MATH.10:4.2 - Construct changes that remain admissible

Build a family x(h) with x(0)=x. State the allowed values of the parameter h and substitute the family into the constraints. The parameter may be a scalar, a vector or a discrete choice.

For a fixed total x1+x2=b, the change (x1+h, x2-h) preserves the total. If both components must stay nonnegative, its range is -x1 <= h <= x2. A further capacity limit can shorten that range.

For a history with fixed endpoint values, try q_h(t)=q(t)+h*eta(t), where eta vanishes at both endpoints and has the regularity required by the functional. Check any additional path constraint as well.

A direction tangent to a constraint may preserve it only to first order. At (1,0) on the unit circle, (1,h) has squared length 1+h^2 and leaves the circle whenever h is nonzero. The curve (cos(h),sin(h)) stays on it. Use an admissible curve for a finite comparison, or keep a tangent calculation at the first-order scope its mathematical argument supports.

When a useful family cannot yet be constructed, the missing result is concrete: a parameterized change that retains the named constraint. Obtain the corresponding mathematical construction rather than continue with an inadmissible substitute.