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Section 2: Finite-Dimensional State Space
Contents
I Matrix Exponentials.
I Existence and Uniqueness Results for ODEs.
I Equilibrium Points and Stability.
I Some Applications.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Matrix Exponentials
Consider the following IVP involving a linear system of n constant-coefficient
ODEs:
◦
u01(t) = l11u1(t) + l12u2(t) + · · · + l1nun(t), u1(0) =u1,
u02(t)
◦
= l21u1(t) + l22u2(t) + · · · + l2nun(t), u2(0) =u2,
..
◦
u0n(t) = ln1u1(t) + ln2u2(t) + · · · + lnnun(t), un(0) =un,
We express the IVP system in the matrix–vector form
◦
u0(t) = Lu(t), u(0) =u,
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Matrix Exponentials
L is the n × n constant real matrix


l11 l12 · · ·
 l21 l22 · · ·
L=
..
...
 ..
ln1 ln2 · · ·
l1n
l2n 
.. 
,
lnn
◦
and u(t) and u are the column vectors


u1(t)
..

,
un(t)

◦

u1
 . 
 . 
◦
un
A solution will be a vector-valued function : u(t) lies in the n-dimensional
Banach space Rn for each t. This means that our state space X is Rn.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Matrix Exponentials
it is tempting to write down a solution in the form
◦
u(t) = etL u .
(∗)
It turns out that this is the unique solution. But this obviously leads to the
following questions.
Q1. What does etL mean when L is an n × n constant matrix?
Q2. How do we verify that (*) is a solution ?
Q3. How do we prove that (*) is the only solution of the IVP?
Q4. For a given n×n constant matrix L, can we actually express etL in terms
of standard scalar-valued functions of t?
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Matrix Exponentials
To answer Q1, we define
eL = exp(L) = I + L +
L2
2!
+
L3
3!
+ ··· .
for any square matrix L.
It can be shown that the infinite series of n × n matrices (or, equivalently,
bounded linear operators in B(Rn)) will always converge (with respect to
the norm on B(Rn)) to a uniquely defined n × n matrix (which can also be
interpreted as an operator in B(Rn)).
Moreover
keLk ≤ ekLk,
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Matrix Exponentials
Thus we define
e
tL
= exp tL = I + tL +
t2L2
2!
+
t3L3
3!
in which case
ketLk ≤ e|t| kLk.
(P1) e0L = I;
(P2) esLetL = e(s+t)L for all s, t ∈ R;
(P3)
d
dt
tL
e f = LetLf for any given vector f ∈ Rn.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
+ ··· ,
Existence and Uniqueness Resuts
We can now answer Q2 and Q3.
◦
On setting u(t) = etLu, it follows immediately from properties (P1) and (P3)
that
◦
◦
u(0) = I u=u
and
u0(t) = Lu(t).
Therefore u(t) = e
◦
tL u
is a solution of the IVP
0
u (t) = Lu(t),
◦
u(0) =u .
This is the only solution.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Existence and Uniqueness Resuts
◦
I Set φ(t, u) := e
◦
tL u
to obtain an n-D CDS φ : [0, ∞) × Rn → Rn.
I The associated semigroup of operators {S(t)}t≥0, S(t) := etL, is referred
to as the semigroup generated by the matrix L.
◦
◦
I The set {S(t) u: t ≥ 0} ⊂ Rn is called the (positive semi-) orbit of u.
I Geometrically, we can regard the orbit as a continuous (with respect to t)
◦
“curve” (or path or trajectory), emanating from u, that lies in the state-space
Rn for all t ≥ 0. The continuity property follows from the fact that
kehL − Ik ≤ e|h| kLk − 1 → 0 as h → 0.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Equilbrium Points
I A constant solution, u(t) ≡ u for all t, where u = (u1, u2, . . . , un) ∈ Rn
is called an equilibrium solution or steady state solution.
I The orbit of such a solution is the single element (or point) u ∈ Rn; u is
called an equilibrium point (or rest point, stationary point or critical point). I If
u is an equilibrium point, then Lu = 0.
I When L is non-singular the only equilibrium point of the system u0(t) =
Lu(t) is u = 0. In this case, when each eigenvalue of L has a negative
real part, then the equilibrium point 0 is globally attractive (or globally
asymptotically stable) since ketLf k → 0 as t → ∞ for all f ∈ Rn.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Diagonalization
I In principle, etL can be computed by using the fact that, if P is a non-singular
matrix and L = P ΛP −1, then etL = P etΛP −1.
I If L has n distinct real eigenvalues λ1, λ2, . . . , λn, then the corresponding
eigenvectors can be used as the columns of a matrix P such that L =
P ΛP −1, where Λ = diag{λ1, λ2, . . . , λn}, in which case
etL = P etΛP −1 with etΛ = diag{eλ1t, eλ2t, . . . , eλnt}.
I In general, it can be shown that each component uj (t), j = 1, 2, . . . , n,
of any given solution u(t) will take the form of a linear combination of the
functions
tketµ cos(νt),
t`etµ sin(νt),
where µ + iν runs through all the eigenvalues of L, with ν ≥ 0, and k, ` run
through all the integers 0, 1, 2, . . . , n − 1.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Nonlinear Autonomous Systems of ODEs
I Consider the IVP
◦
u0(t) = F (u(t)), u(0) =u,
◦
◦
(∗∗)
◦
◦
where u(t) = (u1(t), u2(t), . . . , un(t)), u= (u1, u2, . . . , un) and
F : Rn ⊇ W → Rn is a vector-valued function F = (F1, F2, . . . , Fn) defined
on an open subset W of Rn.
I A solution of (**) is a differentiable function u : J → W defined on some
interval J ⊂ R, with 0 ∈ J such that
◦
u0(t) = F (u(t)) ∀ t ∈ J, u(0) =u .
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Existence and Uniqueness Result
The following theorem provides sufficient conditions for the existence of a
◦
◦
unique solution φ(·, u) = S(t) u to (**) on some interval J = (−a, a).
Theorem 2.1 Let F be continuously differentiable on W .
◦
(i) (Local Existence and Uniqueness) For each u ∈ W , there exists a unique
◦
solution φ(·, u) of (**) defined on some interval (−a, a) where a > 0.
(ii) (Continuous Dependence on Initial Conditions) Let the unique solution
◦
u
φ(·, ) be defined on some closed interval [0, b]. Then there exists a
◦
◦
neighbourhood U of u and a positive constant K such that if v ∈ U , then
◦
the corresponding IVP v 0 = F (v), v(0) =v has a unique solution also
defined on [0, b] and
◦
◦
◦
◦
kφ(t, u) − φ(t, v )k = kS(t) u −S(t) v k ≤ e
Kt
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
◦
◦
ku−vk
∀ t ∈ [0, b].
◦
(iii) (Maximal Interval of Existence) For each u ∈ W , there exists a maximal
◦
open interval Jmax = (α, β) containing 0 (with α and β depending on u)
◦
on which the unique solution φ(t, u) is defined. If β < ∞, then, given any
compact subset K of W , there is some t ∈ (α, β) such that u(t) ∈
/ K.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Existence and Uniqueness Result
I F is differentiable at g ∈ W if there exists a linear operator DF (g) ∈
B(Rn) such that
F (g + h) = F (g) + DF (g)h + khk Eg (h), h ∈ Rn,
where Eg (h) → 0 as h → 0.
I It can be shown that DF (g) can be represented by the n × n Jacobian
matrix


