7 Introduction to Optimisation
249
7.2.4.3 Comparison of Direct and Indirect Methods
Loosely comparing an optimal control problem to a static constrained optimisation,
the direct method’s goal is to pinpoint a local minimum of the performance function,
while an indirect method aims to find a root of the necessary conditions. The
latter shall be preferred when a closed-form solution is aimed for. Indeed, indirect
methods allow to extract the control in an analytical way [73, 74]. However, this
is possible only when several approximations are employed or simplified cases are
considered. When a numerical approach is necessary, a direct method often results
to be the simpler choice due to several considerations [68]:
• The quantities
∂H
∂x
T
and
∂H
∂u
T
needed by indirect methods must be analytically computed and changed when different models are employed Furthermore,
when a problem is divided into phases, these quantities change along the trajectory. This requires an extensive preliminary analytical stage for any different
problem in the matter. On the contrary, a direct method is a flexible approach,
more suitable for black box implementations, and able to handle a problem
divided into different phases.
• Path inequalities, which are quite ordinary in low-thrust applications, represent
a relevant issue for indirect methods. Indeed, a first guess of the activeinactive sequence is needed for practical methods as it changes the form of the
Hamiltonian, by adding the Lagrange multipliers, the number of constrained arcs
and the junction conditions. However, a priori knowledge of the right series is
quite hard to achieve.
• Another issue with first guesses emerges from the initial estimate of the adjoint
variables λ. As remarked by Bryson and Ho [75], the extremal solutions can
be very sensitive to small changes in the unspecified boundary conditions. As
usually the initial state variables are specified, the transversality conditions (7.10)
show that the initial values of the adjoint variables for the optimal trajectory
are not known. Further, these variables are not representing physical quantities.
Hence, setting the right initial conditions, or even reasonable ones, is very
complex, and a bad initialisation often results in numerically ill-conditioned
solutions. On the contrary, direct methods disregard those variables and require
only initial guesses on the physical state and control variables.
7.2.4.4 Practical Techniques for Optimal Control
As stated in numerous occasions, in general the continuous optimal control problem
does not have a closed-form solution, and practical numerical optimisation methods
come into play. Any numerical technique cannot handle an infinite-dimension
problem, but it needs a discrete problem with a finite set of variables and constraints
to work with. This transition can be performed with conceptually different methods
which will be investigated in the present section. It is important to emphasise that
249
7.2.4.3 Comparison of Direct and Indirect Methods
Loosely comparing an optimal control problem to a static constrained optimisation,
the direct method’s goal is to pinpoint a local minimum of the performance function,
while an indirect method aims to find a root of the necessary conditions. The
latter shall be preferred when a closed-form solution is aimed for. Indeed, indirect
methods allow to extract the control in an analytical way [73, 74]. However, this
is possible only when several approximations are employed or simplified cases are
considered. When a numerical approach is necessary, a direct method often results
to be the simpler choice due to several considerations [68]:
• The quantities
∂H
∂x
T
and
∂H
∂u
T
needed by indirect methods must be analytically computed and changed when different models are employed Furthermore,
when a problem is divided into phases, these quantities change along the trajectory. This requires an extensive preliminary analytical stage for any different
problem in the matter. On the contrary, a direct method is a flexible approach,
more suitable for black box implementations, and able to handle a problem
divided into different phases.
• Path inequalities, which are quite ordinary in low-thrust applications, represent
a relevant issue for indirect methods. Indeed, a first guess of the activeinactive sequence is needed for practical methods as it changes the form of the
Hamiltonian, by adding the Lagrange multipliers, the number of constrained arcs
and the junction conditions. However, a priori knowledge of the right series is
quite hard to achieve.
• Another issue with first guesses emerges from the initial estimate of the adjoint
variables λ. As remarked by Bryson and Ho [75], the extremal solutions can
be very sensitive to small changes in the unspecified boundary conditions. As
usually the initial state variables are specified, the transversality conditions (7.10)
show that the initial values of the adjoint variables for the optimal trajectory
are not known. Further, these variables are not representing physical quantities.
Hence, setting the right initial conditions, or even reasonable ones, is very
complex, and a bad initialisation often results in numerically ill-conditioned
solutions. On the contrary, direct methods disregard those variables and require
only initial guesses on the physical state and control variables.
7.2.4.4 Practical Techniques for Optimal Control
As stated in numerous occasions, in general the continuous optimal control problem
does not have a closed-form solution, and practical numerical optimisation methods
come into play. Any numerical technique cannot handle an infinite-dimension
problem, but it needs a discrete problem with a finite set of variables and constraints
to work with. This transition can be performed with conceptually different methods
which will be investigated in the present section. It is important to emphasise that