Chapter 19
Ordinary Di erential Equations
19.1 Classes of ODE
In this chapter we will deal with the class of functions satisfying the following ordinary di erential equation
{˙ (w)=i (w> { (w)) for almost all w 5 [w0>w0 + ] { (w0)={0 (19.1) i : R ×X $ X 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 re- solving 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 di erential 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 Di erential Equations
i (w> {) is continuous in both variables. In this case { (w),satis- fying (19.1), should be continuous di erentiable, i.e.,
1 { (w) 5 F [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 di erential 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 {,thatis, 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 {2k for all w 5 [w0>w0 + ] and all {> {1>{2 5 Eu ({0) where
Eu ({0):={{ 5 X|k{ {0k u} Then the Cauchy’s problem (19.1) has a unique solution on the time- interval [w0>w0 + 1],where © ª 31 1 ? min u@f> Oi > (19.5) 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 di eren- tiable 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 di erentiable 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 k k k { (w) F := max { (w) X (19.7) wM[w0>w0+1]
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 + i (v> { (v)) gv (19.8)
v=w0 574 Chapter 19. Ordinary Di erential Equations
transforms Eu ({0) in to Eu ({0) since ° ° ° Zw ° ° ° ° ° kx ({) {0k =max° i (v> { (v)) gv° wM[w0>w0+1] ° ° v=w0 Zw
max ki (v> { (v))k gv 1f?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 ° ° ° Zw ° ° ° ° ° 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 k k k k 1Oi {1 {2 F = t {1 {2 F where t := 1Oi ? 1 for small enough u. Then, by Theorem (the contractionprinciple)14.17,weconcludethat(19.6)hasaunique 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 di erent 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 k ° 3Oi w ° { (w) max := max h { (w) X (19.9) wM[w0>w0+] 19.2. Regular ODE 575
Then
Zw
k k 3Oi v Oi v k k x ({1) x ({2) Oi h h {1 (v) {2 (v) F = v=w0 Zw ¡ ¢ Oi v 3Oi v k k Oi h h {1 (v) {2 (v) F gv v=w0 Zw
Oi v Oi h k{1 (w) {2 (w)kmax gv =
v=w0 Zw ¡ ¢ Oi v Oi w Oi h gv k{1 (w) {2 (w)kmax = h 1 k{1 (w) {2 (w)kmax
v=w0
Multiplying this inequality by h3Oi w and taking max we get wM[w0>w0+] ¡ ¢ 3Oi kx ({1) x ({2)kmax 1 h 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
2 {˙ (w)={ (w) >{(0) = 1 1 has the exact solution { (w)= that exists only on [0> 1) but not 1 w for all [0> 4). 576 Chapter 19. Ordinary Di erential 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
{ R1+q || | k k } V := (w> {) 5 w w0 w0, { {0 F u (19.10) C and the partial derivative i : V $ Rq is also continuous on V. C{ Define the sets ° ° ° ° ° C ° M := max ki (w> {)k, O := max ° i (w> {)° (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
{ +1 (w)={0 + i (v> { (v)) gv q q (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 k q+1 k k {q+1 (w) { (w) F t {0 { (w) F (19.14)
Proof. Consider the integral equation (19.6) and the integral op- erator 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 ° ° ° Zw ° ° ° k k ° ° x ({ (w)) {0 =max° i (v> {q (v)) gv° wM[w0>w0+] ° ° v=w0 max (w w0)maxki (w> {)k M u wM[w0>w0+] (w>{)MV i.e., x (P) P. By the classical mean value theorem 16.5 ° ° ° ° k k ° C | ° k k i (w> {) i (w> |) = ° i (w> }) }M[{>|] ({ |)° O { | C{ and, hence, ° ° ° Zw ° ° ° ° ° kx ({ (w)) x (| (w))k =max° [i (v> { (v)) i (v> | (v))] gv° wM[w0>w0+] ° ° v=w0 k k k k O max { (w) | (w) = t { (w) | (w) F wM[w0>w0+] Applying now the contraction principle we obtain that (19.6) has a unique solution { 5 Eu ({0).Wealsohave
Zw k k {q+1 (w) { (w) F =max [i (v> {q (v)) i (v> { (v))] gv wM[w0>w0+] v=w0 k k q+1 k k t {q (w) { (w) F t {0 { (w) F 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 Di erential 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. 1 b) Direct proof (Hartman 2002).LetA0 and {0(w) 5 F [w0 > w0] satisfy on [w0 >w0] the following conditions: {0(w)={0, k{0(w) {0k 0 u and k{0(w)k g.For0 ?% 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
0 on [w0>w0 + %].Notethat{%(w) is F -function on [w0 >w0 + %] satis- fying
k{%(w) {0k 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 {% (w) exists uniformly on q<" q [w0 >w0 + ] > that implies that
k k i (w> {%q (w %q)) i (w> { (w)) $ 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> {) -setofE R1+q satisfying ki (w> {)k M.Letalso E0 be a compact subset of E.ThenthereexistsaA0, depending on E, E0 and M,suchthatif(w0>{0) 5 E0, then (19.6) has a solution on |w w0| . 19.2. Regular ODE 579
19.2.2 Di erential inequalities, extension and uniqueness The most important technique in ODE-theory involves the "integra- tion"ofthe,so-called,di erential inequalities. In this subsection we present results dealing with this integration process and extensively used throughout, and, then there will be presented its immediate ap- plication 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 (fA0) and x (y) is a positive non-decreasing continuous function, that is,
0 ? x (y) 5 F [w0> 4) ;y 5 (0> y¯) > y¯ 4 (19.19)
Denote Zy gv [ (y):= (0 ?y?y¯) (19.20) x (v) v=f If in addition
Zw
( ) g ? [ (¯y 0) >w5 [w0> 4) (19.21)
=w0 580 Chapter 19. Ordinary Di erential 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),thatis,
1 | = [ (y) >y = [3 (|) (19.23)
In particular, if y¯ = 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)) 3 4 ( ) Zw w x Cf + ( ) x (y ( )) g D
=w0 Integrating the last inequality, we obtain
Zw Zw (v) x (y (v)) 3 4 ( ) (19.24) Zv gv v gv = = v w0 x Cf + ( ) x (y ( )) g D v w0
=w0 Denote Zw z (w):=f + ( ) x (y ( )) g
=w0 Then evidently z˙ (w)= (w) x (y (w)) 19.2. Regular ODE 581
Hence, in view of (19.20), the inequality (19.24) may be represented as Zw Zz(w) z˙ (v) gz gv = x (z (v)) x (z) v=w0 z=z(w0) Zw
= [ (z (w)) [ (z (w0)) (v) gv
v=w0
Taking into account that z (w0)=f and [ (z (w0)) = 0,fromthelast inequality it follows
Zw [ (z (w)) (v) gv (19.25)
v=w0
Since ´ 1 [ (y)= (0 ?y?y¯) x (y) the function [ (y) has the uniquely defined continuous monotonically increasing inverse function [31 (|) defined within the open interval ([ (+0) > [ (¯y 0)). Hence, (19.25) directly implies 3 4 Zw Zw 1 z (w)=f + ( ) x (y ( )) g [3 C (v) gvD
=w0 v=w0 that, in view of (19.18), leads to (19.22). Indeed, 3 4 Zw Zw 1 y (w) f + ( ) x (y ( )) g [3 C (v) gvD
=w0 v=w0
The case y¯ = 4 and [ (4)=4 is evident. Lemma is proven.
