Understanding Piecewise Smooth Functions in Optimization

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Piecewise Smooth Functions, Admissibility, and Corners: The Hidden Geometry of Optimization

In the world of optimization, where curves define action and functionals guide decisions, not all functions flow like silk. Some break. Some bend. Some come with corners. But that doesn’t make them illegal—it makes them interesting. Welcome to the world of piecewise smooth admissible functions.

What Is an Admissible Function?

An admissible function is a candidate solution in a variational problem. It must satisfy the boundary conditions, belong to the appropriate function space, and follow any constraints posed by the problem.

But not all admissible functions need to be perfectly smooth. Some are allowed to be piecewise smooth, meaning they are smooth (i.e., continuously differentiable) in segments—just not necessarily everywhere.

Enter the Corner

A corner is a point where the function is continuous, but the derivative is not. Imagine a road that doesn’t break—but makes a sharp turn. That sharp turn? That’s your corner.

Mathematically: If y(x) is continuous at x = c, but y′(x) has a discontinuity at x = c, then x = c is a corner point.

Where This Shows Up: Real-World Examples

  • Engineering: The bending of a beam may involve sudden changes in slope (corners), especially at joints or support points.
  • Economics: Optimal consumption paths with taxation or subsidy changes may create kinks in trajectories.
  • City Planning: Road elevation profiles or rail lines may change slope at specific mandated engineering points.

The Weierstrass–Erdmann Corner Conditions

Optimization with corners isn’t wild guessing. There are rules. The Weierstrass–Erdmann conditions tell us how to handle these sharp changes: 

At x = c (corner point):  
  y is continuous  
  p = ∂L/∂y′ is continuous  
  H = y′ ∂L/∂y′ - L is continuous

These conditions ensure that the change in behavior at the corner is still optimal, even though the curve itself has a sharp transition.

Why Allow Corners?

Some variational problems don’t admit perfectly smooth solutions—either due to physical discontinuities or imposed constraints. In such cases, piecewise smooth solutions are not just admissible—they are necessary. Corners are a concession to reality. Life isn’t always smooth, and neither are optimal paths.

A Simple Illustration

Suppose we want to minimize: 

J[y] = ∫₀² |y′(x)| dx  
Subject to: y(0) = 0, y(2) = 1

The optimal solution isn’t smooth. It’s a V-shaped function with a corner at x = 1. A piecewise linear, piecewise smooth path.

Conclusion: The Art of the Corner

In the calculus of variations, not everything needs to glide. Some solutions bend. Some twist. Some hold secrets in their corners. And those corners? They’re not bugs—they’re features.

Embrace piecewise smoothness. It’s where the real story unfolds.