Collocation methods enforce the dynamic equations of an optimal control problem only at a finite set of collocation points. For most problems this works extremely well, but for some classes of problems (notably those with singular arcs, discontinuous controls, or other non-smooth features) collocation can produce solutions that oscillate, ring near discontinuities, or satisfy the dynamics poorly between the collocation points.

Integrated-residual transcription takes a different approach: instead of forcing the residual of the dynamic equations to be zero at selected points, it works with the integral of the squared residual over the whole time horizon. This integrated residual is a global measure of how well the candidate solution satisfies the dynamics. PSOPT supports two ways of using it:

• Residual minimisation, in which the integrated residual is minimised, providing a least-squares fit to the dynamics; and

• Constrained optimisation, in which the original objective is minimised subject to an explicit accuracy constraint on the integrated residual, so that the dynamic equations are satisfied to a specified tolerance.

The second formulation is particularly attractive because it gives direct control over the trade-off between optimality and dynamic feasibility, and it provides a meaningful accuracy certificate for problems where pointwise collocation error estimates are unreliable.

Integrated-residual transcription in PSOPT can be used with the same problem definition interface as the collocation methods, so switching between transcription approaches requires only minor changes to the problem file. It is a useful complement to collocation for problems with non-smooth solutions, and it can also serve as a cross-check on solutions obtained by other methods.