Formulation

This page fixes the mathematical setting and the notation used throughout the documentation. The @def syntax and the functional API both transcribe it into code.

The problem

An optimal control problem (OCP) with fixed initial and final times consists in minimising the cost functional, written in Bolza form,

where the state   and the control   are functions of time  . They are subject to the differential constraint (the dynamics)

and to other constraints such as

The first two lines are box constraints, is a path constraint, and gathers the boundary conditions. An equality constraint is the case where both bounds are equal.

If  , the cost is said to be in Lagrange form; if  , it is in Mayer form. Maximising is the same as minimising , so a cost can be written with → max as well.

When , and do not depend on explicitly, the problem is autonomous. See Time dependence for how to check it on a model.

Free times and extra variables

The initial time and the final time may also be free, that is, part of the optimisation variables:

More generally, a vector   of additional variables can be optimised together with and . It may contain , , or any other free parameter, and the cost, the dynamics and the constraints may all depend on it:

with, in addition, box constraints on :

The control-free case

Nothing above requires a control: taking   (no ) is a legitimate degenerate case. It is used to fit or optimise the parameters of a dynamical system rather than to steer it. See Control-free problems for how to declare and solve such problems.

Notation and conventions

The pages on the indirect method (Flows, Geometry and the examples) use the following conventions. They refer to this section rather than redefine them.

Dimensions.  ,  ,  . A dimension-one quantity is a scalar, in the mathematics as in the code.

Pseudo-Hamiltonian. With the costate   and the constant  ,

The normal case is   ;   is the abnormal case. In the code, is a PseudoHamiltonian.

Maximum principle. Along an optimal trajectory, the control maximises . When this maximisation gives the control in feedback form , the maximised Hamiltonian is

a Hamiltonian in the code. Its Hamiltonian vector field    gives the extremals:

State constraints. A state constraint is a path constraint that does not depend on the control. It is written in the normalised form  . Along a boundary arc, where  , the pseudo-Hamiltonian gains the term   , with a multiplier  :

Minimal time. A free final time to minimise is written as a Mayer cost,  . Then  , so   , and the transversality condition reads

Writing the same cost in Lagrange form,   , gives instead     and   for an autonomous problem. Both are correct; the documentation uses the Mayer form.

Variable costate. When the variable is optimised, the indirect method treats it as a constant state ( ) with its own costate :

for a Mayer term that depends on and no boundary condition that involves . For instance, with the cost term and   ,   .

See also

• Abstract syntax (@def) — write this formulation directly as Julia code.

• Functional API — build the same model without the macro.

• Control-free problems — the   case.

• Flows — build the Hamiltonian flows of the indirect method.