∂1F1 ∂2F1 · · · ∂nF1
 ∂1F2 ∂2F2 · · · ∂nFn 


..
..
..
...


∂1Fn ∂2Fn · · · ∂nFn
evaluated at g. The function F is continuously differentiable on W if all the
partial derivatives ∂j Fi exist and are continuous on W .
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Existence and Uniqueness Result
I The fact that F is continuously differentiable on W means that F is locally
◦
Lipschitz on W ; i.e. for each u ∈ W there is a closed ball
◦
n
◦
Br (u) := {f ∈ R : kf − u k ≤ r} ⊂ W and a constant k, which may
◦
depend on u and r, such that
kF (f ) − F (g)k ≤ k kf − gk
◦
∀f, g ∈ Br (u).
I The proof of (i) involves the Banach Contraction Mapping Principle. The first
step is to note that the IVP (**) is equivalent to the fixed point problem u = T u
where
Z t
◦
F (u(s))ds,
(T u)(t) =u +
0
I The local Lipschitz continuity of F can then be used to establish that T
is a contraction on a suitably defined Banach space of functions; this yields
existence and uniqueness.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Existence and Uniqueness Result
I The proof of (ii) relies on Gronwall’s inequality which states that if ψ :
[0, b] → R is continuous, non-negative and satisfies
Z
t
Kψ(s) ds ∀ t ∈ [0, b],
ψ(t) ≤ C +
0
for constants C ≥ 0, K ≥ 0, then
ψ(t) ≤ CeKt ∀ t ∈ [0, b].
I The operators S(t) have the following semigroup property:
◦
◦
S(t)S(s) u= S(t + s) u,
where this identity is valid whenever one side exists (in which case, the other
side will also exist).
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Equilibrium Points
I When analysisng
u0 = F (u)
the starting point is usually to look for equilibrium points (corresponding to
constant, or steady-state solutions).
I In this case ū is an equilibrium point if
F1(ū) = 0, F2(ū) = 0, . . . , Fn(ū) = 0,
and the local stability properties of an equilibrium are usually determined by
the eigenvalues of the Jacobian matrix (DF )(ū).
I The equilibrium ū is hyperbolic if (DF )(ū) has no eigenvalues with zero
real part.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Stability
I An equilibrium ū is stable if nearby solutions remain nearby for all future
time. More precisely, ū is stable if, for any given neighbourhood U of ū there
is a neighbourhood U1 of ū in U such that
◦
◦
◦
u ∈ U1 ⇒ φ(t, u) exists for all t ≥ 0 and φ(t, u) ∈ U for all t ≥ 0.
I If, in addition,
◦
◦
u ∈ U1 ⇒ φ(t, u) → ū as t → ∞,
then ū is (locally) asymptotically stable.
I Any equilibrium which is not stable is said to be unstable.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Stability
I When ū is hyperbolic then it is either asymptotically stable (when
all eigenvalues of (DF )(ū) have negative real parts) or unstable (when
(DF )(ū) has at least one eigenvalue with positive real part).
I The basic idea behind the proof of these stability results is that of
linearisation.
◦
I Suppose that ū is an equilibrium point and that u is sufficiently close to ū.
◦
Let v(t) = φ(t, u) − ū. Then
v 0(t) = F (ū + v(t)) ≈ F (ū) + DF (ū) v(t)
i.e.
v 0(t) ≈ DF (ū) v(t).
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations
Stability
I Thus, in the immediate vicinity of ū, the non-linear ODE u0 = F (u) can be
approximated by the linear equation
v 0 = Lv,
where L = DF (ū).
I In effect, this means that in order to understand the stability of a hyperbolic
equilibrium point ū of u0 = F (u), we need only consider the linearised
equation v 0 = DF (ū)v.
CIMPA, Muizenberg, 2013. Applying Functional Analytic Techniques to Evolution Equations