# Calculus of Variations and Optimal Control Theory A Concise Introduction

Daniel Liberzon
University of Illinois at Urbana-Champaign

Index

## Index

0 -norm
1.3.1 | 2.2.1
abnormal multiplier
2.5.1 | 4.1.1
absolutely continuous function
3.3.1
accessory equation
see Jacobi equation
action integral
2.4.3
3.4.2 | 4.2.8
1.3.2
applications of optimal control
1.1 | 1.4
arclength
2.1.1
asymptotic stability
6.2.3
augmented cost
1.2.2.1 | 2.5.1 | 2.5.2
augmented system
4.2.1
bang-bang control
4.4.1 | 4.4.2
bang-bang principle
4.4.2 | 4.4.2 | 4.6
basic calculus of variations problem
2.2
basic fixed-endpoint control problem
4.1.1
basic variable-endpoint control problem
4.1.2
Bellman
5.1 | 5.1.5
Bernoulli, Johann
2.1.3 | 2.1.4
Bolza problem
3.3.2
boundary conditions
correct number of
2.3.1 | 2.3.5 | 2.5.1 | 3.1.1 | 4.1.2
brachistochrone
2.1.4 | 2.7
Brouwer's fixed point theorem
4.2.6
canonical coordinates
7.1.1
canonical equations
2.4.1 | 3.4.2 | 7.1.3
as characteristics of HJB equation
7.2.2
canonical variables
2.4.1
Carathéodory
5.1.5
Carathéodory
5.4
catenary
2.1.3 | 2.7 | 3.1.1 | 3.1.1
Cauchy initial value problem
7.2.1
Cauchy-Schwarz inequality
2.6.2
characteristic
7.2.1
characteristic strip
7.2.1
compactness
1.2.1.3 | 1.3.3 | 1.3.4
of reachable sets
4.5 | 4.6
conjugate point
2.6.2
conservation law
2.4.3
conservative force
2.4.3
constrained optimization
1.2.2
constraint
equality
1.2.2 | 4.1.2 | 7.1.1
holonomic
2.5.2.1
integral
2.5.1
non-integral
2.5.2
nonholonomic
2.5.2.1 | 2.7
contravariance
7.1.2
control set
1.1 | 3.3.1
control system
1.1 | 3.3.1
normal
4.4.2
on a manifold
7.1.2
controllability
4.4.2 | 4.6 | 6.2.1
convexity
1.2.1.3 | 1.2.1.3 | 1.3.4 | 1.4 | 4.2.7 | 4.5 | 7.3.3
coordinate chart
7.1.1
corner point
3.1.1
cost functional
1.1
augmented
2.5.1 | 2.5.2
Bolza form
3.3.2
Lagrange form
3.3.2
Mayer form
3.3.2
costate
as covector
7.1.2
cotangent bundle
7.1.1
cotangent space
7.1.1
cotangent vector
see covector
covariance
7.1.2
covector
7.1.1
critical point
1.2.1.3
cycloid
2.1.4
Dido's isoperimetric problem
2.1.1
differentiable manifold
see manifold
differential game
5.1.5 | 7.3.2
differential of a function
7.1.1 | 7.1.2
disturbance attenuation
7.3.2
dual space
7.1.1
dynamic programming
5.1
discrete
5.1.1
equality constraint
1.2.2 | 4.1.2 | 7.1.1
Erdmann
3.1.1
Euler
2.3.2
Euler-Lagrange equation
2.3.1
coordinate invariance
2.3.3 | 2.7
for multiple degrees of freedom
2.3.3
integral form
2.3.3
existence and uniqueness of solution
3.3.1
existence of optimal solution
2.3.1 | 2.7 | 4.5 | 4.6
for Mayer problems
4.5
for nonlinear time-optimal problems
4.6
exponential stability
6.2.3
extremal
2.3.1
broken
3.1.1
extremum
1.2 | 1.3.1
strong
2.2.1
weak
2.2.1
feasible direction
1.2.1.3
Fermat's principle
2.1.2
Filippov's theorem
4.5 | 4.6
final state
1.1 | 3.3.2
final time
1.1 | 3.3.2
finite escape time
2.6.2 | 6.1.4
finite-dimensional optimization
1.2
first variation
1.3.2
alternative definition
1.3.2
first-order necessary condition
1.2.1.1 | 1.3.2
for constrained optimality
1.2.2.1 | 2.5.1 | 2.5.2
for weak extremum
2.3.1
fixed-time, free-endpoint problem
3.3.3
Fréchet derivative
