Ordinary differential equations: analysis, qualitative theory and control

342 Pages·2014·3.438 MB·English

by  Logemann, Hartmut;Ryan, Eugene P

Springer Undergraduate Mathematics Series
Hartmut Logemann
Eugene P. Ryan
Ordinary
Differential
Equations
Analysis, Qualitative Theory
and Control
Springer Undergraduate Mathematics Series
Advisory Board
M.A.J.ChaplainUniversityofDundee,Dundee,Scotland,UK
K.ErdmannUniversityofOxford,Oxford,England,UK
AngusMacIntyreQueenMaryUniversityofLondon,London,England,UK
EndreSüliUniversityofOxford,Oxford,England,UK
M.R.TehranchiUniversityofCambridge,Cambridge,England,UK
J.F.TolandUniversityofCambridge,Cambridge,England,UK
For furthervolumes:
http://www.springer.com/series/3423
Hartmut Logemann Eugene P. Ryan
•
Ordinary Differential
Equations
Analysis, Qualitative Theory and Control
123
HartmutLogemann Eugene P.Ryan
Department of Mathematical Sciences Department of Mathematical Sciences
Universityof Bath Universityof Bath
Bath Bath
UK UK
ISSN 1615-2085 ISSN 2197-4144 (electronic)
ISBN 978-1-4471-6397-8 ISBN 978-1-4471-6398-5 (eBook)
DOI 10.1007/978-1-4471-6398-5
Springer LondonHeidelberg New YorkDordrecht
LibraryofCongressControlNumber:2014933526
Mathematics Subject Classification: 34A12, 34A30, 34A34, 34C25, 34D05, 34D20, 34D23, 34H05,
34H15,93C15,93D05,93D10,93D15,93D20
!Springer-VerlagLondon2014
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Preface
This text is based on various courses taught, over many years, by the au-
thors at the University of Bath. The intention is a rigorous – and essentially
self-contained – treatment of initial-value problems for ordinary differential
equations.Thematerialispresentedatatechnicallevelaccessiblebyfinalyear
undergraduatestudentsofmathematicsandappropriatealsoforstudentsinthe
early stages of postgraduate study, both in mathematics and mathematically-
oriented engineering. Only a basic grounding in linear algebra (e.g. finite-
dimensional vector spaces, norms, inner products, linear transformations and
matrices, Jordan form) and analysis (e.g. uniform continuity, uniform conver-
gence, compactness in a finite-dimensional setting, elementary differential and
integral calculus) is assumed: the typical UK undergraduate attains this level
ofmathematicalmaturitybytheendofhis/hersecondyearofstudyinmathe-
matics.Inanappendix,thesebasicsareassembledtoprovidethemathematical
frameworkunderpinningthebook.Inthemainbodyofthetext,diverseresults
arepresentedpertainingtoexistenceanduniquenessofsolutionsofinitial-value
problems, continuous dependence on initial data, flows, qualitative behaviour
ofsolutions,limitsets,stabilitytheory,invarianceprinciples,introductorycon-
trol theory, stabilization by feedback. The latter aspects, namely the coverage
ofcontroltheoreticconcepts,isadistinguishingfeature.Thisthreadrunsfrom
essentially classical linear control theory, through developments in absolute
stability of feedback systems, and terminates with an introductory account of
morerecentnotionsoffeedbackstabilizabilityandinput-to-statestability.The
book has no pretensions to comprehensiveness. On the one hand, the perme-
ating thread of control reflects a bias towards synthesis: the bringing of stable
behaviour to potentially or inherently unstable processes through appropriate
choice of inputs. On the other hand, the book does not contain material relat-
ingtothetheoryofbifurcationsorchaos(thesetopicsaretreatedinnumerous
other texts on ordinary differential equations).
Finally,wewouldliketothankMrElvijsSarkansforhisvaluablecomments
on an earlier draft of the book.
Hartmut Logemann & Eugene P. Ryan
Bath
November 2013
Notation
N the natural numbers 1,2,3,...
