  A problem of computing time-fuel optimal control for state transfer of a
single input linear time invariant (LTI) system to the origin is considered.
The input is assumed to be bounded. Since, the optimal control is bang-off-bang
in nature, it is characterized by sequences of +1 , 0 and -1 and the
corresponding switching time instants. All (candidate) sequences satisfying the
Pontryagin's maximum principle (PMP) necessary conditions are characterized.
The number of candidate sequences is obtained as a function of the order of
system and a method to list all candidate sequences is derived. Corresponding
to each candidate sequence, switching time instants are computed by solving a
static optimization problem. Since the candidate control input is a piece-wise
constant function, the time-fuel cost functional is converted to a linear
function in switching time instants. By using a simple substitution of
variables, reachability constraints are converted to polynomial equations and
inequalities. Such a static optimization problem can be solved separately for
each candidate sequence. Finally, the optimal control input is obtained from
candidate sequences which gives the least cost. For each sequence, optimization
problem can be solved by converting it to a generalized moment problem (GMP)
and then solving a hierachical sequence of semidefinite relaxations to
approximate the minima and minimizer [1]. Lastly, a numerical example is
presented for demonstration of method.
