies the conditions 19.1, is called the Carathéodory’s type ODE.
b) A function { (w), defined on an open or closed interval o, is called
a solution of the Carathéodory’s type ODE if
- it is absolutely continuous on each interval [> ] 5 o;
- it satisfies almost everywhere this equation or, which under the
conditions 19.1 is the same thing, satisfies the integral equation
(19.108).
19.3.2
Existence and uniqueness theorems
Theorem 19.11 ((Filippov 1988)) For w 5 [w0 > w0 + d] and
{ : k{ {0 k e let the function i (w> {) satisfies the Carathéodory’s
conditions 19.1. Then on a closed interval [w0 > w0 + g] there exists a
solution of the Cauchy’s problem
{˙ (w) = i (w> { (w)) > { (w0 ) = {0
(19.110)
In this case one can take an arbitrary number g such that
Zw
0 ? g d> * (w0 + g) e where * (w) :=
p (v) gv
(19.111)
v=w0
(p (w) is from (19.109)).
Proof. For integer n 1 define k := g@n, and on the intervals
[w0 + lk> w0 + (l + 1) k] (l = 1> 2> ===> n) construct iteratively an approximate solution {n (w) as
Zw
{n (w) := {0 +
i (v> {n31 (v)) gv (w0 ? w w0 + g)
(19.112)
v=w0
(for any initial approximation {0 (v) > for example, {0 (v) = const).
Remember that if i (w> {) satisfies the Carathéodory’s conditions 19.1
608
Chapter 19. Ordinary Dierential Equations
then and { (w) is measurable on [d> e], then the composite function
i (w> { (w)) is summable (integrable in the Lebesgue sense) on [d> e]. In
view of this and by the condition (19.111) we obtain k{n (w) {0 k e.
Moreover, for any > : w0 ? w0 + g
Zw
k{n () {n ()k p (v) gv = * () * ()
(19.113)
v=w0
The function * (w) is continuous on the closed interval [w0 > w0 + g] and
therefore uniformly continuous. Hence, for any % A 0 there exists a =
(%) such that for all | | ? the right-hand side of (19.113) is less
than %. Therefore, the functions {n (w) are equicontinuous (see (14.18))
and uniformly bounded (see (14.17)). Let us choose (by the Arzelà’s
theorem 14.16) from them a uniformly convergent subsequence having
a limit { (w). Since
k{n (v k) { (v)k k{n (v k) {n (v)k + k{n (v) { (v)k
and the first term on the right-hand side is less then % for k = g@n ? ,
it follows that {n (v k) tends to { (v) > by the chosen subsequence. In
view of continuity of i (w> {) in {, and the estimate ki (w> {)k p (w)
(19.109) one can pass to the limit under the integral sign in (19.112).
Therefore, we conclude that the limiting function { (w) satisfies the
equation (19.108) and, hence, it is a solution of the problem (19.110).
Theorem is proven.
Corollary 19.12 If the Carathéodory’s conditions 19.1 are satisfied
for w0 d w w0 and k{ {0 k e, then a solution exists on the
closed interval [w0 g> w0 ] where g satisfies (19.111).
Proof. The case w w0 is reduced to the case w w0 by the simple
substitution of (w) for w.
Corollary 19.13 Let (w0 > {0 ) 5 D R1+q and let there exists a summable function o (w) (in fact, this is a Lipschitz constant) such that for
any two points (w> {) and (w> |) of D
ki (w> {) i (w> |)k o (w) k{ |k
(19.114)
Then in the domain D there exists at most one solution of the
problem (19.110).
19.3. Carathéodory’s Type ODE
609
Proof. Using (19.114) it is su!cient to check the Carathéodory’s
conditions 19.1.
Theorem 19.12 (on the uniqueness) If in Corollary 19.13 instead
of (19.114) there is fulfilled the inequality
(i (w> {) i (w> |) > { |) o (w) k{ |k2
(19.115)
then in the domain D there exists the unique solution of the problem
(19.110).
Proof. Let { (w) and | (w) be two solutions of (19.110). Define for
w0 w w1 the function } (w) := { (w) | (w) for which it follows
¶
μ
g
g
2
k}k = 2 }> } = 2 (i (w> {) i (w> |) > { |)
gw
gw
g
k}k2 o (w) k}k2 and,
gw
Zw
g ¡ 3O(w)
2¢
hence,
h
k}k 0 where O (w) =
o (v) gv. Thus, the abgw
almost everywhere. By (19.115) we obtain
v=w0
solutely continuous function (i.e., it is a Lebesgue integral of some
other function) h3O(w) k}k2 does not increase, and it follows from } (w0 ) =
0 that } (w) = 0 for any w w0 . So, the uniqueness is proven.
Remark 19.7 The uniqueness of the solution of the problem (19.110)
implies that if there exists two solutions of this problem, the graphs
of which lie in the domain D, then these solutions coincide on thew
common part of their interval of existence.
Remark 19.8 Since the condition (19.114) implies the inequality
(19.115) (this follows from the Cauchy-Bounyakoski-Schwartz inequality), thus the uniqueness may be considered to be proven for w w0 also
under the condition (19.114).
610
Chapter 19. Ordinary Dierential Equations
19.3.3
Variable structure and singular perturbed
ODE
Variable structure ODE
In fact, if by the structure of ODE (19.107) {˙ (w) = i (w> { (w)) we will
understand the function i (w> {), then evidently any nonstationary system may be considered as a dynamic system with a variable structure,
since for dierent w1 6= w2 we will have i (w1 > {) 6= i (w2 > {). From this
point of view such treatment seems to be naive and having no correct mathematical sense. But if we consider the special class of ODE
(19.107) given by
{˙ (w) = i (w> { (w)) :=
Q
X
" (w 5 [wl31 > wl )) i l ({ (w))
(19.116)
l=1
where " (·) is the characteristic function of the corresponding event,
namely,
½
1 if w 5 [wl31 > wl )
" (w 5 [wl31 > wl )) :=
(19.117)
> wl31 ? wl
0 if w 5
@ [wl31 > wl )
then ODE (19.116) can be also treated as ODE with "jumping" parameters (coe!cients). Evidently, that if i l ({) are continuous on a
compact D> and, hence, are bounded, that is,
°
°
(19.118)
max max °i l ({)° P
l=1>===>Q {MD
then the third Carathéodory’s condition (19.109) will be fulfilled on
the time interval [> ], since
p (w) = P
Q
X
" (w 5 [wl31 > wl )) = PQ ? 4
(19.119)
l=1
Therefore, such ODE equation (19.116) has at most one solution. If,
in the addition, for each l = 1> ===> Q the Lipschitz condition holds, i.e.,
(i l ({) i l (|) > { |) ol k{ |k2
then, as it follows from Theorem 19.12, this equation has a unique
solution.