{ }
N the non-negative integers N 0
0
∪{ }
Z the integers
Q the field of rational numbers
R the field of real numbers
C the field of complex numbers
R the non-negative reals [0, )
+
∞
C the open right half complex plane z C: Rez >0
+
{ ∈ }
C the open left half complex plane z C: Rez <0
−
{ ∈ }
F either R or C
FN the vector space of ordered N-tuples of numbers from F
FP×N the vector space of P N matrices with entries from F
×
GL(N,F) the group of invertible FN×N matrices (general linear group)
* (in superscript) Hermitian/conjugate transpose of a matrix
direct sum (of subspaces of FN)
⊕
S⊥ orthogonal complement of a subspace S of FN
im image (of an element of FN×P): imM = Mx FN: x FP
{ ∈ ∈ }
ker kernel (of an element of FN×P): kerM = x FP: Mx=0
{ ∈ }
rk matrix rank
tr trace of a square matrix: the sum of the diagonal entries
det determinant (of a square matrix)
adj adjugate (of a square matrix)
σ spectrum (set of eigenvalues of a square matrix):
σ(M)= λ C: λ is an eigenvalue of M
{ ∈ }
span span of a set of vectors
generic symbol for a norm
∥·∥
, inner product (on FN)
⟨· ·⟩
viii Ordinary Differential Equations: Analysis, Qualitative Theoryand Control
B(x,r) the open ball of radius r>0 centred at x X, X a metric space
∈
∂S boundary of a set S in a metric space
cl(S), S closure of a set S in a metric space
dist distance from a point to a set or distance between two sets:
for z RN and non-empty sets X,Y RN,
∈ ⊂
dist(z,X):=inf z x : x X
{∥ − ∥ ∈ }
dist(X,Y):=inf x y : x X, y Y
{∥ − ∥ ∈ ∈ }
dom domain of a map
C(K,Y) vector space of continuous functions K X Y,
⊂ →
(X,Y normed spaces)
the supremum norm on C(K,Y) with K compact:
∞
∥·∥
f :=sup f(x) , a norm on Y
∥ ∥∞ x∈K∥ ∥ ∥·∥
the class of continuous and strictly increasing functions
K
a: R R with a(0)=0.
+ +
→
the class of unbounded functions
∞
K K
the class of functions b: R R R such that,
+ + +
KL × →
for all t R , b(,t) and, for all s R , b(s, ) is
+ +
∈ · ∈K ∈ ·
decreasing with b(s,t) 0 as t
→ →∞
PC(I,Y) vector space of piecewise continuous functions I Y =FP×Q,
→
I R an interval
⊂
PC1(I,Y) vector space of piecewise continuously differentiable functions
⋆ convolution (of functions)
composition (of functions)
◦
∂ i-th partial derivative: for a function X RN R,
i
⊂ →
(x ,...,x )=x f(x), (∂ f)(x) denotes the derivative at x
1 N i
-→
of f with respect to the i-th component x of its argument x
i
gradient of a function X RN R, (x ,...,x )=x f(x):
1 n
∇ ⊂ → -→
( f)(x)= ∂ f,...,∂ f (x)
1 N
∇
D differentiation in the Fr´echet sense.
! "
For a function f: X RN RM, (Df)(x):= (∂ f )(x) is the
j i
⊂ →
M N matrix of partial derivatives at x of components f of f
× ! "i
with respect to components x of its argument
j
C1(X,RM) vector space of continuously differentiable functions defined on
X RN with values in RM.
⊂
f−1(Y) pre-image of a set Y RM under the map f: X RN RM,
⊂ ⊂ →
that is, the set x X :f(x) Y
{ ∈ ∈ }
f−1(y) pre-image of a point y RM under the map f: X RN RM,
∈ ⊂ →
that is, the set x X :f(x)=y =f−1( y )
{ ∈ } { }
I(τ,ξ) maximal interval of existence of the solution of the
non-autonomous initial-value problem x˙(t)=f(t,x(t)), x(τ)=ξ
(assuming uniqueness)
Notation ix
I maximal interval of existence of the solution of the autonomous
ξ
initial-value problem x˙(t)=f(x(t)), x(0)=ξ
(assuming uniqueness)
F(s) the field of rational functions in s with coefficients in F
Laplace transform
L
[[a,b]] line segment joining two points a,b R2:
∈
[[a,b]]:= (1 µ)a+µb: 0 µ 1
{ − ≤ ≤ }
✷ indicates end of proof
indicates end of example
△
Throughout,byanintervalJ Rwemeananon-degenerateinterval,that
⊂
is, an interval with endpointsa and b satisfying a<b. An interval
≥−∞ ≤∞
may be open or closed, neither open nor closed, bounded or unbounded.