Corollary 19.3 Taking in (19.22)
x (y)=yp (pA0>p=1)6 582 Chapter 19. Ordinary Di erential Equations it follows 1 5 6 Zw p 1 y (w) 7f13p +(1 p) ( ) g 8 for 0 ?p?1 (19.26)
=w0 and 1 5 6 Zw 3 p 1 y (w) f 71 (1 p) fp31 ( ) g 8
=w0 for Zw 1 pA1 and ( ) g ? (p 1) fp31 =w0
Corollary 19.4 ((Gronwall 1919) ) If y (w) and (w) are nonnega- tive 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 4 Zw y (w) f exp C (v) gvD (19.28)
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 fA0 Zy ³ ´ gv y [ (y):= =ln v f v=f 19.2. Regular ODE 583 and 1 [3 (|)=f · exp (|) that implies (19.28). The case f =0follows from (19.28) applying f $ 0.
Di erential inequalities Here we completely follow (Hartman 2002).
Definition 19.1 Let i (w> {) be a continuous function on a plane (w> {)- set E.Byamaximal 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).Theminimal solution is similarly defined.
Lemma 19.2 Let i (w> {) be a continuous function on a rectangle © ª + R2 | k k V := (w> {) 5 w0 w w0 + w0, { {0 F 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.Let0 ?0 ?. Then, by the Peano’s existence theorem 19.3, the Cauchy problem 1 {˙(w)=i (w> {)+ >{(w0)={0 (19.32) q 0 has a solution { = {q(w) on [w0>w0 + ] if q is su!ciently large. Then { } 0 there exists a subsequence qn n=1>2>== such that the limit { (w)= 584 Chapter 19. Ordinary Di erential Equations
0 0 lim {q (w) exists uniformly on [w0>w0 + ] and { (w) is a solution of n<" n 0 (19.29). To prove that (19.30) holds on [w0>w0 + ] it is su!cient to verify 0 { (w) {q(w) on [w0>w0 + ] (19.33) for large enough q. If this is not true, then there exists a w = w1 5 0 (w0>w0 + ) 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) 0 0 1 { (w2)={ (w2)+ h,sothat{ (w) A{(w) for wAw2 near w2.This q q 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 2 and (w0>{0) 5 E R . Then the Cauchy’s problem (19.29) a maximal and minimal solution near (w0>{0).
Right derivatives Lemma 19.3 1. If q =1and { 5 F1 [d> e] then |{ (w)| has a right derivative 1 GU |{ (w)| := lim [|{ (w + k)| |{ (w)|] (19.34) 0?k<0 k such that ½ {0 (w)sign({ (w)) if { (w) =06 G |{ (w)| = 0 (19.35) U |{ (w)| if { (w)=0 and 0 |GU |{ (w)|| = |{ (w)| (19.36)
2. If qA1 and { 5 F1 [d> e] then k{ (w)k has a right derivative 1 GU k{ (w)k := lim [k{ (w + k)k k{ (w)k] (19.37) 0?k<0 k such that on w 5 [d> e)
kGU k{ (w)kk =maxGU |{n (w)| n=1>===>q (19.38) k 0 k {| 0 | | 0 |} { (w) := max {1 (w) > ===> {q (w) 19.2. Regular ODE 585
Proof. The assertion 1) is clear when { (w) =06 and the case { (w)=0follows from the identity
0 0 { (w + k)={ (w)+k{ (w)+r (k)=k [{ (w)+r (1)] so that, in general, when k $ 0
0 |{ (w + k)| = |{ (w)| + k [|{ (w)| + r (1)]
The multidimensional case 2) follows from 1) if take into account that
| | | | | 0 | {n (w + k) = {n (w) + k [ {n (w) + r (1)] Taking the max of these identities, we obtain n=1>===>q ¸ k k k k | 0 | { (w + k) = { (w) + k max {n (w) + r (1) n=1>===>q whereas k $ 0.Thisproves(19.38).
2 0 Example 19.1 Let { (w):=(w w0) .Then{ (w):=2(w w0) is con- 1 tinuous and, hence, { (w) 5 F . By Lemma 19.3 it follows GU |{ (w)| = 2 |w w0|.
Di erential inequalities The next theorem deals with the integration of di erential inequalities andismostusedintheODE-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 ¾ y (w0) {0> (w> y) 5 E (19.39) GUy (w) i (w> y (w))
Then, on the common interval of existence of {0(w) and y (w)
y (w) {0(w) (19.40) 586 Chapter 19. Ordinary Di erential Equations
Remark 19.3 If the inequalities (19.39) are reversed with the left- derivative GOy (w) instead of GUy (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.Itissu!cient to show that there exists a A0 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 di erent from . Let in Lemma 19.2 qA0 be large enough and be chosen independent of q such that (19.32) has a solution { = {q(w) on [w0>w0 + ].Inview ofLemma19.2itissu!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 GUy (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 di erential inequalities) Let i (w> {), {0(w) be as in Theorem 19.4 and j (w> {) be continuous on an open 2 (w> {)-set E R satisfying
j (w> {) i (w> {) (19.42)
Let also y (w) be a solution of the following ODE:
y˙ (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 0 {˙ (w)=i (w> { (w)) >{(w0):={ 5 R and {0(w) be the minimal solution of
{˙ (w)=i (w> { (w)) >{(w0):={0 0 1 Let also | = | (w) be a F vector valued function on [w0>w0 + ] such that 0 {0 k| (w0)k { E R2 (w> |) 5 (19.45) g (k| (w)k) i (w> k| (w)k) gw Then the first (second) of two inequalities
0 {0 (w) k| (w)k { (w) (19.46)
0 holds on any common interval of existence of {0 (w) and | (w) (or { (w) and | (w)). Corollary 19.9 Let i (w> {) be continuous and non-decreasing on { 0 when w 5 [w0>w0 + ].Let{ (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 non- decreasing on {. Proof.DenotebyY (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 Y˙ (w)=i (w> y (w)) i (w> Y (w)) 0 By Theorem 19.4 we have Y (w) { (w) on [w0>w0 + ].Thusy (w) {0 (w) that completes the proof. 588 Chapter 19. Ordinary Di erential 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
¡ 2¢ ({> i (w> {)) [ k{k (19.49) where the function [ satisfies the condition
Z" gv = 4, [ (v) A 0 as v v0 0 (19.50) [ (v) v=v0
Then the Cauchy’s problem
q {˙ (w)=i (w> { (w)) >{(w0)={0 5 U has a solution on the complete semi-axis [w0> 4) for any {0 5 Rq.
