A "HUM" conjugate gradient algorithm for constrained nonlinear optimal control : terminal and regular problems

Abstract

Optimal control problems often arise in engineering applications when a known desired behavior is to be imposed on a dynamical system. Typically, there is a performance and controller use trade-off that can be quantified as a total cost functional of the state and control histories. Problems stated in such a manner are not required to follow an exact desired behavior, alleviating potential controllability issues. We present a method for solving large deterministic optimal control problems defined by quadratic cost functionals, nonlinear state equations, and box-type constraints on the control variables. The algorithm has been developed so that systems governed by general parabolic partial differential equations can be solved. The problems addressed are of the regulator-terminal type, in which deviations from specified state variable behavior are minimized over the entire trajectory as well as at the final time. The core of the algorithm consists of an extension of the Hilbert Uniqueness Method which, we show, can be considered a statement of the dual. With the definition of a problem-specific inner-product space, a formulation is constructed around a well-conditioned, stable, SPD operator, thus leading to fast rates of convergence when solved by, for instance, a conjugate gradient procedure (denoted here TRCG). Total computational time scales roughly as twice the order of magnitude of the computational cost of a single initial-value problem.

(cont.) Standard logarithmic barrier functions and Newton methods are employed to address the hard constraints on control variables of the type Umin < U < Umax. We have shown that the TRCG algorithm allows for the incorporation of these techniques, and that convergence results maintain advantageous properties found in the standard (linear programming) literature. The TRCG operator is shown to maintain its symmetric positive-definiteness for temporal discretizations, a property that is crucial to the practical implementation of the proposed algorithm. Sample calculations are presented which illustrate the performance of the method when applied to a nonlinear heat transfer problem governed by partial differential equations.