revenue, 204

river crossing, 188–189, 199–202

saddle point, see equilibrium point

scaling, 127–130, 163–164, 188

second variation, see variation

semi-ﬁeld, 107, 113

separatrix, 212

simple broken extremal, see

extremal

simply connected, 91

singular control, see control

smooth, 8

state, 128, 135

restricted, 196–197

Stokes’ theorem, 94

strengthened Legendre condition,

see Legendre condition

strong minimum, 67, 73, 110, 116

strong variation, see variation, 81

suﬃcient condition

fundamental, 102, 219

surface of revolution

minimum area of, 6, 14–15, 17,

25, 32, 126

switching

curve, 152, 157, 178

function, 150, 160

synthesis of optimal control, see

control

target, 137

general, 167–176

Taylor series, 52, 53

Taylor’s theorem, 137, 171

with remainder, 78, 103

terminal cost, see cost

terminality condition, 136, 141

time-optimal control, see control

total variation, see variation

trajectory

optimal, 136–142, 144, 209

nonunique, 202

transversality conditions, 86–87,

174–176, 179–180

trial curve, 4, 9, 41, 51, 73

two-parameter, 121

trial function, 4, 46, 74, 81

tumor growth, 163

variation

admissible, 10–12

ﬁrst, 53

second, 53

strong, 67, 73–74, 81

total, 52, 53, 102

weak, 9, 67, 73–74, 81

velocity, 2

weak minimum, 67, 73

weak variation, see variation

Weierstrass’s excess function, see

excess function

Weierstrass’s necessary condition,

see necessary condition

Weierstrass-Erdmann conditions,

see corner

wildlife management, 203, 218

Zermelo’s problem, 183–189,

197–202

river crossing, 188–189, 199–202

saddle point, see equilibrium point

scaling, 127–130, 163–164, 188

second variation, see variation

semi-ﬁeld, 107, 113

separatrix, 212

simple broken extremal, see

extremal

simply connected, 91

singular control, see control

smooth, 8

state, 128, 135

restricted, 196–197

Stokes’ theorem, 94

strengthened Legendre condition,

see Legendre condition

strong minimum, 67, 73, 110, 116

strong variation, see variation, 81

suﬃcient condition

fundamental, 102, 219

surface of revolution

minimum area of, 6, 14–15, 17,

25, 32, 126

switching

curve, 152, 157, 178

function, 150, 160

synthesis of optimal control, see

control

target, 137

general, 167–176

Taylor series, 52, 53

Taylor’s theorem, 137, 171

with remainder, 78, 103

terminal cost, see cost

terminality condition, 136, 141

time-optimal control, see control

total variation, see variation

trajectory

optimal, 136–142, 144, 209

nonunique, 202

transversality conditions, 86–87,

174–176, 179–180

trial curve, 4, 9, 41, 51, 73

two-parameter, 121

trial function, 4, 46, 74, 81

tumor growth, 163

variation

admissible, 10–12

ﬁrst, 53

second, 53

strong, 67, 73–74, 81

total, 52, 53, 102

weak, 9, 67, 73–74, 81

velocity, 2

weak minimum, 67, 73

weak variation, see variation

Weierstrass’s excess function, see

excess function

Weierstrass’s necessary condition,

see necessary condition

Weierstrass-Erdmann conditions,

see corner

wildlife management, 203, 218

Zermelo’s problem, 183–189,

197–202