2 Proof. Notice that for the function z (w):=k{ (w)k in view of (19.49) we have g z (w)=2({ (w) > {˙ (w)) = 2 ({ (w) >i(w> { (w))) gw ¡ 2¢ 2[ k{ (w)k =2[ (z (w))
Then by Theorem 19.4 (see (19.44)) it follows that z (w0) v0 implies z (w) v (w),wherev (w) satisfies 2 v˙ (w)=2[ (v (w)) >v(w0)=v0 := k{0k
But the solution of the last ODE is always bounded for any finite w w0. Indeed, Zv(w) gv =2(w w0) (19.51) [ (v) v=v0 and [ (v) A 0 as v v0 implies that v˙ (w) A 0, and, hence, v (w) A 0 for all wAw0.Butthesolutionv (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 ?dand 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 con- tinuous on [w0>w0 + d] vector function, then the Cauchy’s problem
q {˙ (w)=D (w) { (w)+j (w) >{(w0)={0 5 U (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 FA0.
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 Di erential 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 bytheassumption.Thisprovesthetheorem.
19.2.3 Linear ODE Linear vector ODE Lemma 19.4 The solution { (w) of the linear ODE (or, the corre- sponding 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 (19.56) gw 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 di erentiation of (19.55) im- plies 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 ; < ? Zw @ det x (w> w0)=exp trD (v) gv (19.58) = > v=w0
Proof. The usual expansion for the determinant det x (w> w0) and therulefordi erentiating the product of scalar functions show that
Xq g det x (w> w0)= det x˜ (w> w0) gw m m=1
˜ where xm (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 xm>1 (w> w0) > ===> ˙ xm>q (w> w0).Butsince
Xq ˙ 0 0 k k xm>n (w> w )= dm>l (w) xl>n (w> w ) >D(w)= dm>l (w) m=l=1>===>q l=1 it follows ˜ det xm (w> w0)=dm>m (w)detx (w> w0) that gives
Xq g g det x (w> w0)= det x˜ (w> w0) gw gw m m=1 Xq = dm>m (w)detx (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 Di erential 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 Lemma 19.5 If (19.59) is fulfilled, namely, trD (v) gv A 4,
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 1 { (w)=x (w> w0) 7{0 + x3 (v> w0) j (v) gv8 (19.62)
v=w0
1 where x3 (w> w0) exists for all w 5 [w0>W] and satisfies
g 1 1 1 x3 (w> w0)=x3 (w> w0) D (w), x3 (w0>w0)=L (19.63) gw 1 Proof. By the previous corollary, x3 (w> w0) exists within the in- terval [w0>W]. The direct derivation of (19.62) implies 5 6 Zw 1 {˙ (w)=x˙ (w> w0) 7{0 + x3 (v> w0) j (v) gv8
v=w0 31 +5x (w> w0) x (w> w0) j (w)= 6 Zw 1 D(w)x (w> w0) 7{0 + x3 (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
1 x (w> w0) x3 (w> w0)=L g 1 1 [x (w> w0) x3 (w> w0)] = x˙ (w> w0) x3 (w> w0) gw g 1 + x (w> w0) x3 (w> w0)=0 gw it follows h i g 1 1 1 x3 (w> w0)=x3 (w> w0) x˙ (w> w0) x3 (w> w0)= gw
1 1 1 x3 (w> w0)[D (w) x (w> w0)] x3 (w> w0)=x3 (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 sinceby(19.57)
1 x (w> v)=x (w> w0) x3 (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 Zw
k{ (w)k (k{ (w0)k+ kj (v)k gv)exp( kD (v)k gv) (19.66)
v=w0 v=w0 594 Chapter 19. Ordinary Di erential Equations
Proof. By (19.61) it follows k{˙ (w)k kD (w)kk{ (w)k + kj (w)k Let y (w) be the unique solution of the following ODE:
y˙ (w)=kD (w)k y (w)+kj (w)k >y(w0)=k{ (w0)k which solution is Zw Zv Zw
y (w) =[y (w0) + ki (v)k exp(- kD (u)k gu)] exp( kD (v)k gv)
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
y˙ (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 Zw
k{ (w)k (k{ (w0)k kj (v)k gv)exp( kD (v)k gv) (19.67)
v=w0 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 x (w> w0):=hD w3w0 where hDw = Dnwn (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 (19.69)
( 0) ( ) = hD w3w {0 + hD w3v j (v) gv
v=w0 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 ,thatis,foranyw
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)=x˜ (w w0)=] (w w0) hU w3w0 (19.72) ] ( )=] ( + W ) and U is a constant q × q matrix.
Proof.Sincex˜ ( ) is fundamental matrix of (19.70), then x˜ ( + W) is fundamental too. By the group property (19.57) it follows x˜ ( + W ) = x˜ ( ) x˜ (W).Sincedet x˜ (W ) =06 one can represent x˜ (W) as x˜ (W )= hUW and hence x˜ ( + W)=x˜ ( ) hUW (19.73) So, defining ] ( ):=x˜ ( ) h3U > we get
( + ) h ] ( + W )=i x˜ ( + W ) h3U W = x˜ ( + W) h3UW h3U = x˜ ( ) h3U = ] ( ) that completes the proof.
First integrals and related adjoint linear ODE
Definition 19.2 AfunctionI = I (w> {):R × Cq $ C, belonging to F1 [R × Cq], is called the first integral toODE(19.1)ifitisconstant 596 Chapter 19. Ordinary Di erential 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 Cwμ 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 F1 [Cq]. The existence of the first integral for ODE (19.1) permits to de- crease 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).Ifone 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)) + (19.78) (DW (w) } (w) >{(w)) = (} ˙ (w)+DW (w) } (w) >{(w)) 19.2. Regular ODE 597
0 0 0 0 Suppose that }˙ (w )+DW (w ) } (w ) =06 for some w w0.Put
0 0 0 0 { (w ):=˙} (w )+DW (w ) } (w )
0 0 1 0 Since { (w )=x (w >w0) {0 and x3 (w >w0) always exists, then for {0 = 1 0 0 x3 (w >w0) { (w ) we obtain
g I (w0>{(w0)) = (} ˙ (w0)+DW (w0) } (w0) >{(w0)) gw 2 = k}˙ (w0)+DW (w0) } (w0)k =06 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 ad- joint to (19.54). For the corresponding inhomogeneous system (19.61) the adjoint system is
× }˙ (w)=DW (w) } (w) j˜ (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 Amatrixx (w> w0) is a fundamental matrix for the linear 31 1 W ODE (19.54) if and only if (xW (w> w0)) =(x3 (w> w0)) is a funda- mental matrix for the adjoint system (19.77).
