hyperbola
rectangular, 20, 24
hyperplane, 168
separating, 168, 172, 174
hypersurface, 167
index set, 135
index theorem, 212
interior minimum, 11, 83
constrained, 122
invariant integral
Hilbert’s, 93–95, 101, 114
isoperimetrical problem, 119–124,
147–148
classical, 119, 123–124
Jacobi’s equation, 59–65
Jacobi’s necessary condition, see
necessary condition
Lagrange multiplier, 123, 125
Lagrangian, 26
least action, principle of, 26, 27
Legendre condition
strengthened, 58, 110
Legendre’s necessary condition, see
necessary condition
Legendre-Clebsch condition
generalized, 209
limit cycle, 212
little oh, 52, 137
Mathematica , 3, 63, 187, 189,
213, 222, 223
maximization
versus minimization, 13, 22, 54,
104, 119, 160, 190, 205
maximum principle
Pontryagin’s, 141–143, 146, 150,
159, 160, 162, 167
Michaelis-Menten kinetics, 163
minimization
versus maximization, 13, 22, 54
natural dynamics, 203
navigation
boat, 183–189
plane, 189–193
necessary condition
Jacobi’s, 59, 115, 209
Legendre’s, 54, 79, 147
Weierstrass’s, 78, 110
necessary conditions
for optimal control, 135–143
negative phase-plane, 144, 151
Newton’s method
simple versus double root, 201
node, see equilibrium point
nonautonomous control problem,
159, 162
normal case, 177
notation, xi, 15, 72, 99
numerical integration, 188–189,
201, 214–215
ODE, 29
open-loop control, see control
optimal control, see control
optimal control problems, 127
optimal harvesting, 129, 203–218
optimal trajectory, see trajectory
orthogonally, 89
parabola, 61–62, 96, 98, 105, 144
pencil
of extremals, 91–93, 112, 117
point, 91, 102
perturbation cone, 172
phase-plane analysis, 144–146,
151–157, 203–216
piecewise-continuous control, see
control
piecewise-smooth, 8
Pontryagin’s principle, see
maximum principle
positive phase-plane, 144, 151
present value, 129, 205
Problem B, 164, 166
Problem E, 130, 132–134, 136,
159–162, 204
Problem P, 128, 131–132, 136,
143–146, 149, 202
proof by contradiction, 16, 55, 59
pseudo-optimal, 196, 199, 202
regular extremal, see extremal
regular problem, 39, 58, 78, 103,
116
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