19.3. Carathéodory’s Type ODE
611
Singular perturbed ODE
Consider the following ODE containing a singular-type of perturbations:
{˙ (w) = i ({ (w)) +
Q
X
l (w wl ) > w> wl w0
(19.120)
l=1
where (w wl ) is the "Dirac delta-function" (15.128), l is a real
constant and i is a continuous function. The ODE (19.120) must be
understand as the integral equation
Zw
{ (w) = { (w0 ) +
i ({ (v)) gw +
Q
X
Zw
l
l=1
v=w0
(w wl ) gw
(19.121)
v=w0
The last term, by the property (15.134), can be represented as
Q
X
Zw
l
l=1
g" (v A wl ) =
Q
X
l " (w A wl )
l=1
v=w0
where " (w wl ) is the "Heavyside’s (step) function" defined by (19.117).
Let is apply the following state transformation:
{˜ (w) := { (w) +
Q
X
l " (w A wl )
l=1
New variable {˜ (w) satisfies (with 0 := 0) the following ODE:
Ã
!
Q
X
g
{˜ (w) = i {˜ (w) l " (w A wl )
gw
!
à l=1
Q
l
X
X
(19.122)
" (w A wl ) i { (w) l
=
l=1
=
Q
X
v=1
" (w A wl ) i˜l ({ (w))
l=1
Ã
where
i˜l ({ (w)) := i
{ (w) l
X
v=1
!
l
612
Chapter 19. Ordinary Dierential Equations
Claim 19.1 This exactly means that the perturbed ODE (19.120)
are equivalent to a variable structure ODE (19.116).
19.4
ODE with DRHS
In this chapter we will follow (Utkin 1992)), (Filippov 1988) and (Gelig
et al. 1978).
19.4.1
Why ODE with DRHS are important in
Control Theory
Here we will present some motivating consideration justifying our further study of ODE withy DRHS. Let us start with the simplest scalar
case dealing with the following standard ODE which is a!ne (linear)
on control:
{˙ (w) = i ({ (w)) + x (w) > { (w) = {0 is given
(19.123)
where { (w) > x (w) 5 R are interpreted here as the state of the system
(19.123) and, respectively, the control action applied to it at time
w 5 [0> W ]. The function i : R $ R is a Lipschitz function satisfying
the, so-called, Lipschitz condition, that is, for any {> {0 5 R
|i ({) i ({0 )| O |{ {0 | > 0 O ? 4
(19.124)
Problem 19.1 Let us try to stabilize this system at the point {W = 0
using the, so-called, feedback control
x (w) := x ({ (w))
(19.125)
considering the following informative situations
- the complete information case when the function i ({) is exactly
known;
- the incomplete information case when it is only known that the
function i ({) is bounded as
|i ({)| i0 + i + |{| > i0 ? 4> i + ? 4
(this inequality is assumed to be valid for any { 5 R).
(19.126)
19.4. ODE with DRHS
613
There are two possibilities to do that:
1. use any continuous control, namely, take x : R $ R as a
continuous function, i.e., x 5 F;
2. use a discontinuous control which will be defined below.
The complete information case
Evidently that at the stationary point {W = 0 any continuous control
x (w) := x ({ (w)) should satisfy the following identity
i (0) + x (0) = 0
(19.127)
For example, this property may be fulfilled if use the control x ({)
containing the nonlinear compensating term
xfrps ({) := i ({)
and the linear correction term
xfru ({) := n{> n A 0
that is, if
x ({) = xfrps ({) + xfru ({) = i ({) n{
(19.128)
The application of this control (19.128) to the system (19.123) implies
that
{˙ (w) = n{ (w)
and, as the result, one gets
{ (w) = {0 exp (nw) $ 0
w<0
So, this continuous control (19.128) in the complete information case
solves the stabilization problem (19.1).
614
Chapter 19. Ordinary Dierential Equations
The incomplete information case
Several informative situations may be considered.
1. i ({) is unknown, but a priory it is known that i (0) = 0.
In this situation the condition the Lipschitz condition (19.124)
is transformed in to
|i ({)| = |i ({) i (0)| O |{|
that for the Lyapunov function candidate Y ({) = {2 @2 implies
Ẏ ({ (w)) = { (w) {˙ (w) =
{ (w) [i ({ (w)) + x ({ (w))] |{ (w)| |i ({ (w))|
+ { (w) x ({ (w)) O |{ (w)|2 + { (w) x ({ (w))
(19.129)
Since i ({) is unknown let us select x ({) in (19.128) as
x ({) = xfru ({) = n{
xfrps ({) := 0
(19.130)
The use of (19.130) in (19.129) leads to the following identity:
Ẏ ({ (w)) O{2 (w) + { (w) x ({ (w)) =
(O n) {2 (w) = 2 (n O) Y ({ (w))
Selecting n big enough (this method is known as the "high-gain
control") we get
Ẏ ({ (w)) 2 (n O) Y ({ (w)) 0
Y ({ (w)) Y ({0 ) exp (2 [n O] w) $ 0
w<0
This means that in the considered informative situation the "highgain control" solves the stabilization problem.