1.3.2 | 1.4
free-time, fixed-endpoint problem
3.3.3
Fuller's phenomenon
4.4.4
Fuller's problem
4.4.4 | 4.6
function
1.3.1
absolutely continuous
3.3.1
locally Lipschitz
3.3.1
measurable
3.3.1
piecewise
2.2.1
piecewise continuous
3.3.1
functional
1.3
bilinear
1.3.3
linear
1.3.2
1.3.3
Gateaux derivative
1.3.2 | 1.4
global minimum
1.2 | 1.2.1.3 | 1.3.1 | 1.3.4
1.2.1.1
guard
see switching surface
Hamilton
2.4 | 5.1.5
Hamilton-Jacobi-Bellman (HJB) equation
5.1.3
and maximum principle
5.2 | 7.2
characteristics of
7.2.2
for infinite-horizon problems
5.1.3.1
in calculus of variations
5.1.5 | 5.4
solving
5.4
Hamiltonian
2.3.4 | 2.4.1 | 3.4.1 | 4.1.1
as energy
2.4.3
classical vs. modern formulation
2.4.2
Hamiltonian matrix
6.1.1
Hamiltonian maximization condition
2.4.1 | 2.6.1 | 3.1.2 | 3.4.3 | 4.1.1 | 4.2.9.1 | 5.1.3
Hessian matrix
1.2.1.2
holonomic constraint
2.5.2.1
Hurwitz matrix
7.3.1
hybrid system
7.4
discrete state
7.4.1
hyperplane
separating
4.2.7 | 4.2.10
supporting
4.4.2
infinite-dimensional optimization
1.3
infinite-horizon problem
3.3.3 | 5.1.3.1 | 6.2
integral constraint
2.5.1
inverse function theorem
1.2.2.1
Isaacs
5.1.5
Jacobi
2.6.2 | 5.1.5
Jacobi equation
2.6.2
Jacobian matrix
1.2.2.1
Kalman
5.1.5 | 6.3
Lagrange
1.2.2.1 | 2.2 | 2.3.2 | 2.5.1 | 2.6.2
Lagrange multiplier
1.2.2.1 | 2.5.1
distributed
2.5.2
Lagrange problem
3.3.2
Lagrangian
augmented
2.5.1 | 2.5.2
differentiability assumptions
2.3.3 | 2.7
Legendre
2.6.1 | 2.6.2
Legendre transformation
2.4.2 | 2.7
Legendre's condition
2.6.1
for multiple degrees of freedom
2.6.1 | 2.7
Legendre-Clebsch condition
3.4.3
light
reflection
2.1.2 | 7.4.3
refraction
2.1.2 | 2.1.4 | 2.7
linear matrix inequality (LMI)
7.3.3
finite-horizon
6.1
infinite-horizon
6.2
linearization
3.4.1 | 4.2.4
Lipschitz condition
3.3.1 | 3.3.1
local coordinates
7.1.1
canonical
7.1.1
local minimum
1.2 | 1.3.1
locally Lipschitz function
3.3.1
Lyapunov-like function
6.2.2 | 7.3.1
manifold
7.1
embedded
7.1.1
maximum
1.2 | 1.3.1
vs. minimum
1.2 | 2.2 | 3.4.4
maximum principle
and HJB equation
5.2 | 7.2
for basic fixed-endpoint control problem
4.1.1
for basic variable-endpoint control problem
4.1.2
for fixed-time problems
4.3.1.1
for Mayer problems
4.6
for problems with terminal cost
4.3.1.3
for time-varying problems
4.3.1.2
higher-order
4.6
hybrid
7.4.2
in discrete time
4.6
local optimality
4.3 | 4.6
nonsmooth
4.6
on manifolds
7.1.2
sign convention
3.4.4
stochastic
4.6
Mayer problem
3.3.2
McShane
3.1.2 | 4.2.3
measurable function
3.3.1
minimum
global
1.2 | 1.2.1.3 | 1.3.1 | 1.3.4
local
1.2 | 1.3.1
strict
1.2 | 1.3.1
vs. maximum
1.2 | 2.2 | 3.4.4
momentum
2.3.4 | 2.4.1
angular
2.4.3
multiple degrees of freedom
2.2
necessary condition
first-order
1.2.1.1 | 1.3.2
for constrained optimality
1.2.2.1 | 2.5.1 | 2.5.2
for weak extremum
2.3.1
for strong extremum
3.1.1
for strong maximum
3.1.2
for strong minimum
3.1.2
second-order
1.2.1.2 | 1.3.3 | 2.6.1
for constrained optimality
1.2.2.2
needle perturbation
4.2.3
Newton's second law
2.4.3
Noether's theorem
2.4.3 | 2.7
non-integral constraint
2.5.2
nonholonomic constraint
2.5.2.1 | 2.7
nontriviality condition
