second, 41–47, 79, 147
Weierstrass-Erdmann, 47, 72,
147
cost
axis, 142, 168
functional, 136
general, 174–176
terminal, 176
cycloid, 5, 6, 32
Dido’s problem, 123
differentiability, 8
dimensionless variables and
parameters, 127–130, 163–164,
188
direct method, 20, 27, 28, 132
direction field, 91
discontinuous control, 134, 167
discount factor or rate, 129, 205
du Bois-Reymond equation, 37
ecosystem
two-species, 203
eigenvalue, 150
complex, 155
of Jacobian matrix, 210
real, 152, 153, 155
ellipse, 39, 112
polar equation of, 192
embedded extremal, see extremal
endpoint minimum, 42, 46
energy, 2, 27
envelope, 107, 113–114
equations of, 108
equilibrium point, 151, 204
node
stable, 152, 154, 155
unstable, 153–155, 211
saddle point, 154–155, 211
Euler’s rule, 123, 148
Euler-Lagrange equation, 13, 26,
29, 37, 59, 72, 85, 123, 147
excess function
Weierstrass’s, 78, 102, 106
existence of optimal control, see
control
extinction, 206, 216
extremal, 13
admissible, 15, 17–18, 20, 24,
31–34
nonunique, 17, 49, 62, 111
unique, 25
broken, 23, 38, 48–50
simple, 48–50
embedded, 99, 102–108
regular, 58
straight-line, 21
extremization, 13
feedback control, see control
field
of extremals, 91, 101–107
of semi-extremals, 107
first integral of the Euler-Lagrang
equation, 30, 32
first variation, see variation
fishery, 129, 203
free
endpoint, 86
terminal time, 136
frictionless bead, 1, 34
functional, 3
functional analysis, ix, 220
fundamental lemma, 15
fundamental problem, 7–17
fundamental sufficient condition,
see sufficient condition
global
maximum, 134
minimum, 68, 116
Green’s theorem, 94, 133
growth rate
per capita
maximum, 129, 204
Hamiltonian, 29, 45, 140, 142, 147
160, 164
invariance of, 180–181
modified
for state variable restrictions,
197
for terminal cost, 175
Hilbert’s differentiability condition
38
Hilbert’s invariant integral, see
invariant integral
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