2. i ({) is unknown and it is admissible that i (0) 6= 0. In
this situation the condition (19.127) never can be fulfilled since
we do not know exactly the value i (0) and, hence, neither the
control (19.128) no the control (19.130) can be applied. Let us
try to apply a discontinuous control, namely, let us take x ({) in
the form of the, so-called, sliding-mode(or relay) control :
x ({) = nw sign ({) > nw A 0
(19.131)
19.4. ODE with DRHS
where
615
;
?
sign({):=
=
if { A 0
1
1
if { ? 0
5 [1> 1] if { = 0
(19.132)
(see Fig.19.1). Staring from some {0 6= 0> analogously to (19.129)
Figure 19.1: The signum function.
and using (19.126), we have
Ẏ ({ (w)) = { (w) {˙ (w) = { (w) [i ({ (w)) + x ({ (w))] |{ (w)| |i ({ (w))| + { (w) x ({ (w)) |{ (w)| (i0 + i + |{ (w)|)
nw { (w) sign ({ (w)) = |{ (w)| i0 + i + |{ (w)|2 nw |{ (w)|
Taking
nw = n({ (w)) := n0 + n1 |{ (w)|
0
1
n A i0 > n A i
+
(19.133)
one has
Ẏ ({ (w)) |{ (w)| (n0 i0 ) (n1 i + ) |{ (w)|2 p
s
|{ (w)| (n0 i0 ) = 2 (n 0 i0 ) Y ({ (w)) 0
Hence,
s ¡
¢
gY ({ (w))
p
2 n0 i0 gw
Y ({ (w))
616
Chapter 19. Ordinary Dierential Equations
that leads to the following identity
³p
´
p
s ¡
¢
2
Y ({ (w)) Y ({0 ) 2 n0 i0 w
or, equivalently,
p
p
n0 i0
Y ({ (w)) Y ({0 ) s w
2
This means that the, so-called, "reaching phase", during which
the system (19.123) controlled by the sliding-mode algorithm
(19.131)-(19.133) reaches the origin, is equal to
p
2Y ({0 )
wW = 0
n i0
(19.134)
Conclusion 19.1 As it follows from the considerations above, the
discontinuous (in this case, sliding-mode) control (19.131)(19.133) can stabilize the class of the dynamic systems (19.123),
(19.124), (19.126) in finite time (19.134) without the exact knowledge of its model. Besides, the reaching phase may be done as
small as you wish by the simple selection of the gain parameter n0 in
(19.134). In other words, the discontinuous control (19.131)-(19.133)
is robust with respect to the presence of unmodelled dynamics
in (19.123) that means that it is capable to stabilize a wide class of
"black/grey-box" systems.
Remark 19.9 Evidently, that using such discontinuous control, the
trajectories of the controlled system can not stay in the stationary
point {W = 0 since it arrives to it in finite time but with a nonzero
rate, namely, with {˙ (w) such that
½
i (0) + n 0 if { (w) $ +0
{˙ (w) =
i (0) n0 if { (w) $ 0
that provokes the, so-called, "chattering eect" (see Fig.19.2). Simple engineering considerations show that some sort of smoothing (or,
low-pass filtering) should be applied to keep dynamics close to the stationary point {W = 0.
19.4. ODE with DRHS
617
Figure 19.2: The chattering eect.
Remark 19.10 Notice that when { (w) = {W = 0 we only know that
{˙ (w) 5 [i (0) n0 > i (0) + n0 ]
(19.135)
This means that we deal with a dierential inclusion (not an equation) (19.135). So, we need to define what does it mean mathematically
correctly a solution of a dierential inclusion and what is it itself.
All these questions, arising in the remarks above, will be considered
below in details and be illustrated by the corresponding examples and
figures.
19.4.2
ODE with DRHS and dierential inclusions
General requirements to a solution
As it is well known, a solution of the dierential equation
{˙ (w) = i (w> { (w))
(19.136)
with a continuous right-hand side is a function { (w) which has a derivative and satisfies (19.136) everywhere on a given interval. This definition is not, however, valid for DE with DRHS since in some points
of discontinuity the derivative of { (w) does not exists. That’s why
the consideration of DE with DRHS requires a generalization of the
concept of a solution. Anyway, such generalized concept should necessarily meet the following requirements:
618
Chapter 19. Ordinary Dierential Equations
- For dierential equations with a continuous right-hand side the definition of a solution must be equivalent to the usual (standard)
one.
- For the equation
{˙ (w) = i (w) the solution should be the functions
R
{ (w) = i (w) gw + f only.
- For any initial data { (w0 ) = {lqlw within a given region the solution
{ (w) should exist (at least, locally) for any w A w0 and admit the
possibility to be continued up to the boundary of this region or
up to infinity (when (w> {) $ 4).
- The limit of a uniformly convergent sequences of solutions should
be a solution too.
- Under the commonly used changes of variables a solution must be
transformed into a solution.
The definition of a solution
Definition 19.9 A vector-valued function i (w> {), defined by a mapping i : R × Rq $ Rs , is said to be piecewise continuous in a
finite domain G Rq+1 if J consists of a finite numbers of a domains
Gl (l = 1> ===> o), i.e.,
o
[
Gl
G=
l=1
such that in each of them the function i (w> {) is continuous up to the
boundary
Ml := Ḡl \Gl (l = 1> ===> o)
(19.137)
of a measure zero.
The most frequent case is the one where the set
M=
o
[
Ml
l=1
of all discontinuity points consists of a finite number of hypersurfaces
0 = Vn ({) 5 F 1 > n = 1> ===> p
where Vn ({) is a smooth function.
19.4. ODE with DRHS
619
Definition 19.10 The set M defined as
M = {{ 5 Rq | V ({) = (V1 ({) > ===> Vp ({))| = 0}
(19.138)
is called a manifold in Rq . It is referred to as a smooth manifold
if Vn ({) 5 F 1 > n = 1> ===> p.
Now we are ready to formulate the main definition of this section.
Definition 19.11 (A solution in the Filippov’s sense) A solution { (w) on a time interval [w0 > wi ] of ODE {˙ (w) = i (w> { (w)) with
DRHS in the Filippov’s sense is called a solution of the dierential inclusion
{˙ (w) 5 F (w> { (w))
(19.139)
that is, an absolutely continuous on [w0 > wi ] function { (w) (which can
be represented as a Lebesgue integral of another function) satisfying
(19.139) almost everywhere on [w0 > wi ], where the set F (w> {) is the
smallest convex closed set containing all limit values of the
vector-function i (w> {W ) for (w> {W ) 5
@ M, {W $ {, w = const.
Remark 19.11 The set F (w> {)
1) consists of one point i (w> {) at points of continuity of the function i (w> {);
2) is a segment (a convex polygon, or polyhedron), which in the case
when (w> {) 5 Ml (19.137) has the vertices
il (w> {) :=
lim
(w>{W )MGl >
i (w> {W )
{W <{
(19.140)
All points il (w> {) are contained in F (w> {), but it is not obligatory that all of them are vertices.
Example 19.4 For the scalar dierential inclusion
{˙ (w) 5 sign ({ (w))
the set F (w> {) is as follows (see Fig.19.3):
1. F (w> {) = 1 if { A 0;
2. F (w> {) = 1 if { ? 0;
3. F (w> {) = [1> 1] if { = 0.
620
Chapter 19. Ordinary Dierential Equations
Figure 19.3: The right-hand side of the dierential inclusion {˙ w =
vljq ({w ).
