with y(t) = 0; the solution is given by 
t 
y(t) = { O(t, 0) g(o) do (il, 1177) 
1E 
which can be verified by substitution into (1.16). For the control 
problem, (1.17) has two consequences: it can be used in conjunction 
with (1.6) to obtain an estimate for the order of magnitude of 6x and 
it can be used to solve (1.9). 
In the first case, 
e t 
| dx | i [o(t, o)|| £(@) - £~)| do + { 0(6x7) do 
0 0 
[A 
ie t 
M { | £(u) - £()|do + \ 0(8x7) do 
0 0 
[A 
where M is a bound for ®. From (1.3) 
t 
t 
| Sx| LM i | Su| do + { 0 (6x2) do 
0 0 
JA 
t 
IMe + | 0(éx") do 
0 
[A 
By iteration 
t 
|ox| < LMe + { 0(e7) do = O(c) 
0 
The second case is of more interest, of course, for it gives an 
approximation of 6x good to the second order in €, namely, 
t 
6x = il (Es (6) ea) FG) lindo (1.18) 
0 
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