Abstract

this paper is to show the application of these canonical formulas to optimal control processes being integrated by the Runge-Kutta family of numerical methods. There are many papers concerning numerical comparisions between automatic differentiation, finite differences and symbolic differentiation. See, for example, [1, 2, 6, 7, 21] among others. Another objective is to test the behavior of the spectral projected gradient methods introduced in [5]. These methods combine the classical projected gradient with two recently developed ingredients in optimization: (i) the nonmonotone line search schemes of Grippo, Lampariello and Lucidi ([24]), and (ii) the spectral steplength (introduced by Barzilai and Borwein ([3]) and analyzed by Raydan ([30, 31])). This choice of the steplength requires little computational work and greatly speeds up the convergence of gradient methods. The numerical experiments presented in [5], showing the high performance of these fast and easily implementable methods, motivate us to combine the spectral projected gradient methods with automatic differentiation. Both tools are used in this work for the development of codes for numerical solution of optimal control problems. In Section 2 of this paper, we apply the canonical formulas to the discrete version of the optimal control problem. In Section 3, we give a concise survey about spectral projected gradient algorithms. Section 4 presents some numerical experiments. Some final remarks are presented in Section 5. 2 CANONICAL FORMULAS The basic optimal control problem can be described as follows: Let a process governed by a system of ordinary differential equations be dx(t) dt = f(x(t); u(t); ); T 0 t T f ; (1) where x : [T 0 ; T f ] ! IR nx , u : [T 0 ; T f ] ! U ` IR nu , U compact, and 2 V ...

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