Named after I. Michael Ross and F. Fahroo, the Ross–Fahroo lemma is a fundamental result in

optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...

theory. I. M. Ross and F. Fahroo, A Pseudospectral Transformation of the Covectors of Optimal Control Systems, Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001. I. M. Ross and F. Fahroo, Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems, ''Proceedings of the American Control Conference, Invited Paper'', June 2004, Boston, MA.N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", ''

IEEE Spectrum ''IEEE Spectrum'' is a magazine edited and published by the Institute of Electrical and Electronics Engineers. The first issue of ''IEEE Spectrum'' was published in January 1964 as a successor to ''Electrical Engineering''. In 2010, ''IEEE Spe ...

'', November 2012. It states that dualization and

discretization In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...

are, in general, non-commutative operations. The operations can be made commutative by an application of the covector mapping principle.

Description of the theory

A continuous-time optimal control problem is information rich. A number of interesting properties of a given problem can be derived by applying the

Pontryagin's minimum principle Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that i ...

or the

Hamilton–Jacobi–Bellman equation The Hamilton-Jacobi-Bellman (HJB) equation is a nonlinear partial differential equation that provides necessary and sufficient conditions for optimality of a control with respect to a loss function. Its solution is the value function of the opti ...

s. These theories implicitly use the continuity of time in their derivation.B. S. Mordukhovich, Variational Analysis and Generalized Differentiation: Basic Theory, Vol.330 of Grundlehren der Mathematischen Wissenschaften undamental Principles of Mathematical SciencesSeries, Springer, Berlin, 2005. When an optimal control problem is discretized, the Ross–Fahroo lemma asserts that there is a fundamental loss of information. This loss of information can be in the primal variables as in the value of the control at one or both of the boundary points or in the dual variables as in the value of the Hamiltonian over the time horizon.F. Fahroo and I. M. Ross, Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems, AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA. To address the information loss, Ross and Fahroo introduced the concept of closure conditions which allow the known information loss to be put back in. This is done by an application of the covector mapping principle.

Applications to pseudospectral optimal control

When pseudospectral methods are applied to discretize optimal control problems, the implications of the Ross–Fahroo lemma appear in the form of the discrete covectors seemingly being discretized by the transpose of the differentiation matrix. When the covector mapping principle is applied, it reveals the proper transformation for the adjoints. Application of the transformation generates the Ross–Fahroo pseudospectral methods.A. M. Hawkins, ''Constrained Trajectory Optimization of a Soft Lunar Landing From a Parking Orbit,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2005.
/ref>J. R. Rea, ''A Legendre Pseudospectral Method for Rapid Optimization of Launch Vehicle Trajectories,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2001.
/ref>

References

{{DEFAULTSORT:Pseudospectral Optimal Control Optimal control Numerical analysis Control theory

Description of the theory

Applications to pseudospectral optimal control

See also

References