Appendix 1 Existence of Solutions to Periodic and Almost Periodic Differential Systems

Appendix 1 EXISTENCE OF SOLUTIONS TO PERIODIC AND ALMOST PERIODIC DIFFERENTIAL SYSTEMS

Existence Conditions for Periodic Systems

Al

Consider the general homogeneous periodic system x where

B

f(t,x),

=

f:D

+

f(t+w,x)

Rn

for D

(x:IIx-xojI < f3, x

=

E

f(t,x),

=

J = {t: It-t

= JxB,'

Rn}.

w > 0

I

(AI. 1)

< a, t

E

R}

and

Assume that if any solutions to

(Al.1) exist then they are unique.

By utilising Schauders fixed

point theorem (for generalisations appropriate to periodic systems see Browder ( 1 9 5 9 ) ) , Massera (1950) has shown that if the system (Al.l) is scalar, a solution which exists and remains bounded in the future implies the existence of a p e r i o d i c solu-

t i o n of period w. f(t,x) with

A:J

period a.

= -f

Additionally if the system is linear such that

A(t)x

Mn

and

+ h(t),

h:J

+

(Al.2) both continuous and periodic of

R"

Then Massera's result extends to this n-dimension li-

near periodic system.

The proof follows by assuming that

0, then given some initial condition (x

t ) a solution

0) 0

x(w;x ,t ) 0 .

x

w

0

=

where X(t)

h(t) # x

w

=

through (x ,t ) i s by ( 3 . 5 7 ) as 0

X(w)X-'

0

(to) + X(w>

r

X-l (s)h(s)ds,

(A1 . 3 )

0

is the fundamental matrix of the linear homogeneous

STABILITY OF LINEAR SYSTEMS

208 equation x = A(t)x.

Setting x

w

=

Gxo

+

b

Tx

where G and b

are defined by the right hand side of (A1.3) and T is a transformation of x

*

0'

equations

now suppose that the system of linear algebraic

(G-I)x+b

=

0 have no solution then (G-I) must be sin-

gular and there exists a fixed vector y such that y'b # 0 .

and

=

y'

then a l s o

y'Gk

=

0

y'

for k

=

and by applying the transformation T repeatedly we have

l,Zy...y

k

x = T x

=

0

ky'b.

Since y'G

=

y'(G-I)

G xo + (Gk-1+Gk-2+. ..+I)b

But since y'h # 0 then as k

-

and hence +

y'x

=

+

y'x 0

y'\

which implies

+ m

\

that the solution to (A1.3) is unbounded. But

=

x(kw;xoyto)

is bounded by definition so by contradiction the linear equation must have a periodic solution. Consider now the more general case of (Al.1) for n an arbitrary positive integer.

By applying Browders (1959) fixed point theorem

to the periodic system ( A l . I ) ,

the following is a generalisation

of Cartwright's (1950) result for second order systems:

Theorem A 1 . I : Existence o f general periodic solution If the solutions of ( A l . l )

are bounded for some bound N, then

there exists a periodic solution of period w such that

N

for

t

E

//x(t)ll

5

R.

So far an assumption on the uniqueness of solution has been

necessary; however by imposing a s t a b i l i t y condition upon the periodic solution, the uniqueness requirement can be dropped: Theorem A 1 . 2 : (Yoshizawa, 1975) Given that f(t,x)

is continuous on D and that the periodic

B* < B Ily(t) 11 Then there exists a periodic solution of ( A l . l )

system (AI. 1 ) has a solution y(t) such that for all

t E R+.

of period ru(r lly(to)-xoII

< N

2

l , Z , . . . ) if there exists a

implies that

N > 0 such that

Ily(t)-x(t;xo,to)ll

+

0 as

t

+ m.

We note that the existence of bounded uniformly asymptotically stable solution to (AI.1) does not necessarily imply the existence of a periodic solution of the same period w.

Continuing the sta-

bility approach, Deysach and Sell (1965) have shown that if y(t)

AI. EXISTENCE OF SOLUTION

209

is uniformly stable then there exists an almost periodic solution to (A1.1);

that is we cannot necessarily obtain a periodic solu-

tion to a dynamical system with periodic coefficients without imposing additional constraints. Theorem 5 . 1 7 has demonstrated that if y(t) is uniformly stable, then y(t) is stable under disturbances from the hull H(f) and

so

the following theorem follows:

Theorem A 1 . 3 : (HaZanay, 1 9 6 2 )

If the solution y(t) to (Al.1) is uniformly stable then y(t) is asymptoticaZZy almost periodic and the system (Al.1) has an almost periodic solution which is also uniformly stable. Finally, Sell (1966) has similarly shown that a periodic solution of period rw (r

2 I)

to ( A l . 1 ) exists if there is a bounded

solution y(t) which is weakly uniformly asymptotically stable (equivalent to uniform asymptotic stability in the case of periodic systems).

