Formulation

An optimal control problem (OCP) with fixed initial and final times can be described as minimising the cost functional (in Bolza form)

where the state and the control are functions of time , subject for   to the differential constraint

and other constraints such as

If  , the cost is said to be in Lagrange form; if  , it is in Mayer form.

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 introduced (it may contain , , or any other free parameter). The cost, dynamics, and constraints then all depend on :

with, in addition, box constraints on :

The control-free case

Nothing above requires a control to be present: taking   (no ) is a legitimate degenerate case, used to fit or optimise parameters of a dynamical system rather than to steer it. See Problems without a control for how to declare and solve such problems.

See also

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

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

• Problems without a control — the   case.