Semi-continuous sets as functions
Definition 19.12 A multi-valued function (or, a set) F = F (w> {)
(w 5 R> { 5 Rq ) is said to be
• a semi-continuous in the point (w0 > {0 ) if for any % A 0 there
exists = (w0 > {0 > %) such that the inclusion
(w> {) 5 {} | k} (w0 > {0 )k }
(19.141)
F (w> {) 5 {i | ki i (w0 > {0 )k %}
(19.142)
implies
• a continuous in the point (w0 > {0 ) if it is a semi-continuous and,
additionally, for any % A 0 there exists = (w0 > {0 > %) such that
the inclusion
(w0 > {0 ) 5 {} | k} (w0 > {0 )k }
(19.143)
F (w0 > {0 ) 5 {i | ki i (w0 > {0 )k %}
(19.144)
implies
Example 19.5 Consider the multi-valued functions F (w> {) depicted
at Fig.19.4.
Here the functions (sets) F (w> {), corresponding the plots 1)-4),
are semi-continuous.
19.4. ODE with DRHS
621
Figure 19.4: Multivalued functions.
Theorem on the local existence of solution
First, let us formulate some useful result which will be applied in the
following considerations.
Lemma 19.14 If { (w) is absolutely continuous on the interval w 5
[> ] and within this interval k{˙ (w)k f, then
1
({ { ) Conv
a.a. wM[>]
^ {˙ (w)
(19.145)
where Conv is a convex closed set containing ^ {˙ (w) for almost all
w 5 [> ].
Proof. By the definition of the Lebesgue integral
1
1
({ { ) =
Z
w=
{˙ (w) gw = lim vn
n<"
622
Chapter 19. Ordinary Dierential Equations
where
vn =
n
X
l=1
n
X
l
{˙ (wl ) > l 0>
l = | |
| |
l=1
are Lebesgue sums of the integral above. But vn 5
Conv
a.a. wM[>]
^ {˙ (w).
Hence, the same fact is valid for the limit vectors lim vn that proves
n<"
the lemma.
Theorem 19.13 (on the local existence) Suppose that
A1) a multi-valued function (set) F (w> {) is a semi-continuous at
each point
(w> {) 5 G> (w0 > {0 ) := {(w> {) | k{ {0 k > |w w0 | }
A2) the set F (w> {) is a convex compact and sup k|k = f whenever
| 5 F (w> {) and (w> {) 5 G> (w0 > {0 )
Then for any w such that |w w0 | := @f there exists an absolutely continuous function { (w) (may be, not unique) such that
{˙ (w) 5 F (w> {) > { (w0 ) = {0
that is, the ODE {˙ (w) = i (w> { (w)) with DRHS has a local solution
in the Filippov’s sense (see Definition 19.11).
(p)
Proof. Divide the interval [w0 > w0 + ] into 2p-parts wl :=
w0 +m (l = 0> ±1> == ± p) and construct the, so-called, partially linear
p
Euler’s curves
³
´ ³
´
³
´
h
i
(p)
(p)
(p)
(p)
(p) (p+1)
ˆ
{p (w) := {p wl
+ w wl
il
wl
> w 5 wl > wl
³
´
³
´
³
³
´´
(p)
(p)
(p)
(p)
(p)
ˆ
= {0 > il
wl
5 F wl > {p wl
{p w0
By the assumption A2) it follows that {p (w) is uniformly bounded and
continuous on G> (w0 > {0 ). Then, by the Arzelà’s theorem 14.16 there
exists a subsequence {{pn (w)} which uniformly converges to some vector function { (w). This limit evidently has a Lipschitz constant on
19.4. ODE with DRHS
623
G> (w0 > {0 ) and satisfies the initial condition { (w0 ) = {0 . In view of
Lemma 19.14, for any k A 0 one has
31
k
Conv
[{pn (w + k) {pn (w)] pn
[
a.a.[w0 3 >w0 + ]
pn
[
k Conv
l
a.a. M w0 3 p >w0 + p +k l=3p
n
n
n
(p )
iˆl n
l=3pn
(p )
iˆl n () := Dn
Since F (w> {) is semi-continuous, it follows that sup inf k{ |k $ 0
whenever n $ 4 (here D :=
po
[
Conv
a.a. M[w>w+k]
ity of F (w> {) implies also that sup
{MDn |MD
i (> { )). The convex-
l=3po
inf
{MD |MF(w>{)
k{ |k $ 0 when k $ 0
that, together with previous property, proves the theorem.
Remark 19.12 By the same reasons as for the case of regular ODE,
we may conclude that the solution of the dierential inclusion (if it
exists) is continuously dependent on w0 and {0 .
19.4.3
Sliding mode control
Sliding mode surface
Consider the special case where the function i (w> {) is discontinuous
on a smooth surface V given by the equation
v ({) = 0> v : Rq $ R> v (·) 5 F 1
(19.146)
The surface separates its neighborhood (in Rq ) into domains G + and
G 3 . For w = const and for the point {W approaching the point { 5 V
from the domains G + and G 3 let us suppose that the function i (w> {W )
has the following limits:
i (w> {W ) = i 3 (w> {)
lim
(w>{W )MG 3 > {W <{
lim
(w>{W )MG + >
i (w> {W ) = i + (w> {)
(19.147)
{W <{
Then by the Filippov’s definition, F (w> {) is a linear segment joining
the endpoints of the vectors i 3 (w> {) and i + (w> {). Two situations are
possible.
624
Chapter 19. Ordinary Dierential Equations
- If for w 5 (w1 > w2 ) this segment lies on one side of the plane S tangent
to the surface V at the point {, the solutions for these w pass
from one side of the surface V to the other one (see Fig.19.5
depicted at the point { = 0);
Figure 19.5: The sliding surface and the rate vector field at the point
{ = 0.