This result is a special case of Theorem A 1 . 3

when uniform convergence is used. Existence Conditions for Almost Periodic Systems

A2

Consider the almost periodic system x

=

where

f(t,x) f:RxD

-f

(A1 .4)

(f

Rn,

formly for x E B"

=

E

AP(RxD))

{x: x

E

is almost periodic in t uni-

Rn, /lx-xoII < B * }

and for all

t E

Let F be a compact subset of B*, and let y(t) be a solution * Ily(t) 11 < B , and y(t) E F for all t E

R+.

of (A1.4) such that

For some positive sequence {a } iet k

R,.

uniformly on RxF, then h a,)

=

x

z(t) .=

E

H(f).

lirn f(t+ak,x) = h(t,x) kMoreover assume that lim y(t+ k-

uniformly on R+ where z(t) is a solution to

h(t,x),

h

E H(f)

(A1 .5)

the dual differential system (A1.4), (A1.5) leads us to the definition of inheritad properties.

210

STABILITY OF LINEAR SYSTEMS

Definition A . l : Inherited properties of a h o s t periodic systems (Fink, 1 9 7 2 )

If y(t) has a particular property with respect to system (A1.4), and z(t) has the same property with respect to the dual system (Al.5), then this property is said to be inherited. For almost periodic systems total stability and stability under disturbances are inherited properties; in addition for periodic systems, uniform stability and uniform asymptotic stability are also inherited properties.

However it should be noted that

for almost periodic systems, uniform stability and uniform asymptotic stability are inherited properties only if the uniqueness of solution is assumed.

An example that demonstrates this res-

triction is given by Kato (1970), it also shows that uniform asymptotic stability does not necessarily imply total stability in almost periodic systems whilst it does for periodic systems. For periodic systems the boundedness of solution implies the existence of a periodic solution, whilst for almosc periodic systems this is not the case.

Indeed, several examples of almost

periodic systems have been constructed by Opial (1961) and Fink and Frederickson (1971) such that the almost periodic system (A1.4) has no almost periodic solutions yet its solutions are uniformly ultimately bounded.

Thus in discussing the existence

of almost periodic solutions, stability properties of some kind

must be implied.

Based upon the assumption of uniqueness of solu-

tion to (A1.4), Miller (1965), required that the bounded solution is totally stable for the existence of solution, Seifert (1966) required the C-stability of the bounded solution, whilst Sell (1967) required stability under disturbances from the hull (see

also section 5.6).

All of these results can be achieved without

the condition of uniqueness of solution by utilising the property of asymptotically almost periodic functions (see section 2 . 6 ) . Typical of these results are the following two theorems which are stated without proof:

A l . EXISTENCE OF SOLUTION

21 I

Theorem A1 - 4 : Existence of an almost p e r i o d i c solution (Coppel, 1967)

If the almost periodic system x = f ( t , x ) ,

f E AP(RxD),

has

a bounded solution on R, which is asymptotically almost periodic, then it has an almost periodic solution. Theorem A l .5 : (Yoshizawa, 1 9 7 5 ) If the bounded solution y(t) (A1.4)

of the almost periodic system

is asymptotically almost periodic then for any

there exists a positive sequence

such that

h

E

H(f)

z(t) = lim y(t+ k-

is an almost periodic solution of the system ( A 1 . 5 ) uniformly ak) on R+. These theorems tell us that if the almost periodic system ( A 1 . 4 ) has a bounded asymptotically almost periodic solution then the dual system ( A 1 . 4 - 5 )

also have almost periodic solutions. A l s o

since stability under disturbances from the hull is by Theorem

5.17 a sufficient condition for asymptotic almost periodicity then

if the system ( A 1 . 4 ) has a bounded soluiion which is stable under disturbances fromH(f),

then it also has an almost periodic soluA

tion y(t) which is also stable under disturbances from H ( f ) .

corollary to this result is that if the above solution y(t) is totally stable then y(t) is asymptotically almost periodic and the system ( A 1 . 4 ) has an almost periodic solution which is totally stable. In the previous section we have seen that if a periodic system has a bounded and stable solution then there exists an almost periodic solution.