We denote, by FN, the set of all ordered N-tuples x with components
x ,...,x inF,Anelementx FN canbeviewedasacolumn(N 1matrix),
1 N
∈ ×
that is,
x
1
.
x= . ,
⎛ . ⎞
x
⎜ N⎟
⎝ ⎠
or, alternatively, x FN can be viewed as a row (1 N matrix), that is,
∈ ×
x= x ,...,x .
1 n
Throughout,weexploitthenotation!alflexibilit"yaffordedbythetwoequivalent
representationsofelementsofFN:insomesituations,weadoptthecolumnform
and, in other situations, we opt for the row alternative. For example, in linear
algebraic contexts, the column form is appropriate in matrix manipulation:
for a P N matrix M FP×N and x FN, both x and Mx FP should
× ∈ ∈ ∈
be interpreted in column form. On the other hand, if f is a (nonlinear) map
FN FP andwewishtoexpressf(x) FP,x FN,inexplicitcomponentwise
→ ∈ ∈
form, then we adopt
f(x)= f (x ,...,x ),...,f (x ,...,x )
1 1 N P 1 n
as the preferred alternat!ive to its typographically cumberso"me column form.
Our view is that the benefits to typography and layout available through se-
lective use of the equivalent representations of elements of FN outweigh any
potential for confusion.
Contents
1. Introduction................................................ 1
1.1 Examples................................................ 2
1.1.1 An example from circuit theory ...................... 2
1.1.2 The nonlinear pendulum ............................ 4
1.1.3 The controlled inverted pendulum .................... 5
1.1.4 Satellite dynamics .................................. 6
1.1.5 Population dynamics................................ 9
1.2 Initial-value problems ..................................... 12
1.2.1 Continuous righthand side........................... 13
1.2.2 Righthand side with discontinuous time dependence .... 16
1.2.3 Linear systems..................................... 18
1.3 Related texts ............................................ 19
2. Linear differential equations ................................ 21
2.1 Homogeneous linear systems ............................... 22
2.1.1 Transition matrix function........................... 24
2.1.2 Solution space ..................................... 30
2.1.3 Autonomous systems ............................... 33
2.2 Inhomogeneous linear systems.............................. 40
2.3 Systems with periodic coefficients: Floquet theory ............ 43
2.4 Proof of Theorem 2.19 and Proposition 2.29 ................. 60
3. Introduction to linear control theory ....................... 65
3.1 Controllability ........................................... 67
3.2 Observability ............................................ 81
3.3 Impulse response and transfer function ...................... 89
xii Ordinary Differential Equations: Analysis, Qualitative Theoryand Control
3.4 Realization theory ........................................ 94
4. Nonlinear differential equations.............................101
4.1 Peano existence theory ....................................102
4.2 Maximal interval of existence ..............................106
4.3 The Lipschitz condition and uniqueness of solutions...........115
4.4 Contraction-mapping approach to existence and uniqueness....119
4.5 Periodic solutions ........................................135
4.6 Autonomous differential equations ..........................138
4.6.1 Flows and continuous dependence ....................138
4.6.2 Limit sets .........................................141
4.6.3 Equilibria and periodic points........................145
4.7 Planar systems...........................................151
4.7.1 The Poincar´e-Bendixson theorem.....................151
4.7.2 First integrals and periodic orbits ....................161
4.7.3 Limit cycles .......................................164
5. Stability and asymptotic behaviour .........................167
5.1 Lyapunov stability theory .................................168
5.2 Invariance principles ......................................176
5.3 Asymptotic stability ......................................182
5.4 Stability of linear systems .................................189
5.5 Nonlinearly perturbed linear systems........................193
5.6 Linearization of nonlinear systems ..........................194
5.7 Nonlinear systems and exponential stability..................200
5.8 Input-to-state stability ....................................201
5.8.1 Linear prototype ...................................202
5.8.2 Nonlinear systems ..................................203
6. Stability of feedback systems and stabilization ..............215
6.1 Linear systems and state feedback ..........................218
6.1.1 Eigenvalue assignment by state feedback ..............219
6.1.2 Stabilizability of linear systems.......................228
6.2 Nonlinear systems and feedback ............................230
6.2.1 Stabilizability and linearization ......................231
6.2.2 Feedback stabilization of smooth input-affine systems ...233
6.2.3 Feedback stabilization of bilinear systems..............236
6.3 Lur’e systems and absolute stability ........................240
6.4 Proof of Lemmas 6.16 and 6.18.............................251