Normality and second-order optimality conditions in state-constrained optimal control problems with bounded minimizers

Abstract

A smooth optimal control problem with endpoint, mixed and state constraints is considered. The controllability matrix is constructed and the normality condition is presented as the condition of full rank for this matrix. The approach based on the controllability matrix allows one to examine the case of state-constrained control problems with fixed endpoints. As an application of normality, the second-order necessary optimality conditions are derived. The connection between different forms of second-order conditions is indicated.