  Studies regarding the computation of Optimal Control Problems (OCPs) with
terminal inequality constraint, under the frame of the Variation Evolving
Method (VEM), are carried out. The attributes of equality constraints and
inequality constraints in the generalized optimization problem is traversed,
and the intrinsic relations to the multipliers are uncovered. Upon these
preliminaries, the right Evolution Partial Differential Equation (EPDE) is
derived, and the costate-free optimality conditions are established. Besides
the analytic expression for the costates in the classic treatment, they also
reveal the analytic relations between the states, the controls and the
(Lagrange and KKT) multipliers, which adjoin the terminal (equality and
inequality) constraints. Moreover, in solving the transformed Initial-value
Problems (IVPs) with common Ordinary Differential Equation (ODE) integration
methods, the numerical soft barrier is proposed to eliminate the numerical
error resulting from the suddenly triggered inequality constraint and it is
shown to be effective.