1 Proof. Since x (w> w0) x3 (w> w0)=L by di erentiation it follows that
g 31 31 g 31 31 x (w> w0)=x (w> w0) x (w> w0) x (w> w0)=x (w> w0) D (w) gw gw and, taking the complex conjugate transpose of the last identity gives ¡ ¢ ¡ ¢ g 31 W W 31 W x (w> w0) = D (w) x (w> w0) gw Theconverseisprovedsimilarly. 598 Chapter 19. Ordinary Di erential 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.,
C C {˙ (w)= K (}>{) > }˙ (w)= K (}>{) (19.80) C} C{ 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,wehave Cw μ ¶ μ ¶ g C C C K (w> }> {)= K (w> }> {)+ K (}>{) > }˙ + K (}>{) > {˙ gw μ Cw ¶C}μ C{ ¶ 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 (19.83) w
Proof. One has directly
g 2 k{ (w)k =(˙{ (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 j˜ (w) be con- tinuous 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) j˜ (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 Anumber is called a Lyapunov order number (or the index of the increment) for a vector function { (w) de- 0 fined for w w0, if for every %A0 there exists positive constants F% and F% such that
( +%)w k{ (w)k F%h for all large w (19.85) k k 0 ( 3%)w { (w) F% h for some arbitrary large w that equivalently can be formulated as
=limsupw31 ln k{ (w)k (19.86) w<"
Lemma 19.12 If { (w) is the solution of (19.61), then it has the Lya- punov order number 3 4 Zw 1 lim sup w3 ln Ck{ (w0)k + ki (v)k gvD w<" v=w0 Zw (19.87) + lim sup w31 kD (v)k gv w<" v=w0
Proof. It follows directly from (19.66). 600 Chapter 19. Ordinary Di erential Equations
19.2.5 Riccati di erential equation Let us introduce the symmetric q × q matrix function S (w)=S | (w) 5 F1 [0>W] which satisfies the following ODE:
| < S˙ (w)=S (w) D (w)+D (w) S (w) A @A S (w) U (w) S (w)+T (w) (19.88) A >A S (W)=J 0 with × × D (w) >T(w) 5 Rq q>U(w) 5 Rp p (19.89)
Definition 19.5 We call ODE (19.88) the matrix Riccati di er- ential equation.
Theorem 19.9 (on the structure of the solution) Let S (w) be a symmetric nonnegative solution of (19.88) defined on [0>W].Then there exist two functional q × q matrices [ (w) >\ (w) 5 F1 [0>W] sat- isfying the following linear ODE μ ¶ μ ¶ [˙ (w) [ (w) = K (w) ˙ ( ) \ (w) \ w (19.90) [ (W )=L> \ (W)=S (W )=J with ¸ D (w) U (w) K (w)= | (19.91) T (w) D (w) 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,sodet [ (W)=1A 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.Asaresult,[31 (w) exists within the small semi-open interval (W >W]. Then, directly using (19.90) and in view of the identities £ ¤ 1 g 1 1 [3 (w) [ (w)=L> [3 (w) [ (w)+[3 (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)=S˙ (w)
g that implies [\ (w) [31 (w) S (w)] = 0,or, gw 1 \ (w) [3 (w) S (w)= const wM(W 3 >W]
But for w = W we have
1 const = \ (W ) [3 (W) S (W)=\ (W ) S (W)=0 wM(W 3 >W]
So, for all w 5 (W >W] it follows S (w)=\ (w) [31 (w). 602 Chapter 19. Ordinary Di erential 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 ; < ? WZ3 @ det [ (W )=det[ (0) exp wu [D (w) U (w) S (w)] gw = > w=0 ; < ?ZW @ 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)=0for any w 5 [0>W].So, \ | (w)=[| (w) \ (w) [31 (w)=[| (w) S (w) and, hence, by the trans- position 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 di erential 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 603
19.2.6 Linear first order partial DE Consider the following linear first order partial DE
Xq C} [ ({> }) = ] ({> }) (19.94) l C{ l=1 l where { 5 Rq is a vector of q independent real variables and } = } ({) is a real-valued function of the class F1 (X ), { 5 X Rq. Defining μ ¶| | C} C} C} [ ({> }):=([1 ({> }) > ===> [q ({> })) > := > ===> C{ C{1 C{q theequation(19.94)canberewrittenasfollows μ ¶ C} [ ({> }) > = ] ({> }) (19.95) C{ Any function } = } ({) 5 F1 (X ) satisfying (19.95) is its solution. If so, then its full di erential g} is μ ¶ Xq C} C} g} = >g{ = g{ (19.96) C{ C{ l l=1 l Consider also the following auxiliary system of ODE:
g{1 g{ g} = ···= q = (19.97) [1 ({> }) [q ({> }) ] ({> }) or, equivalently,
1 31 g{ 31 g{q 31 [1 ({> }) = ···= [ ({> }) = ] ({> }) (19.98) g} q g} or, < g{1 1 A = [1 ({> }) ]3 ({> }) A g} @ ··· (19.99) g{ A q = [ ({> }) ]31 ({> }) >A g} q 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 Di erential Equations
Lemma 19.13 If } = } ({) satisfies (19.97), then it satisfies (19.95) too.
Proof. Indeed, by (19.99) and (19.96) we have
31 g{l = [l ({> }) ] ({> }) g} Xq Xq C} C} g} = g{ = [ ({> }) ]31 ({> }) g} C{ l C{ l l=1 l l=1 l 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 characteris- tics of (19.94).
Assume that this solution can be resolved with respect to the con- stants 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# ({> }) det l =06 (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. ByTheorem16.8onanimplicitfunction,thesystems (19.74) can be uniquely resolved with respect to { if (19.104) is ful- filled. 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 Xq C} { = s} (s is a constant) (19.106) l C{ l=1 l The system (19.97)
g{1 g{ g} = ···= q = {1 {q s} has the following first integrals
s {l } = fl (l =1> ===> q) So, } = } ({) can be found from the algebraic equation
s s x ({1 }> ===> {q })=0 where x (#1> ===> #q) is an arbitrary function, for example, Xq Xq 6 x (#1> ===> #q):= l#l, l =0 l=1 l=1 that gives à ! 1 Xq 3 Xq s } = l l{l l=1 l=1 606 Chapter 19. Ordinary Di erential Equations 19.3 Carathéodory’s Type ODE
19.3.1 Main definitions The di erential 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 inte- gral 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 con- ditions 19.1, is called the Carathéodory’s type ODE. b) Afunction{ (w),definedonanopenorclosedintervalo,iscalled asolutionof the Carathéodory’s type ODE if
-itisabsolutely 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{ {0k 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 approx- imate 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 Di erential 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) {0k 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 %A0 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{ {0k e, then a solution exists on the closed interval [w0 g> w0] where g satisfies (19.111).
