Optimization of nonlinear discrete systems with finite set of controls (Q1921529)

A deterministic discrete optimal control problem of the form \[ J(u)= \sum^T_{t=1} f_t[y(t-1), u(t)]+ f[y(t)]\to\min, \] where the evolution is described by the equation \(y(t)= \Phi_t[y(t-1), u(t)]\), \(t=1,\dots,T\), \(y(0)\) is given and \(u(t)\) is chosen from a finite set of controls, is considered. Some additional constraints on states and controls are also presented. The author presents a general outline of the discrete \(\Psi\)-transformation method which consists in replacing the minimization of \(J\) by the minimization of a suitable continuous function on one real variable. Moreover, the realization of this method leads to an algorithm for dynamic discrete optimization. Numerical results for some problems of industrial planning (which were solved by using this algorithm) are given.