For a mathematical program in standard form, x* in X (the domain) is an optimal solution if it is a maximum (or a minimum): x* is feasible; f(x*) = f(x) for all feasible x (maximum value). Some authors refer to an optimal solution when they mean a local optimum; others mean a member of the optimality region (which are global optima). In either case, the optimal value is the objective value, evaluated at an optimal solution.
Pertaining to a trajectory, path, or control motion, one that minimizes or maximizes some quantity or combination of quantities such as fuel, time, energy, distance, heat transfer, etc. This optimum condition, or path, is commonly calculated by a type of mathematics known as calculus of variations.