- If this segment intersects the plane S , the intersection point is the
endpoint of the vector i 0 (w> {) which defines the velocity of the
motion
{˙ (w) = i 0 (w> { (w))
(19.148)
along the surface V in Rq (see Fig.19.6 depicted at the point
{ = 0). Such a solution, lying on V for all w 5 (w1 > w2 ), is often
called a sliding motion ( or, mode). Defining the projections
of the vectors i 3 (w> {) and i + (w> {) to the surface V (uv ({) 6= 0)
as
s3 (w> {) :=
(uv ({) > i 3 (w> {)) +
(uv ({) > i + (w> {))
> s (w> {) :=
kuv ({)k
kuv ({)k
one can find that when s3 (w> {) ? 0 and s+ (w> {) A 0
i 0 (w> {) = i 3 (w> {) + (1 ) i + (w> {)
Here can be easily found from the equation
¢
¡
uv ({) > i 0 (w> {) = 0
19.4. ODE with DRHS
625
Figure 19.6: The velocity of the motion.
or, equivalently,
0 = (uv ({) > i 3 (w> {) + (1 ) i + (w> {)) =
s3 (w> {) + (1 ) s+ (w> {)
that implies
=
s+ (w> {)
s+ (w> {) s3 (w> {)
Finally, we obtain that
s+ (w> {)
i 3 (w> {) +
s+ (w> {) s3 (w> {)
¶
μ
s+ (w> {)
i + (w> {)
1 +
s (w> {) s3 (w> {)
i 0 (w> {) =
(19.149)
Sliding mode surface as a desired dynamics
Let us consider in this subsection several examples demonstrating that
a desired dynamic behavior of a controlled system may be expressed
not only in the traditional manner, using some cost (or payo) functionals as possible performance indices, but also representing a nominal (desired) dynamics in the form of a surface (or, manifold) in a
space of coordinates.
626
Chapter 19. Ordinary Dierential Equations
First-order tracking system Consider a first-order system given
by the following ODE:
{˙ (w) = i (w> { (w)) + x (w)
(19.150)
where x (w) is a control action and i : R×R $ R is supposed to be
bounded, that is,
|i (w> { (w))| i + ? 4
Assume that the desired dynamics (signal), which should be tracked,
is given by a smooth function u (w) (|u̇ (w)| ), such that the tracking
error hw is (see Fig.19.7)
h (w) := { (w) u (w)
Select a desired surface v as follows
Figure 19.7: A tracking system.
v (h) = h = 0
(19.151)
that exactly corresponds to an "ideal tracking" process. Then, designing the control x (w) as
x (w) := nsign(h (w))
we derive that
ḣ (w) = i (w> { (w)) u̇ (w) nsign (h (w))
and for Y (h) = h2 @2 one has
Ẏ (h (w)) = h (w) ḣ (w) = h (w) [i (w> { (w)) u̇ (w) nsign (h (w))]
= h (w) [i (w> { (w)) u̇ (w)] n |h (w)| |h (w)| [i + + ] n |h (w)|
p
s
= |h (w)| [i + + n] = 2 [n i + ] Y (h (w))
19.4. ODE with DRHS
627
and, hence,
p
p
¤
1 £
Y (hw ) Y (h0 ) s n i + w
2
So, taking n A i + + implies
p the finite time convergence of hw (with
2Y (h0 )
the reaching phase wi =
) to the surface (19.151) (see Fig.
n i+ 19.8, and Fig. 19.9).
Figure 19.8: The finite time error cancellation.
Figure 19.9: The finite time tracking.
Stabilization of a second order relay-system Let us consider a
second order relay-system given by the following ODE
628
Chapter 19. Ordinary Dierential Equations
{¨ (w) + d2 {˙ (w) + d1 { (w) = x (w) + (w)
x (w) = nsign (ṽ (w)) - the relay-control
ṽ (w) := {˙ (w) + f{ (w) > f A 0
| (w)| + - a bounded unknown disturbance
(19.152)
One may rewrite the dynamic ({1 := {) as
{˙ 1 (w) = {2 (w)
{˙ 2 (w) = d1 {1 (w) d2 {2 (w) + x (w) + (w)
x (w) = nsign ({2 (w) + f{1 (w))
(19.153)
Here the sliding surface is
v ({) = {2 + f{1
So, the sliding motion, corresponding the dynamics ṽ (w) := {˙ (w) +
f{ (w) = 0> is given by (see Fig.19.10)
{ (w) = {0 h3fw
Let us introduce the following Lyapunov function candidate:
Figure 19.10: The sliding motion on the sliding surface v ({) = {2 +f{1 =
Y (v) = v2 @2
19.4. ODE with DRHS
629
for which the following property holds:
¸
Cv ({ (w))
Cv ({ (w))
Ẏ (v) = vv̇ = v ({ (w))
{˙ 1 (w) +
{˙ 2 (w) =
C{1
C{2
v ({ (w)) [f{2 (w) d1 {1 (w) d2 {2 (w) + x (w) + (w)] £
¤
|v ({ (w))| |d1 | |{1 (w)| + (f+ |d2 |) |{2 (w)| + + -nv ({ (w)) sign (v ({ (w)))
£
¤
= n |d1 | |{1 (w)| (f + |d2 |) |{2 (w)| + |v ({ (w))| 0
if take
n = |d1 | |{1 (w)| + (f + |d2 |) |{2 (w)| + + + > A 0
(19.154)
p
This implies Ẏ (v) 2Y (v), and, hence, the reaching time wi
(see Fig.19.9) is
p
2Y (v0 )
|{˙ 0 + f{0 |
=
wi =
(19.155)
Sliding surface and a related LQ-problem Consider a linear
multi-dimensional plant given by the following ODE
{˙ (w) = D (w) { (w) + E (w) x (w) + (w)
{0 is given, { (w) 5 Rq , x (w) 5 Ru
E | (w) E (w) A 0 rank [E (w)] = u for any w 5 [wv > w1 ]
(w) is known external perturbation
(19.156)
A sliding mode is said to be taking place in this system (19.156) if there
exists a finite reaching time wv , such that the solution { (w) satisfies
({> w) = 0 for all w wv
(19.157)
where ({> w) : Uq × U+ $ Uu is a sliding function and (21.65) defines
a sliding surface in Uq+1 . For each w1 A 0 the quality of the system
(19.156) motion in the sliding surface (21.65) is characterized by the
performance index (Utkin 1992)
Mwv >w1 =
1 Rw1
({ (w) > T{ (w)) gw> T = T| 0
2 wv
(19.158)
630
Chapter 19. Ordinary Dierential Equations
Below we will show that the system motion in the sliding surface
(21.65) does not depend on the control function x, that’s why (19.158)
is a functional of { and ({> w) only. Let us try to solve the following
problem.