Proof.Thecasew w0 is reduced to the case w w0 by the simple substitution of (w) for w. 1+ Corollary 19.13 Let (w0>{0) 5 D R q and let there exists a sum- mable 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
2 (i (w> {) i (w> |) >{ |) o (w) k{ |k (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 2 g k}k =2 }> } =2(i (w> {) i (w> |) >{ |) gw gw
g 2 2 almost everywhere. By (19.115) we obtain k}k o (w) k}k and, gw Zw ¡ ¢ g 2 hence, h3O(w) k}k 0 where O (w)= o (v) gv.Thus,theab- gw v=w0 solutely continuous function (i.e., it is a Lebesgue integral of some ( ) 2 other function) h3O w k}k does not increase, and it follows from } (w0)= 0 that } (w)=0for 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 inequal- ity), thus the uniqueness may be considered to be proven for w w0 also under the condition (19.114). 610 Chapter 19. Ordinary Di erential 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 sys- tem may be considered as a dynamic system with a variable structure, since for di erent w1 =6 w2 we will have i (w1>{) =6 i (w2>{).Fromthis point of view such treatment seems to be naive and having no cor- rect mathematical sense. But if we consider the special class of ODE (19.107) given by
XQ l {˙ (w)=i (w> { (w)) := " (w 5 [wl31>wl)) i ({ (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)) := >wl31 ?wl (19.117) 0 if w@5 [wl31>wl) then ODE (19.116) can be also treated as ODE with "jumping" pa- rameters (coe!cients). Evidently, that if i l ({) are continuous on a compact D> and, hence, are bounded, that is, ° ° ° ° max max i l ({) P (19.118) l=1>===>Q {MD then the third Carathéodory’s condition (19.109) will be fulfilled on the time interval [> ],since
XQ p (w)=P " (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.,
l l 2 (i ({) i (|) >{ |) ol k{ |k 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 perturba- tions: XQ
{˙ (w)=i ({ (w)) + 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 Zw XQ
{ (w)={ (w0)+ i ({ (v)) gw + l (w wl) gw (19.121) l=1 v=w0 v=w0 The last term, by the property (15.134), can be represented as
Zw XQ XQ
l g" (vAwl)= l" (wAwl) l=1 l=1 v=w0 where " (w wl) is the "Heavyside’s (step) function" defined by (19.117). Let is apply the following state transformation: XQ
{˜ (w):={ (w)+ l" (wAwl) l=1
New variable {˜ (w) satisfies (with 0 := 0) the following ODE: Ã ! XQ g {˜ (w)=i {˜ (w) " (wAw) gw l l Ãl=1 ! XQ Xl
= " (wAwl) i { (w) l (19.122) l=1 v=1 XQ ˜l = " (wAwl) i ({ (w)) l=1 where à ! Xl ˜l i ({ (w)) := i { (w) l v=1 612 Chapter 19. Ordinary Di erential 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 fur- ther 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].Thefunctioni : R $ R is a Lipschitz function satisfying the, so-called, Lipschitz condition, that is, for any {> {0 5 R 0 0 |i ({) i ({ )| O |{ { | > 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 (19.126) (this inequality is assumed to be valid for any { 5 R). 19.4. ODE with DRHS 613
There are two possibilities to do that:
1. use any continuous control,namely,takex : 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 =0any 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 Di erential 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
Y˙ ({ (w)) = { (w)˙{ (w)= { (w)[i ({ (w)) + x ({ (w))] |{ (w)||i ({ (w))| (19.129) 2 + { (w) x ({ (w)) O |{ (w)| + { (w) x ({ (w)) Since i ({) is unknown let us select x ({) in (19.128) as x ({)=x ({)=n{ fru (19.130) xfrps ({):=0 The use of (19.130) in (19.129) leads to the following identity:
Y˙ ({ (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
Y˙ ({ (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 "high- gain control" solves the stabilization problem.
2. i ({) is unknown and it is admissible that i (0) =06 .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 615
where ; ? 1 if {A0 sign({):= 1 if {?0 (19.132) = 5 [1> 1] if { =0
(see Fig.19.1). Staring from some {0 =06 > analogously to (19.129)
Figure 19.1: The signum function.
and using (19.126), we have
Y˙ ({ (w)) = { (w)˙{ (w)={ (w)[i ({ (w)) + x ({ (w))]
+ |{ (w)||i ({ (w))| + { (w) x ({ (w)) |{ (w)| (i0 + i |{ (w)|)
+ 2 nw{ (w)sign({ (w)) = |{ (w)| i0 + i |{ (w)| nw |{ (w)| Taking 0 1 nw = n({ (w)) := n + n |{ (w)| (19.133) 0 1 + n Ai0>n Ai one has
0 1 + 2 Y˙ ({ (w)) |{ (w)| (n i0) (n i ) |{ (w)| p 0 s 0 |{ (w)| (n i0)= 2(n i0) Y ({ (w)) 0 Hence, ¡ ¢ gY ({ (w)) s 0 p 2 n i0 gw Y ({ (w)) 616 Chapter 19. Ordinary Di erential Equations
that leads to the following identity ³p p ´ ¡ ¢ s 0 2 Y ({ (w)) Y ({0) 2 n i0 w
or, equivalently, p p 0 n 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 0 W 2Y ({ ) w = 0 (19.134) n i0
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 knowl- edge of its model. Besides, thereachingphasemaybedoneas 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 =0sinceitarrivestoitinfinitetimebutwithanonzero rate, namely, with {˙ (w) such that ½ i (0) + n0 if { (w) $ +0 {˙ (w)= 0 i (0) n if { (w) $ 0 that provokes the, so-called, "chattering e ect" (see Fig.19.2). Sim- ple engineering considerations show that some sort of smoothing (or, low-pass filtering) should be applied to keep dynamics close to the sta- tionary point {W =0. 19.4. ODE with DRHS 617
Figure 19.2: The chattering e ect.
Remark 19.10 Notice that when { (w)={W =0we only know that
{˙ (w) 5 [i (0) n0>i(0) + n0] (19.135)
This means that we deal with a di erential inclusion (not an equa- tion) (19.135). So, we need to define what does it mean mathematically correctly a solution of a di erential 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 di erential inclu- sions General requirements to a solution As it is well known, a solution of the di erential equation
{˙ (w)=i (w> { (w)) (19.136) withacontinuousright-handsideisafunction{ (w) which has a deriv- ative and satisfies (19.136) everywhere on a given interval. This defi- nition 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 nec- essarily meet the following requirements: 618 Chapter 19. Ordinary Di erential Equations
- For di erential equations with a continuous right-hand side the de- finition of a solution must be equivalent to the usual (standard) one.
