First-Order Necessary Conditions in Optimal Control

In an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided conditions under which accumulation points satisfy a condition requiring a certain optimality function, used in the algorithms to generate search directions, to be nonnegative for all controls. The aim of this paper is to clarify the nature of this optimality condition, which we call the first-order minimax condition, and of a related integrated form of the condition, which, also, is implicit in past algorithm convergence analysis. We consider these conditions, separately, when a pathwise state constraint is, and is not, included in the problem formulation. When there are no pathwise state constraints, we show that the integrated first-order minimax condition is equivalent to the minimum principle and that the minimum principle (and equivalent integrated first-order minimax condition) is strictly stronger than the first-order minimax condition. For problems with state constraints, we establish that the integrated first-order minimax condition and the minimum principle are, once again, equivalent. But, in the state constrained context, it is no longer the case that the minimum principle is stronger than the first-order minimax condition, or vice versa. An example confirms the perhaps surprising fact that the first-order minimax condition is a distinct optimality condition that can provide information, for problems with state constraints, in some circumstances when the minimum principle fails to do so.

On the adjoint group of some radical rings

A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (see also [2, Theorem 6.1.2]). However, we will show that the converse holds if we restrict to the class of nil rings, i.e. the rings R such that for any a є R there exists an n = n(a) with an = 0.