Introduction to Modern Economic Growth More formally let us assume that (ˆ x (t) , yˆ (t)) is an admissible pair such that yˆ (·) is continuous over [0, t1 ] and yˆ (t) ∈IntY (t), and we have W (ˆ x (t) , yˆ (t)) ≥ W (x (t) , y (t)) for any other admissible pair (x (t) , y (t)) The important and stringent assumption here is that (ˆ x (t) , yˆ (t)) is an optimal solution that never hits the boundary and that does not involve any discontinuities Even though this will be a feature of optimal controls in most economic applications, in purely mathematical terms this is a strong assumption Recall, for example, that in the previous chapter, we did not make such an assumption and instead started with a result on the existence of solutions and then proceeded to characterizing the properties of this solution (such as continuity and differentiability of the value function) However, the problem of continuous time optimization is sufficiently difficult that proving existence of solutions is not a trivial matter We will return to a further discussion of this issue below, but for now we follow the standard practice and assume that an interior continuous solution yˆ (t) ∈IntY (t), together with the corresponding law of motion of the state variable, xˆ (t), exists Note also that since the behavior of the state variable x is given by the differential equation (7.2), when y (t) is continuous, x˙ (t) will also be continuous, so that x (t) is continuously differentiable When y (t) is piecewise continuous, x (t) will be, correspondingly, piecewise smooth We now exploit these features to derive necessary conditions for an optimal path of this form To this, consider the following variation y (t, ε) ≡ yˆ (t) + εη (t) , where η (t) is an arbitrary fixed continuous function and ε ∈ R is a scalar We refer to this as a variation, because given η (t), by varying ε, we obtain different sequences of controls The problem, of course, is that some of these may be infeasible, i.e., y (t, ε) ∈ / Y (t) for some t However, since yˆ (t) ∈IntY (t), and a continuous function over a compact set [0, t1 ] is bounded, for any fixed η (·) function, we can always find εη > such that y (t, ε) ≡ yˆ (t) + εη (t) ∈ IntY (t) 316