Minimizing an integral subject to terminal equality constraints

Question

How can I determine the minimum value that the following integral can take

$$ J (y) = \int_0^1 \left( x^4 \left(y''\right) + 4 x^2 \left(y'\right)^2 \right) {\rm d} x $$

knowing that $y$ is not singular in $x=0$ and that $y(1) = y'(1) = 1$?

Answer 1

Let's suppose that $y\in C^{2}([0,1])$. Integrating the first term of $J$ by parts, we obtain \begin{align} J&=x^4y'(x)\bigg|_0^1+\int_0^1[-4x^3y'+4x^2(y')^2]\,dx \\ &=1+\int_0^14x^2[(y')^2-xy']\,dx, \tag{1} \end{align} where we have used the condition $y'(1)=1$ and the assumption that $y'$ is not singular at $x=0$. Now, let's complete the square in the integral: \begin{align} J&=1+\int_0^1\left[4x^2\left(y'-\frac{x}{2}\right)^2-x^4\right]dx, \\ &=1-\frac{1}{5}+\int_0^1 4x^2\left(y'-\frac{x}{2}\right)^2dx. \tag{2} \end{align} It follows from $(2)$ that $J\geq\frac{4}{5}$, the equality ocurring iff $y'(x)=\frac{x}{2}$ for $x\in[0,1]$. Unfortunately, this is not compatible with the condition $y'(1)=1$, so we cannot say that the minimum value of $J$, given the conditions on $y$ and $y'$, is $\frac{4}{5}$. On the other hand, it's possible to show that $$ \inf J[y]=\frac{4}{5}. \tag{3} $$ Indeed, for $0<\delta<1$, let's define $$ y_{\delta}'(x):= \begin{cases} \frac{x}{2},&\text{if}\,\,0\leq x\leq 1-\delta, \\ \frac{x}{2}+\frac{(x-1+\delta)^2}{2\delta^2},&\text{if}\,\,1-\delta<x\leq 1. \end{cases} \tag{4} $$ It's straightforward to verify that $y_{\delta}'(x)$ and $y_{\delta}''(x)$ are continuous at $x=1-\delta$ and $y_{\delta}'(1)=1$; besides, $$ J[y_{\delta}]=\frac{4}{5}+\int_{1-\delta}^1 4x^2\frac{(x-1+\delta)^4}{4\delta^4}\,dx =\frac{4}{5}+\frac{\delta(\delta^2-7\delta+21)}{105}, \tag{5} $$ which shows that we can make $J[y_{\delta}]$ arbitrarily close to $\frac{4}{5}$ by choosing $\delta$ sufficiently small.

Answer 2

The following SymPy script

from sympy import *

x = Symbol('x')
y = Function('y')(x)

# define Lagrangian
L = x**4 * Derivative(y, x, x) + 4 * x**2 * (Derivative(y, x))**2

# Euler-Lagrange equation
ELE = euler_equations(L, y, x) 

# solve the ODE
solution = dsolve(ELE[0], y)

# print LaTeX
print(latex(ELE[0]))
print(latex(solution))

produces the following ODE

$$12 x^{2} - 8 x \left(x \frac{d^{2}}{d x^{2}} y{\left(x \right)} + 2 \frac{d}{d x} y{\left(x \right)}\right) = 0$$

and its general solution

$$y{\left(x \right)} = C_{1} + \frac{C_{2}}{x} + \frac{x^{2}}{4}$$

If $y$ is not singular at $x = 0$, then $C_2 = 1$. However, the quadratic function $y{\left(x \right)} = C_{1} + \frac{x^{2}}{4}$ cannot satisfy $y(1) = y'(1) = 1$. It is tempting to conclude that the problem is ill-posed.