Problem formulation: for the given linear system (19.156) and w1 A
0 define the optimal sliding function = ({> w) (21.65) providing the
optimization in the sense of (19.158) in the sliding mode, that is,
Mwv >w1 $ inf
Ms
(19.159)
where s is the set of the admissible smooth (dierentiable on all arguments) sliding functions = ({> > w). So, we wish to minimize the
performance index (19.158) varying (optimizing) the sliding surface
5 s.
Introduce new state vector } defined by
} = W (w) {
(19.160)
where the linear nonsingular transformations W (w) are given by
¸
L(q3u)×(q3u) E1 (w) (E2 (w))31
W (w) :=
(19.161)
0
(E2 (w))31
(q3u)×(q3u)
Here E1
(w) 5 Ru×u and E2 (w) 5 Ru×u represent the matrices
E (w) in the form
¸
E1 (w)
, det [E2 (w)] 6= 0 ; w 0
(19.162)
E (w) =
E2 (w)
Applying (19.162) to the system (19.156), we obtain (below we will
omit the time-dependence)
¶ μ
¶ μ ¶ μ
¶
μ
˜ 1
0
}˙1
D̃11 }1 + D̃12 }2
=
+
+ ˜
(19.163)
}˙ =
x
}˙2
D̃21 }1 + D̃22 }2
2
where }1 5 Uq3u > }2 5 Uu and
¶
¸
μ
˜ 1 (w)
D̃11 D̃12
31
31
= W DW + Ẇ W >
D̃ =
˜ 2 (w) = W (w) (19.164)
D̃21 D̃22
19.4. ODE with DRHS
631
Using the operator W 31 , it follows { = W 31 } and, hence, the performance index (19.158) in new variables } may be rewritten as
´
1 Rw1
1 Rw1 ³
Mwv >w1 =
({> T{) gw =
}> T̃ } gw =
2 wv
2 wv
h³
´
³
´ ³
´i
w
1
R
1
}1 > T̃11 }1 + 2 }1 > T̃12 }2 + }2 > T̃22 }2 gw
2 wv
¸
T̃11 T̃12
31 |
31
T̃ := (W ) TW =
T̃21 T̃22
and the sliding function = ({> w) becomes
¡
¢
= W 31 }> w := ˜ (}> w)
(19.165)
(19.166)
Remark 19.13 The matrices T̃11 > T̃12 > T̃21 and T̃22 are supposed to
be symmetric. Otherwise, they can be symmetrized as follows:
Mwv >w1 =
¢
¡
¢ ¡
¢¤
1 Rw1 £¡
}1 > T̄11 }1 + 2 }1 > T̄12 }2 + }2 > T̄22 }2 gw
2 wv ³
´
´
³
T̄11 := T̃11 + T̃|11 @2> T̄22 := T̃22 + T̃|22 @2
³
´
T̄12 = T̃12 + T̃|12 + T̃21 + T̃|21 @2
(19.167)
Assumption (A1). We will look for the sliding function (19.166) in
the form
˜ (}> w) := }2 + ˜ 0 (}1 > w)
(19.168)
If the sliding mode exists for the system (19.163) in the sliding surface ˜ (}> w) = 0 under the assumption A1, then for all w wv the
corresponding sliding mode dynamics, driven by the unmatched disturbance ˜ 1 (w), are given by
}˙1 = D̃11 }1 + D̃12 }2 + ˜ 1
}2 = ˜
0 (}1 > w)
(19.169)
with the initial conditions }1 (wv ) = (W { (wv ))1 . Defining }2 as a virtual
control, that is,
0 (}1 > w)
(19.170)
y := }2 = ˜
632
Chapter 19. Ordinary Dierential Equations
the system (19.169) may be rewritten as
}˙1 = D̃11 }1 + D̃12 y + ˜ 1
(19.171)
and the performance index (19.165) becomes
Mwv >w1
1
=
2
Zw1 h³
´
³
´ ³
´i
}1 > T̃11 }1 + 2 }1 > T̃12 y + y> T̃22 y gw
(19.172)
wv
In view of (19.171) and (19.172), the sliding surface design problem
(19.159) is reduced to the following one:
Mwv >w1 $ infu
yMU
(19.173)
But this is the standard LQ-optimal control problem. This means
that the optimal control ywW = yW (}1>w > w), optimizing the cost functional
(19.172), defines the optimal sliding surface W ({> w) (see (19.171) and
(19.168)) by the following manner:
yW (}1>w > w) = ˜
0 (}1 > w)
W
˜ (}> w) = }2 y (}1>w > w) = 0
or, equivalently,
W ({> w) = (W {)2 y W ((W {)1 > w) = 0
(19.174)
Equivalent control method
Equivalent control construction Here a formal procedure will be
described to obtain sliding equations along the intersection of sets of
discontinuity for a nonlinear system given by
{˙ (w) = i (w> { (w) > x (w))
{0 is given
{ (w) 5 Rq > x (w) 5 Ru
(19.175)
and the manifold M (19.138) defined as
V ({) = (V1 ({) > ===> Vp ({))| = 0
(19.176)
representing an intersection of p submanifolds Vl ({) (l = 1> ===> p).
19.4. ODE with DRHS
633
Definition 19.13 Hereinafter the control x (w) will be referred to (according to V.Utkin) as the equivalent control x(ht) (w) in the system
(19.175) if it satisfies the equation
V̇ ({ (w)) = J ({ (w)) {˙ (w) = J ({ (w)) i (w> { (w) > x (w)) = 0
J ({ (w)) 5 R
p×q
C
> J ({ (w)) =
V ({ (w))
C{
(19.177)
It is quite obvious that, by virtue of the condition (19.177), a
motion starting at V ({ (w0 )) = 0 in time w0 will proceed along the
trajectories
¡
¢
{˙ (w) = i w> { (w) > x(ht) (w)
(19.178)
which lies on the manifold V ({) = 0.
Definition 19.14 The above procedure is called the equivalent control method (Utkin 1992), (Utkin, Guldner & Shi 1999) and the
equation (19.178), obtained as a result of applying this method, will be
regarded as the sliding mode equation describing the motion on the
manifold V ({) = 0.
From the geometric viewpoint, the equivalent control method implies a replacement of the undefined discontinued control on the discontinuity boundary with a continuous control which directs the velocity vector in the system state space along the discontinuity surface
intersection.
In other
¢ words, it exactly realizes the velocity
¡
i 0 w> { (w) > x(ht) (w) (19.149) corresponding to the Filippov’s definition
of the dierential inclusion in the point { = { (w).