- For the equationR {˙ (w)=i(w) the solution should be the functions { (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 wAw0 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 map- ping i : R × Rq $ Rs,issaidtobepiecewise 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 G = Gl l=1 such that in each of them the function i (w> {) is continuous up to the boundary ¯ Ml := Gl\Gl (l =1> ===> o) (19.137) of a measure zero. Themostfrequentcaseistheonewheretheset
[o M = Ml l=1 of all discontinuity points consists of a finite number of hypersurfaces 1 0=Vn ({) 5 F >n=1>===>p where Vn ({) is a smooth function. 19.4. ODE with DRHS 619
Definition 19.10 The set M defined as
q | M = {{ 5 R | V ({)=(V1 ({) > ===> Vp ({)) =0} (19.138) is called a manifold in Rq.Itisreferredtoasa smooth manifold 1 if Vn ({) 5 F >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) Asolu- tion { (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 di eren- tial 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 ],wherethesetF (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 func- tion i (w> {); 2) is a segment (a convex polygon, or polyhedron), which in the case when (w> {) 5 Ml (19.137) has the vertices W il (w> {) := lim i (w> { ) (19.140) W W (w>{ )MGl>{ <{
All points il (w> {) are contained in F (w> {), but it is not obliga- tory that all of them are vertices. Example 19.4 For the scalar di erential inclusion {˙ (w) 5sign ({ (w)) the set F (w> {) is as follows (see Fig.19.3): 1. F (w> {)=1 if {A0; 2. F (w> {)=1if {?0; 3. F (w> {)=[1> 1] if { =0. 620 Chapter 19. Ordinary Di erential Equations
Figure 19.3: The right-hand side of the di erential inclusion {˙ w = vljq ({w).
Semi-continuoussetsasfunctions 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 %A0 there exists = (w0>{0>%) such that the inclusion
(w> {) 5 {} |k} (w0>{0)k } (19.141) implies
F (w> {) 5 {i |ki i (w0>{0)k %} (19.142)
• a continuous in the point (w0>{0) if it is a semi-continuous and, additionally, for any %A0 there exists = (w0>{0>%) such that the inclusion 0 0 (w0>{0) 5 {} |k} (w >{)k } (19.143) implies
0 0 F (w0>{0) 5 {i |ki i (w >{)k %} (19.144)
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 ^ {˙ (w) (19.145) a.a. wM[>] where Conv is a convex closed set containing ^ {˙ (w) for almost all w 5 [> ].
Proof. By the definition of the Lebesgue integral
Z 1 1 ({ {)= {˙ (w) gw = lim vn n<" w= 622 Chapter 19. Ordinary Di erential Equations where Xn Xn v = l {˙ (w ) > 0> = | | n | | l l l l=1 l=1 are Lebesgue sums of the integral above. But vn 5 Conv ^ {˙ (w). a.a. wM[>] 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{ {0k >|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 ab- solutely 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 w := l w0 +m (l =0> ±1>==± p) and construct the, so-called, partially linear p Euler’s curves ³ ´ ³ ´ ³ ´ h i (p) (p) ˆ(p) (p) (p) (p+1) {p (w):={p wl + w wl il wl >w5 wl >wl ³ ´ ³ ´ ³ ³ ´´ (p) ˆ(p) (p) F (p) (p) {p w0 = {0> il wl 5 wl >{p wl
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 vec- tor 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.Inviewof Lemma 19.14, for any kA0 one has
[pn ( ) 31 Conv ˆ pn k [{pn (w + k) {pn (w)] il a.a.[ 0 0+ ] w 3 >w = l 3pn [pn ( ) Conv ˆ pn k l il ():=Dn a.a. M w03 >w0+ +k = pn pn l 3pn Since F (w> {) is semi-continuous, it follows that sup inf k{ |k $ 0 |MD {MDn [po whenever n $4(here D := Conv i (> {)). The convex- a.a. [ + ] M w>w k = l 3po ity of F (w> {) implies also that sup inf k{ |k $ 0 when k $ 0 {MD |MF(w>{) that, together with previous property, proves the theorem. Remark 19.12 BythesamereasonsasforthecaseofregularODE, we may conclude that the solution of the di erential 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 F1 (19.146)
The surface separates its neighborhood (in Rq) into domains G+ and G3.Forw =constand for the point {W approaching the point { 5 V from the domains G+ and G3 let us suppose that the function i (w> {W) has the following limits: lim i (w> {W)=i 3 (w> {) ( W) G3 W w>{ M >{ <{ (19.147) lim i (w> {W)=i + (w> {) (w>{W)MG+>{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 Di erential 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),isoften called a sliding motion ( or, mode). Defining the projections of the vectors i 3 (w> {) and i + (w> {) to the surface V (uv ({) =0)6 as 3 + (uv ({) >i (w> {)) + (uv ({) >i (w> {)) s3 (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 canbeeasilyfoundfromtheequation ¡ ¢ 0 uv ({) >i (w> {) =0 19.4. ODE with DRHS 625
Figure 19.6: The velocity of the motion.
or, equivalently, 0=(uv ({) >i3 (w> {)+(1 ) i + (w> {)) =
s3 (w> {)+(1 ) s+ (w> {) that implies s+ (w> {) = s+ (w> {) s3 (w> {) Finally, we obtain that
s+ (w> {) i 0 (w> {)= i 3 (w> {)+ s+ (w> {) s3 (w> {) μ ¶ (19.149) s+ (w> {) 1 i + (w> {) s+ (w> {) s3 (w> {)
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 )func- tionals as possible performance indices, but also representing a nom- inal (desired) dynamics in the form of a surface (or, manifold) in a space of coordinates. 626 Chapter 19. Ordinary Di erential 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)(|u˙ (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, design- ing the control x (w) as x (w):=nsign(h (w)) we derive that h˙ (w)=i (w> { (w)) u˙ (w) nsign (h (w)) and for Y (h)=h2@2 one has Y˙ (h (w)) = h (w)˙h (w)=h (w)[i (w> { (w)) u˙ (w) nsign (h (w))]