Examples of periodic systems can be generated

(Yoshizawa, 1975) whereby the system has a quasi-periodic solution and thus the module of the almost periodic solution is not contained within the module of the system. This demonstrates that uniform stability and stability under disturbances from the hull do not give module containment for almost periodic systems.

In

the following we develop the conditions, based upon uniform stability of solutions, for the existence of almost periodic solutions

212

STABILITY OF LINEAR SYSTEMS

to the almost periodic linear inhomogeneous system: x where

=

x

=

(Al.6)

M and f:R Rn are almost periodic uniformly in n Corresponding to (A1.6) is the homogeneous linear system

A:R

t on R.

+ f(t),

A(t)x

-f

-f

A(t)x,

(A1 .7)

and the equation on the hull x

=

G

G(t)x,

E

H(A).

(Al.8)

Clearly if the almost periodic system (A1.6) has a bounded solution on R, which is uniformly stable then the null solution o f (A1.7) and (A1.8) are also uniformly stable.

Under these condi-

tions the following theorem due to Favard (1933) shows that (A1.6) has an almost periodic solution: Theorem A 1 . 6 If every nontrivial solution x(t) of (A1.8) on the homogeneous hull of the linear almost periodic system (A1.6) is bounded on R and satisfies

Inf Ilx(t) t&R

solution on

E

y(t)

t

R+,

11

> 0,

then if (Al.6) has a bounded

there exists an almost periodic solution

of (A1.6) such that mod(y) c mod(A,f),

module of y and mod(A,f)

where mod(y)

is the

is the set module of A on f.

Proof: (Outline) It is straightforward to show that a bounded solution y (t) to (A1.6) exists and each equation in the hull of

(A1.6) has a unique solution with minimum norm. simple matter to show that

y(t),

yo(t)

E

It is then a

AP(RxRn),

and finally

by module containment (see definition 2.4) that mod(y) c mod(A,f) By similar reasoning Bochner (1962) derived as a corollary to Farvard’s theorem (A1.6): Theorem A 1 . 7 If every equation in the homogeneous hull of the linear almost periodic system (A1.6) is almost periodic, then every bounded solution of (A1.6) i s almost periodic. Bochner’s result also holds for a variety of important special

Al. EXISTENCE OF SOLUTION

213

cases; (i) the Bohr-Neugebauer theorem (1926) for linear constant coefficient almost periodic systems when

f(t)

0 and

periodic, that is

A(t)

E

AP(Mn),

A(t+w)

=

A(t)

(A E En, f E AP(Rn)), (ii) and (iii) when A(t) i s purely

for some w > 0.

References Bochner, S. (1962). Proc.Nat.Acad.Sci.(USA) 48, 2039-2043 Bohr, H. and Neugebauer, 0. (1926). "Uber lineare differentialgleichungen mit konstanten koeffizienten und fastperiodischer rechter seite", Nachr.Ges. Wiss.Ggttingen, Math.-Phys.KZass. , 8-22 Browder, F.E. (1959). Duke Math. J. 26, 291-303 Cartwright, M.L. (1950). "Forced oscillations in nonlinear systems", contri. to "The theory of nonlinear oscillations", Ed. S. Lefschetz 1 , Princeton University Press Coppel, W.A. (1967). Ann.Mat.Pura AppZic. 76, 27-50 Deysach, L.G. and Sell, G.R. (1965). Michigan Math.J. 12, 87-95 Favard, J. (1933). "LeGons sur les Fonctions Presque Pdriodiques", Gauthier Villars, Paris Fink, A . M . (1972). SIAM Review 14, 572-581 Fink, A.M. and Frederickson,'P.O. (1971). J.Diff.Eqns. 9 , 280-284 Halanay, A. (1962). Uspeh Mat.Nauk. 17, 231-233 Kato, J. (1970). Tohoku Math. J. 22, 254-269 Massera, J.L. (1950). Duke Math.J. 17, 457-475 Miller, R.K. (1965). J.Diff.Eqns. 1 , 293-305 Opial, 2 . (1961). BuZ2.Acad.PoZon.Sci.Ser.Sci.Math.Astron.Phys. 9 , 673-676 Seifert, G. (1966). J.Diff.Eqns. 2, 305-319 Sell, G.R. (1966). J.Diff.Eqns. 2, 143-157 Sell, G.R. (1967). Trans.Math.Soc. 127, 241-283 Yoshizawa, T. (1975). "Stability theory and the existence of periodic solutions and almost periodic solutions". Appl.Maths.Sci. No.14, Springer Verlag, New York