Computation of exact gradients in distributed dynamic systems

A new and unified methodology for computing first order derivatives of functions obtained in complex multistep processes is developed on the basis of general expressions for differentiating a composite function. From these results, we derive the formulas for fast automatic differentiation of elementary functions, for gradients arising in optimal control problems, nonlinear programming and gradients arising in discretizations of processes governed by partial differential equations. In the proposed approach we start with a chosen discretization scheme for the state equation and derive the exact gradient expression. Thus a unique discretization scheme is automatically generated for the adjoint equation. For optimal control problems, the proposed computational formulas correspond to the integration of the adjoint system of equations that appears in Pontryagin's maximum principle. This technique appears to be very efficient, universal, and applicable to a wide variety of distributed controlled dynamic systems and to sensitivity analysis.