What is transversality condition?

The transversality condition is a key concept in optimization theory and is used to ensure that an optimal solution is well-behaved in relation to a given constraint. Mathematically, the transversality condition requires that the tangent space to the constraint curve is not tangent to the direction of optimization at the optimal solution. This means that the optimal solution will not be located at a point where the constraint is tangent to the direction of optimization, which could result in an unstable or unbounded solution. The transversality condition is commonly used in the study of optimal control and dynamic optimization, where it is used to ensure that a solution is optimal over a finite horizon. In addition to this, the transversality condition is also used in the study of differential geometry, where it is used to study the behavior of curves and surfaces in higher dimensional space.