Consider now the equivalent control procedure for an important
particular case of a nonlinear system which is a!ne on x, the righthand side of whose dierential equation is a linear function of the
control, that is,
{˙ (w) = i (w> { (w)) + E (w> { (w)) x (w)
(19.179)
where i : R × Rq $ Rq and E : R × Rq $ Rq×u are all argument
continuous vector and matrix, respectively, and x (w) 5 Ru is a control
634
Chapter 19. Ordinary Dierential Equations
action. The corresponding equivalent control should satisfies (19.177),
namely,
V̇ ({ (w)) = J ({ (w)) {˙ (w) = J ({ (w)) i (w> { (w) > x (w))
(19.180)
= J ({ (w)) i (w> { (w)) + J ({ (w)) E (w> { (w)) x (w) = 0
Assuming that the matrix J ({ (w)) E (w> { (w)) is nonsingular for all
{ (w) and w, one can find the equivalent control from (19.180) as
x(ht) (w) = [J ({ (w)) E (w> { (w))]31 J ({ (w)) i (w> { (w))
(19.181)
Substitution of this control into (19.179) yields the following ODE:
{˙ (w) = i (w> { (w)) E (w> { (w)) [J ({ (w)) E (w> { (w))]31 J ({ (w)) i (w> { (w))
(19.182)
which describes the sliding mode motion on the manifold V ({) = 0.
Below the corresponding trajectories in (19.182) will be referred to as
{ (w) = {(vo) (w).
Remark 19.14 If we deal with an uncertain dynamic model (19.175)
or, particularly, with (19.179), then the equivalent control x(ht) (w) is
not physically realizable.
Below we will show that x(ht) (w) may be successfully approximated
(in some sense) by the output of the first order low-pass filter with the
input equal to the corresponding sliding mode control.
Sliding mode control design Let us try to stabilize the system
(19.179) applying sliding mode approach. For the Lyapunov function
Y ({) := kV ({)k2 @2> considered on the trajectories of the controlled
system (19.179), one has
³
´
Ẏ ({ (w)) = V ({ (w)) > V̇ ({ (w)) =
(V ({ (w)) > J ({ (w)) i (w> { (w)) + J ({ (w)) E (w> { (w)) x (w)) =
(V ({ (w)) > J ({ (w)) i (w> { (w))) + (V ({ (w)) > J ({ (w)) E (w> { (w)) x (w)) kV ({ (w))k kJ ({ (w)) i (w> { (w))k + (V ({ (w)) > J ({ (w)) E (w> { (w)) x (w))
19.4. ODE with DRHS
635
Taking x (w) as a sliding mode control, i.e.,
x (w) = x(vo) (w)
x(vo) (w) := nw [J ({ (w)) E (w> { (w))]31 SIGN (V ({ (w)))
(19.183)
nw A 0> SIGN (V ({)) := (sign (V1 ({)) > ===> sign (Vp ({)))|
we obtain
Ẏ ({ (w)) kV ({)k kJ ({ (w)) i (w> { (w))k nw
p
X
|Vl ({ (w))|
l=1
that, in view of the inequality, kVk p
P
l=1
|Vl |, implies
Ẏ ({ (w)) kV ({)k (nw kJ ({ (w)) i (w> { (w))k)
Selecting
nw = kJ ({ (w)) i (w> { (w))k + > A 0
(19.184)
p
gives Ẏ ({ (w)) kV ({)k = 2Y ({ (w)) that provides the reaching phase in time
p
2Y ({0 )
kV ({0 )k
wi =
=
(19.185)
Remark 19.15 If the sliding motion on the manifold V({) = 0 is
stable then there exists a constant n0 5 (0> 4) such that
kJ ({ (w)) i (w> { (w))k n0
and, hence, nw (19.184) may be selected as a constant
nw := n = n0 + (19.186)
636
Chapter 19. Ordinary Dierential Equations
Low-pass filtering To minimize the influence of the chattering effect arising after the reaching phase let us consider the property of the
signal obtained as an output of a low-pass filter with the input equal
to the sliding mode control, that is,
(dy)
ẋ(dy) (w) + x(dy) (w) = x(vo) (w) > x0
= 0> A 0
(19.187)
where x(vo) (w) is given by (19.183). The next simple lemma states the
relation between the, so-called, averaged control x(dy) (w), which is the
filtered output, and the input signal x(vo) (w).
Lemma 19.15 If
|
|
j+ kJE (w> { (w))k := 1@2
max ([E (w> { (w)) J ] [JE (w> { (w))])
1@2
min ([E | (w> { (w)) J| ] [JE (w> { (w))]) yLu×u > y A 0
(19.188)
then for the low-pass filter (19.187) the following properties hold:
1. The dierence between the input and output signals are bounded,
i.e.,
x(dy) (w) = x(vo) (w) + (w)
k (w)k 2f> f := (j+ + ) p@y
(19.189)
° (dy) °
°ẋ (w)° 2f@
2. The amplitude-frequency characteristic D ($) of the filter is
1
D ($) = q
, $ 5 [0> 4)
1 + ($)2
(19.190)
whose plot is depicted at Fig.19.11 for = 0=01, where | = D ($)
and { = $.
Proof. 1) The solution of the ODE (19.183) and its derivative are
as follows:
°
°
°
°
°nw [JE (w> {w )]31 ° (kJi (w> {w )k + ) °[JE (w> {w )]31 °
=
(kJi (w> {w )k + )
1@2
min ([E |
(w> {w ) J| ] [JE (w> {w )])
y31 (j + + )
19.4. ODE with DRHS
y
637
1
0.9988
0.9975
0.9963
0
2.5
5
7.5
10
x
Figure 19.11: The amplitude-phase characteristic of the low-pass filter.
°
°
¢
¡
° (vo) °
°xw ° y31 j+ + p := f
and by (19.187)
1 Rw 3(w3v)@ (vo)
=
h
xv gv
v=0
¸
1 (vo) 1 Rw 3(w3v)@ (vo)
x =
h
xv gv
w
v=0
(dy)
xw
(dy)
ẋw
(19.191)
that implies
° °
°
°
°
°
° (dy) ° ° (vo) ° 1 Rw 3(w3v)@ ° (vo) °
h
°ẋw ° °xw ° +
°xv ° gv v=0
Rw 3(w3v)@
f Rw 3(w3v)@
h
gv = f + f
h
g (v@) =
f+
v=0
v=0
w@
¡
¢
R 3(w@3ṽ)
h
gṽ = f + f 1 h3w@ 2f
f+f
ṽ=0
Hence, (19.189) holds.