= h (w)[i (w> { (w)) u˙ (w)] n |h (w)| |h (w)| [i + + ] n |h (w)| s p = |h (w)| [i + + n]= 2[n i + ] Y (h (w)) 19.4. ODE with DRHS 627 and, hence, p p £ ¤ 1 + Y (h ) Y (h0) s n i w w 2 + So, taking nAi + impliesp the finite time convergence of hw (with 2Y (h0) the reaching phase w = ) to the surface (19.151) (see Fig. i 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 Di erential Equations
{¨ (w)+d2{˙ (w)+d1{ (w)=x (w)+ (w) x (w)= nsign (˜v (w)) -therelay-control (19.152) v˜(w):=˙{ (w)+f{ (w) >fA0 + | (w)| - a bounded unknown disturbance
One may rewrite the dynamic ({1 := {) as
{˙ 1 (w)={2 (w) {˙ 2 (w)= d1{1 (w) d2{2 (w)+x (w)+ (w) (19.153) x (w)= nsign ({2 (w)+f{1 (w)) Here the sliding surface is
v ({)={2 + f{1
So, the sliding motion, corresponding the dynamics v˜(w):=˙{ (w)+ f{ (w)=0> is given by (see Fig.19.10)
3fw { (w)={0h 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)) Y˙ (v)=vv˙ = 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 Y (v) 2Y (v), and, hence, the reaching time wi (see Fig.19.9) is p 2Y (v0) |{˙ 0 + f{0| w = = (19.155) i
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 | (19.156) E (w) E (w) A 0rank[E (w)] = u for any w 5 [wv>w1] (w) is known external perturbation 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)=0for all w wv (19.157) where ({> w):Uq ×U+ $ Uu is a sliding function and (21.65) defines +1 aslidingsurfacein Uq .Foreachw1 A 0 the quality of the system (19.156) motion in the sliding surface (21.65) is characterized by the performance index (Utkin 1992)
Rw1 1 | Mwv>w1 = ({ (w) >T{(w)) gw> T = T 0 (19.158) 2 wv 630 Chapter 19. Ordinary Di erential 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 (19.159) Ms where s is the set of the admissible smooth (di erentiable on all ar- guments) 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 ¸ 31 L(q3u)×(q3u) E1 (w)(E2 (w)) W (w):= 31 (19.161) 0(E2 (w))
(q3u)×(q3u) u×u u×u Here E1 (w) 5 R and E2 (w) 5 R represent the matrices E (w) in the form ¸ E1 (w) E (w)= , det [E2 (w)] =06 ; w 0 (19.162) E2 (w)
Applying (19.162) to the system (19.156), we obtain (below we will omit the time-dependence) μ ¶ μ ¶ μ ¶ μ ¶ }˙1 D˜11}1 + D˜12}2 0 ˜1 }˙ = = + + (19.163) }˙2 D˜21}1 + D˜22}2 x ˜2 where }1 5 Uq3u>}2 5 Uu and ¸ μ ¶ D˜11 D˜12 1 1 ˜1 (w) D˜ = = WDW3 + WW˙ 3 > = W(w) (19.164) D˜21 D˜22 ˜2 (w) 19.4. ODE with DRHS 631
Using the operator W 31,itfollows{ = W 31} and, hence, the perfor- mance index (19.158) in new variables } may be rewritten as ³ ´ 1 Rw1 1 Rw1 ˜ Mwv>w1 = ({> T{) gw = }>T } gw = 2 wv 2 wv Rw1 h³ ´ ³ ´ ³ ´i 1 }1> T˜11}1 +2 }1> T˜12}2 + }2> T˜22}2 gw (19.165) 2 wv ¸ 1 | 1 T˜11 T˜12 T˜ := (W 3 ) TW 3 = T˜21 T˜22 and the sliding function = ({> w) becomes ¡ ¢ 1 = W 3 }>w := ˜ (}>w) (19.166)
Remark 19.13 The matrices T˜11> T˜12> T˜21 and T˜22 are supposed to be symmetric. Otherwise, they can be symmetrized as follows:
Rw1 £¡ ¢ ¡ ¢ ¡ ¢¤ 1 ¯ ¯ ¯ Mwv>w1 = }1> T11}1 +2 }1> T12}2 + }2> T22}2 gw 2 wv ³ ´ ³ ´ ¯ ˜ ˜| ¯ ˜ ˜| (19.167) T11 := T11 +³ T11 @2> T22 := T22 ´+ T22 @2 | | T¯12 = T˜12 + T˜12 + T˜21 + T˜21 @2
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 sur- face ˜ (}>w)=0under the assumption A1, then for all w wv the corresponding sliding mode dynamics, driven by the unmatched dis- turbance ˜1 (w),aregivenby
}˙1 = D˜11}1 + D˜12}2 + ˜1 (19.169) }2 = ˜0 (}1>w) with the initial conditions }1 (wv)=(W{(wv))1. Defining }2 as a virtual control, that is, y := }2 = ˜0 (}1>w) (19.170) 632 Chapter 19. Ordinary Di erential Equations the system (19.169) may be rewritten as
}˙1 = D˜11}1 + D˜12y + ˜1 (19.171) and the performance index (19.165) becomes
Zw1 h³ ´ ³ ´ ³ ´i 1 ˜ ˜ ˜ M 1 = }1> T11}1 +2 }1> T12y + y> T22y gw (19.172) wv>w 2 wv In view of (19.171) and (19.172), the sliding surface design problem (19.159) is reduced to the following one:
Mwv>w1 $ inf (19.173) yMUu But this is the standard LQ-optimal control problem. This means W W that the optimal control yw = y (}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:
W y (}1>w>w)= ˜0 (}1>w) W ˜ (}>w)=}2 y (}1>w>w)=0 or, equivalently,
W W ({> w)=(W{)2 y ((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 (19.175) { (w) 5 Rq>x(w) 5 Ru 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 (ac- cording to V.Utkin) as the equivalent control x(ht) (w) in the system (19.175) if it satisfies the equation
V˙ ({ (w)) = J ({ (w)){ ˙ (w)=J ({ (w)) i (w> { (w) >x(w)) = 0 (19.177) C J ({ (w)) 5 Rp×q>J({ (w)) = V ({ (w)) C{
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 con- trol 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 im- plies a replacement of the undefined discontinued control on the dis- continuity boundary with a continuous control which directs the ve- locity 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 di erential 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,theright- hand side of whose di erential 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 Di erential Equations action. The corresponding equivalent control should satisfies (19.177), namely, V˙ ({ (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
1 x(ht) (w)= [J ({ (w)) E (w> { (w))]3 J ({ (w)) i (w> { (w)) (19.181)
Substitution of this control into (19.179) yields the following ODE: {˙ (w)=i (w> { (w)) 1 (19.182) E (w> { (w)) [J ({ (w)) E (w> { (w))]3 J ({ (w)) i (w> { (w)) 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.