4.3
norm
1.3.1
normal system
4.4.2
observability
6.2.3
optimal control
1.1 | 3.3.3
bang-bang
4.4.1 | 4.4.2
singular
4.4.3
optimal control problem
1.1 | 3.3
Bolza form
3.3.2
hybrid
7.4.1
Lagrange form
3.3.2
Mayer form
3.3.2
on a manifold
7.1.2
parking problem
4.4.1
partial differential equation (PDE)
5.1.3 | 5.3.2 | 7.2.1
pendulum
7.1
4.5
perturbation
1.3.2
needle
4.2.3
spatial
4.2.3
temporal
4.2.2
piecewise continuous function
3.3.1
Pontryagin
4.1.2 | 4.2.3
principle of least action
2.4.3 | 2.7
principle of optimality
4.2.1 | 5.1.2 | 5.1.5
1.3.3
reachable set
4.4.2 | 4.5
regular point
1.2.2.1
Riccati equation
algebraic
6.2.1
unique positive definite solution
6.2.4
differential
2.6.2 | 2.6.2 | 6.1.2
global existence of solution
6.1.4
reduction to linear differential equation
2.6.2 | 2.6.2
solving
6.3
6.2.1
Riccati inequality
7.3.1 | 7.3.2
reduction to LMI
7.3.3
robust control
7.3.2
running cost
1.1 | 2.2 | 3.3.2
Schur complement
7.3.3
second variation
1.3.3
alternative definition
1.3.3
second-order necessary condition
1.2.1.2 | 1.3.3 | 2.6.1
for constrained optimality
1.2.2.2
second-order sufficient condition
1.2.1.2 | 1.3.3 | 2.6.2
for constrained optimality
1.2.2.2
separating hyperplane
4.2.7 | 4.2.10
single degree of freedom
2.2
singular arc
4.4.3
Snell's law
2.1.2 | 2.1.4
spatial control perturbation
4.2.3
stability
6.2.3
state feedback
4.4.1 | 4.6 | 5.1.1 | 5.2 | 6.1.1
stationary point
1.2.1.1
strict minimum
1.2 | 1.3.1
strong extremum
2.2.1
sub-differential
5.3.1
suboptimal control
7.3.2
sufficient condition
1.2.1.2 | 1.3.3 | 3.4.4 | 5.1.4
for strong minimum
3.1.2
for constrained optimality
1.2.2.2
for strong minimum
3.5
for weak minimum
2.6.2
super-differential
5.3.1
supporting hyperplane
4.4.2
surface
1.2.1.3 | 1.2.1.3 | 4.1.2 | 7.1.1
switching condition
7.4.2
switching cost
7.4.1
switching curve
4.4.1 | 4.4.4
switching function
4.4.3
switching surface
7.4.1
switching time
7.4
symplectic form
7.1.3
tangent bundle
7.1.1
tangent space
1.2.2.1 | 7.1.1
characterization of
1.2.2.1 | 1.4 | 4.1.2
tangent vector
1.2.2.1
target set
3.3.3
temporal control perturbation
4.2.2
terminal cone
4.2.5
terminal cost
1.1 | 3.3.2
terminal state
1.1 | 3.3.2
terminal time
1.1 | 3.3.2
test function
5.3.1
time-optimal problem
3.2 | 4.4 | 4.5 | 4.6
transition matrix
4.2.4
transversality condition
2.7 | 4.1.2 | 4.2.10
for Cauchy problem
7.2.1
for initial sets
4.3.1.4 | 7.4.2
unconstrained optimization
1.2.1
completely
1.2.1
value function
5.1.2
as viscosity solution
5.3.3
nondifferentiable
5.2.1
variable initial state
4.3.1.4 | 7.4.1
variable-endpoint problem
4.1.2
in calculus of variations
2.3.5 | 2.7
variation
alternative definition
1.3.2 | 1.3.3
first
1.3.2
second
1.3.3
variational equation
2.6.2 | 4.2.4
viscosity
physical interpretation
5.3.2 | 5.4
solution
5.3.2
of HJB equation
5.3.3
subsolution
5.3.2
supersolution
5.3.2
weak extremum
2.2.1
Weierstrass
3.1.1 | 3.1.2
Weierstrass excess function
3.1.2
Weierstrass necessary condition
3.1.2
Weierstrass theorem
1.2.1.3 | 1.3.4
Weierstrass-Erdmann corner conditions
3.1.1
Zeno behavior
see Fuller's phenomenon

Daniel 2010-12-20