2) Applying the Fourie transformation to (19.187) leads to the
following identity:
m$X (dy) (m$) + X (dy) (m$) = X (vo) (m$)
638
Chapter 19. Ordinary Dierential Equations
or, equivalently,
X (dy) (m$) =
1
1 m$ (vo)
X (m$)
X (vo) (m$) =
1 + m$
1 + ($)2
So, the amplitude-frequency characteristic
q
2
2
D ($) := [Re X (dy) (m$)] + [Im X (dy) (m$)]
of the filter (19.187) is as in (19.190). Lemma is proven.
The realizable approximation of the equivalent control
(dy)
By (19.191) xw
(dy)
may be represented as xw
(dy)
{w
=
Rw
¢
(vo) ¡
xv g h3(w3v)@ .
v=0
(dy)
of the system (19.175) controlled by xw
Consider the dynamics
(19.191) at two time intervals: during the reaching phase and during
the sliding mode regime.
1. Reaching phase (w 5 [0> wi ]). Here the integration by part
implies
(dy)
xw
Zw
=
x(vo)
v g
¡ 3(w3v)@ ¢
(vo)
(vo)
h
= xw x0 h3w@ v=0
Zw
ẋv(vo) h3(w3v)@ gv
v=0
(vo)
Supposing
that xw (19.183) is bounded almost everywhere, i.e.,
°
°
° (vo) °
°ẋw ° g. The above identity leads to the following estimation:
° °
°
°
°
°
Rw 3(w3v)@
° (vo) °
° (vo) °
° (dy)
(vo) °
h
gv = °x0 ° h3w@ +
°xw xw ° °x0 ° h3w@ + g
v=0
°
°
w@
R 3(w@3ṽ)
° (vo) °
h3(w3v)@ g (v@) = °x0 ° h3w@ + g
h
gṽ =
g
v=0 °
ṽ=0
°
¡
¢
¡
¢
° (vo) ° 3w@
+ g 1 h3w@ = g + R h3w@
°x0 ° h
Rw
(dy)
So, xw
may be represented as
(dy)
xw
(vo)
= xw
+ w
(19.192)
19.4. ODE with DRHS
639
where w may be done as small as you wish taking tending to
zero, since
¡
¢
k w k g + R h3w@
(vo)
(dy)
As a result, the trajectories {w and {w will dier a little bit.
Indeed,
´
³
´
³
(vo)
(vo)
(vo)
(vo)
{˙ w = i w> {w
E w> {w
x
´
³
´w
³
(dy)
(dy)
(dy)
(dy)
E w> {w
xw
{˙ w = i w> {w
Defining
´
³
´
³
´
³
(dy)
(dy)
(dy)
> J̃ = J {w
> i˜ = i w> {w
Ẽ = E w> {w
and omitting the arguments for the simplicity, the last equation
may be represented as
(vo)
{˙ w
(vo)
(dy)
= i Exw > {˙ w
(dy)
= i˜ Ẽxw
(vo)
Hence by (19.192), the dierence {w := {w
(dy)
{w
satisfies
´
i
Rw h³
(vo)
(dy)
i i˜ Exv + Ẽxv
gv =
{w = {0 v=0
³
´
´i
Rw h³
(vo)
(vo)
˜
i i Exv + Ẽ xv + v gv
{0 v=0
Taking into account that {0 = 0 (the system starts with the
same initial conditions independently on an applied control) and
that i (w> {) and E ({) are Lipschitz (with the constant Oi and
OE ) on { it follows
°i
° °³
´
Rw h°
°
° °
°
(vo)
k{w k °i i˜° + ° Ẽ E xv + Ẽ v ° gv
v=0
° ° °
°
i
Rw h
° (vo) ° ° °
Oi k{v k + OE k{v k °xv ° + °Ẽ ° k v k gv v=0
°
°´
° °¡
¡
¢¢i
Rw h³
° (vo) °
° °
Oi + OE °xv ° k{v k + °Ẽ ° g + R h3w@
gv
v=0
´
³
¡
¢
Since R h3w@ = R 1 h3w@ = r (1) % and
° °
° (vo) °
° °
(vo)
°xv ° x+
? 4> °Ẽ ° E + ? 4
640
Chapter 19. Ordinary Dierential Equations
we finally have
´
i
Rw h³
(vo)
Oi + OE x+ k{v k + E + (g + %) gv
v=0
´
Rw ³
(vo)
+
E (g + %) wi +
Oi + OE x+ k{v k gv
k{w k v=0
Now let us apply the Gronwall lemma which says that if y (w) and
(w) are nonnegative continuous functions on [w0 > 4) verifying
Zw
y (w) f +
(v) y (v) gv
(19.193)
v=w0
then for any w 5 [w0 > 4) the following inequality holds:
3 w
4
Z
y (w) f exp C (v) gvD
(19.194)
v=w0
This results remains true if f = 0. In our case
(vo)
y (w) = k{w k > f = E + (g + %) wi > (v) = Oi + OE x+
for any v 5 [0> wi ). So,
k{w k :=E + (g + %) wi exp
´ ´
³³
(vo)
Oi + OE x+ wi
(19.195)
Claim. For any finite reaching time wi and any small value
A 0 there exists a small enough such that k{w k is less than
.
2. Sliding mode phase (w A wi ). During the sliding mode phase
we have
³
´
³
´
³
´
(vo)
(vo)
(ht)
V {w
= V̇ {w
= J i Exw
=0
(19.196)
(ht)
(dy)
if xw = xw for all w A wi . Applying xw = xw
guarantee (19.196) already. Indeed,
³
´
³
´ Zw ¡
¢
(dy)
(dy)
V {w
V̇ {(dy)
= V {wi
+
gv
v
v=wi
we can not
19.4. ODE with DRHS
641
and, by (19.195),
° ³
°
´° ° ³
´ ³
´° ° ³ ´
°
°
°
°
(dy) °
(dy)
(vo) °
(vo)
-V {wi ° °J wi {wi ° R()
°V {wi ° = °V {wi
° ³
´°
°
(dy) °
Hence, in view of (19.196), °V {w
° = R().
Claim 19.2 During the sliding-mode phase
° ³
´°
°
(dy) °
°V {w
° = R()
(19.197)