A guide to the understanding of dynamic processes. Calculus of variations was the first step towards understanding the functioning of dynamic process that occur around us, be it the living world or man-made mechanisms.

1. the initial approach towards
dynamic optimization

2. Certain fundamentals
 FUNCTIONAL- a mapping of set of functions to
set of real numbers as opposed to functions
Theses are formulated as definite integrals
involving unknown functions and their
derivatives
Set in infinite-horizon spaces, we try to seek the
critical points using calculus; these are called
boundary conditions

3. The method of solving
 Firstly, the given functional has to be
‘piecewise continuous and differentiable’
 Our objective is minimization of a quadratic
functional associated with linear boundary
value problems
For e.g. finding the shortest distance between
any two points on a plane given the end point
restrictions
We formulate a functional gradient- which will
give the optimal value from the ‘n’ number of
available paths

4.  Given this minimal curve problem, the
calculus of variations uses the Euler-
Lagrange equation (necessary condition)
 ∆J[u]=0 or the gradient vanishes; this gives
the first order condition, which may be a
minimum. Inmost of the cases, the resulting
first-order homogenous differential equation
satisfies the sufficient conditions as well
 In many cases, the boundary conditions are
not defined, i.e. time and state are variable;
we then require transversality conditions

5. The Optimal Control theory
 Another approach to wards dynamic
optimisation problems; gives a direct method to
solving
 We use the Pontryagin’s maximum principle;
form the Hamiltonian and use the boundary
conditions to get the set of first-order conditions
 Akin to the Lagrangian function in static
optimization where the critical values optimize
the function and not functional
 The Hamiltonian thus directly gives the
optimizing values using partial derivatives w.r.t
the boundary/transversility conditions

6. Where lies the difference?
 The Calculus of variations is the precursor to
Optimal Control theory; the former is more of a
crude approach towards solving
 We use differential and integral calculus
extensively in the former while the latter directly
uses the maximum principle as given
 The thematic distinction lies in approach; the
former builds the necessary and sufficient
conditions and finds the optimal path, the latter
needs to just place the given curve in the
conditions to see whether it is optimal or not
 Thus calculus of variations is a more extensive
method of solving; involves rigorous treatment;
and more importantly has led to the
development of Optimal Control Theory.

7. Finally,
we know
what’s
this!
Thank
you