3.1. Necessary Optimality Conditions for Problem (1.1)
To develop the necessary conditions for the extremum for (1.1), assume that
is the desired function, let
, and define a family of curves
since
is a linear operator; then we get (1.1) in the form
and where
is extremum at
, we get by differentiating both sides with respect to
and set
, for all admissible
,
But we have (by integration by parts in classic and fractional calculus)
Substituting in (3.2), we get
Since
is arbitrary, we get
and
which gives the fractional Euler-Lagrange equation in the form
with the natural boundary condition (transversality conditions)
If
is specified, then we have
, but if it is not specified, then we get the boundary condition
Remark 3.1.
These conditions are only necessary for an extremum. The question of sufficient conditions for the existence of an extremum is considered in the next section.
Special Cases
Case 1.
If
is a local extremizer to
by putting
and
in (3.5), (3.6), and (3.7), we get the fractional Euler-Lagrange equation in the form
for all
, with the boundary condition
Case 2.
If
is a local extremizer to
we get similar results as in [18].
3.2. Necessary Optimality Conditions for Problem (1.2)
To develop the necessary conditions for the extremum for (1.2), assume that
is the desired function, let
, and define a family of curves
since
is a linear operator; then we get (1.2) in the form
and where
is extremum at
, we get by differentiating both sides with respect to
and set
, for all admissible
,
But we have (by integration by parts) that
Substituting in (3.13), we get
Since
is arbitrary, we get
and
which gives the fractional Euler-Lagrange equation in the form
with the natural boundary condition (transversality conditions)
If
is specified, then we have
, but if it is not specified, then we get the boundary condition