Algorithms for relaxed control problems with delay

Abstract

Three convergent procedures are presented. These determine a minimizing point of a relaxed optimal control problem defined by delay-differential equations with the control restricted to a compact convex subset of ℝ m In the first of these procedures, a strong variational algorithm due to Virk (1985 a) is extended to relaxed controls. The second uses a steepest descent approach. It is shown that both these procedures generate accumulation controls satisfying first-order necessary conditions of optimality. In the third procedure, the relaxed controls generated at each iteration are approximated using ordinary controls. When this is done, we show that limit points satisfy optimality conditions to within delta. Numerical performances of all three algorithms are also presented.