Belowwewillshowthatx(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 2 Y ({):=kV ({)k @2> considered on the trajectories of the controlled system (19.179), one has ³ ´ Y˙ ({ (w)) = V ({ (w)) > V˙ ({ (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))kkJ ({ (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 slidingmodecontrol,i.e.,
x (w)=x(vo) (w)
(vo) 31 x (w):=nw [J ({ (w)) E (w> { (w))] SIGN (V ({ (w))) (19.183)
| nw A 0> SIGN (V ({)) := (sign (V1 ({)) > ===> sign (Vp ({))) we obtain
Xp ˙ Y ({ (w)) kV ({)kkJ ({ (w)) i (w> { (w))k nw |Vl ({ (w))| l=1
Pp that, in view of the inequality, kVk |Vl|,implies l=1
˙ Y ({ (w)) kV ({)k (nw kJ ({ (w)) i (w> { (w))k)
Selecting
nw = kJ ({ (w)) i (w> { (w))k + > A 0 (19.184) p gives Y˙ ({ (w)) kV ({)k = 2Y ({ (w)) that provides the reach- ing phase in time p 2Y ({0) kV ({0)k w = = (19.185) i
Remark 19.15 If the sliding motion on the manifold V({)=0is stable then there exists a constant n0 5 (0> 4) such that
0 kJ ({ (w)) i (w> { (w))k n and, hence, nw (19.184) may be selected as a constant
0 nw := n = n + (19.186) 636 Chapter 19. Ordinary Di erential Equations
Low-pass filtering To minimize the influence of the chattering ef- fect 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) (dy) (vo) (dy) x˙ (w)+x (w)=x (w) >x0 =0>A0 (19.187) where x(vo) (w) isgivenby(19.183).Thenextsimplelemmastatesthe relation between the, so-called, averaged control x(dy) (w),whichisthe filtered output, and the input signal x(vo) (w). Lemma 19.15 If + 1@2 | | j kJE (w> { (w))k := 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 di erence 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) ° ° ° ( ) ° x˙ dy (w) 2f@
2. The amplitude-frequency characteristic D ($) of the filter is 1 D ($)=q , $ 5 [0> 4) (19.190) 2 1+($)
whose plot is depicted at Fig.19.11 for =0=01,where| = D ($) and { = $. Proof. 1) The solution of the ODE (19.183) and its derivative are as follows:° ° ° ° ° 31° ° 31° nw [JE (w> {w)] (kJi (w> {w)k + ) [JE (w> {w)]
(kJi (w> { )k + ) = w y31 (j+ + ) 1@2 | | min ([E (w> {w) J ][JE (w> {w)]) 19.4. ODE with DRHS 637
y 1
0.9988
0.9975
0.9963
0 2.5 5 7.5 10
x
Figure 19.11: The amplitude-phase characteristic of the low-pass filter. ° ° ° ° ¡ ¢ ° (vo)° 31 + xw y j + p := f and by (19.187)
Rw (dy) 1 3(w3v)@ (vo) xw = h xv gv v=0 ¸ Rw (19.191) (dy) 1 (vo) 1 3(w3v)@ (vo) x˙ w = xw h xv gv v=0 that implies ° ° ° ° ° ° ° ( )° ° ( )° 1 Rw ° ( )° ° dy ° ° vo ° 3(w3v)@ ° vo ° x˙ w xw + h xv gv v=0 f Rw Rw f + h3(w3v)@gv = f + f h3(w3v)@g (v@)= v=0 v=0 w@R ¡ ¢ f + f h3(w@3v˜)gv˜ = f + f 1 h3w@ 2f v˜=0
Hence, (19.189) holds. 2) Applying the Fourie transformation to (19.187) leads to the following identity:
( ) ( ) ( ) m$X dy (m$)+X dy (m$)=X vo (m$) 638 Chapter 19. Ordinary Di erential Equations or, equivalently,
(dy) 1 (vo) 1 m$ (vo) X (m$)= X (m$)= 2 X (m$) 1+m$ 1+($) So, the amplitude-frequency characteristic q 2 2 D ($):= [Re X (dy) (m$)] +[ImX (dy) (m$)] of the filter (19.187) is as in (19.190). Lemma is proven.
The realizable approximation of the equivalent control
Rw ¡ ¢ (dy) (dy) (vo) 3(w3v)@ By (19.191) xw may be represented as xw = xv g h . v=0 (dy) (dy) Consider the dynamics {w of the system (19.175) controlled by xw (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
Zw Zw ¡ ¢ (dy) (vo) 3(w3v)@ (vo) (vo) 3w@ (vo) 3(w3v)@ xw = xv g h = xw x0 h x˙ v h gv v=0 v=0
( ) °Supposing° that x vo (19.183) is bounded almost everywhere, i.e., ° ° w ° (vo)° x˙ w g. The above identity leads to the following estimation: ° ° ° ° ° ° ° ° ° ° Rw ° ° ° (dy) (vo)° ° (vo)° 3w@ 3(w3v)@ ° (vo)° 3w@ xw xw x0 h + g h gv = x0 h + ° °v=0 Rw ° ° w@R 3(w3v)@ (vo) 3w@ 3(w@3v˜) g h g (v@)=°x0 ° h + g h gv˜ = =0 ° ° ˜=0 v ° ° ¡ ¢ v ¡ ¢ (vo) 3w@ 3w@ 3w@ °x0 ° h + g 1 h = g + R h
(dy) So, xw may be represented as
(dy) (vo) xw = xw + w (19.192) 19.4. ODE with DRHS 639
where w maybedoneassmallasyouwishtaking tending to zero, since ¡ ¢ k k 3w@ w g + R h (vo) (dy) As a result, the trajectories {w and {w will di er a little bit. Indeed, ³ ´ ³ ´ ( ) ( ) ( ) ( ) ˙ vo = vo vo vo {w i ³w> {w ´ E ³w> {w ´xw (dy) (dy) (dy) (dy) {˙ w = i w> {w E w> {w xw Defining ³ ´ ³ ´ ³ ´ ˜ (dy) ˜ (dy) ˜ (dy) E = E w> {w > J = J {w > i = i w> {w and omitting the arguments for the simplicity, the last equation may be represented as (vo) (vo) (dy) ˜ ˜ (dy) {˙ w = i Exw > {˙ w = i Exw
(vo) (dy) Hence by (19.192), the di erence {w := {w {w satisfies
Rw h³ ´ i ˜ (vo) ˜ (dy) {w = {0 i i Exv + Exv gv = v=0 Rw h³ ´ ³ ´i ˜ (vo) ˜ (vo) {0 i i Exv + E xv + v gv 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 h° ° °³ ´ °i Rw ° ° ° ° k k ° ˜° ° ˜ (vo) ˜ ° {w i i + E E xv + Ev gv h v=0 ° ° ° ° i Rw ° ° ° ° k k k k ° (vo)° ° ˜° k k Oi {v + OE {v xv + E v gv h³v=0 ° °´ ° ° i Rw ° ° ° ° ¡ ¡ ¢¢ (vo) ˜ 3w@ Oi + OE °xv ° k{vk + °E° g + R h gv v=0 ¡ ¢ ³ ´ 3w@ 1 3w@ Since R h = R h = r (1) % and ° ° ° ° ° ° ° (vo)° (vo) ° ˜° + xv x+ ? 4> E E ? 4 640 Chapter 19. Ordinary Di erential Equations
we finally have
Rw h³ ´ i (vo) + k{wk Oi + OEx+ k{vk + E (g + %) gv v=0 Rw ³ ´ + (vo) E (g + %) wi + Oi + OEx+ k{vk gv 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 4 Zw 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{wk >f= E (g + %) wi >(v)=Oi + OEx+
for any v 5 [0>wi ).So, ³³ ´ ´ + (vo) k{wk :=E (g + %) wi exp Oi + OEx+ wi (19.195)
Claim. For any finite reaching time wi and any small value A0 there exists a small enough such that k{wk is less than .
2. Sliding mode phase (wAwi ). During the sliding mode phase we have ³ ´ ³ ´ ³ ´ (vo) ˙ (vo) (ht) V {w = V {w = J i Exw =0 (19.196)
(ht) (dy) if xw = xw for all wAwi . Applying xw = xw we can not guarantee (19.196) already. Indeed, ³ ´ ³ ´ Zw ( ) ( ) ¡ ¢ dy = dy + ˙ (dy) V {w V {wi V {v gv = v wi 19.4. ODE with DRHS 641
and, by (19.195), ° ³ ´° ° ³ ´ ³ ´° ° ³ ´ ° ° ( ) ° ° ( ) ( ) ° ° ( ) ° ° dy ° = ° dy - vo ° ° vo ° ( ) V {wi V {wi V {wi J wi {wi R ° 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)