Enginering Optimsation Solving Functional Optimisation

Functional optimization involves finding functions (rather than just values) that maximize or minimize a functional. Here’s a comprehensive approach to tackling these problems:

1. Understanding the Problem Structure

A functional optimization problem typically involves minimizing or maximizing:

$J[y]={\int }_{x_1}^{x_2}f(x,y,y^{\prime }),dx$

Subject to certain boundary conditions, where $y=y(x)$ is the unknown function.

2. Classify the Problem Type

• Unconstrained problems: Only boundary conditions at endpoints
• Constrained problems: Additional constraints such as:
  • Isoperimetric constraints: $\int g(x,y,y^{\prime }),dx=constant$
  • Holonomic constraints: $g(x,y)=0$
  • Non-holonomic constraints: $g(x,y,y^{\prime })=0$

3. The Variational Approach

For Unconstrained Problems:

1. Apply the Euler-Lagrange Equation: $\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y^{\prime }}\right)=0$ This is a second-order ODE that the optimal function must satisfy.

2. Special Cases:

   • If $f$ doesn’t explicitly depend on $x$: $f-y^{\prime }\frac{\partial f}{\partial y^{\prime }}=C$ (Beltrami identity)
   • If $f$ doesn’t explicitly depend on $y$: $\frac{\partial f}{\partial y^{\prime }}=C$ (momentum conservation)
   • If $f=f(y^{\prime })$ only: The solution is a straight line

For Constrained Problems:

1. Using Lagrange Multipliers:
   • Form an augmented functional: $J[y]=\int (f+\lambda g),dx$
   • Apply Euler-Lagrange to this functional
   • Solve the resulting system with the constraint equations

4. Solving the Resulting Differential Equations

1. Direct Integration: If the ODE can be directly integrated
2. Numerical Methods: For complex ODEs
3. Series Solutions: For cases where analytic solutions are difficult

5. Boundary Value Problem Solution

Most functional optimization problems lead to boundary value problems (BVPs) rather than initial value problems. Methods include:

1. Shooting Method: Convert the BVP to an initial value problem and iterate
2. Finite Difference Method: Discretize the domain and solve a system of equations
3. Spectral Methods: Express the solution as a sum of basis functions

6. Verification

1. Second Variation: Check if the second variation is positive (for minimization) or negative (for maximization)
2. Numerical Verification: Compare with numerical solutions
3. Known Solutions: Compare with known solutions for special cases

Example: Shortest Path Problem

Let’s revisit our shortest path problem as an example of this approach:

1. Identify the functional: $L[y]={\int }_{x_1}^{x_2}\sqrt{1+(y^{\prime }{)}^2},dx$

2. Apply Euler-Lagrange: Since $f=\sqrt{1+(y^{\prime }{)}^2}$ doesn’t depend on $y$, we know $\frac{\partial f}{\partial y^{\prime }}=C$:

   $\frac{y^{\prime }}{\sqrt{1+(y^{\prime }{)}^2}}=C$

3. Solve the differential equation:

   Squaring both sides: $\frac{(y^{\prime }{)}^2}{1+(y^{\prime }{)}^2}=C^2$

   Solving for $y^{\prime }$: $y^{\prime }=\frac{C}{\sqrt{1-C^2}}$ (assuming $|C|<1$)

   This gives $y^{\prime }=m$ (constant), meaning $y=mx+b$

   So the solution is a straight line, which aligns with our intuition about the shortest path between two points.

4. Apply boundary conditions: $y(x_1)=y_1$ and $y(x_2)=y_2$, which uniquely determines the values of $m$ and $b$.

This approach can be extended to more complex functionals and constraints, following the general framework outlined above.

Extra Problems

1. Minimal Surface of Revolution Problem

Given two points $(x_1,y_1)$ and $(x_2,y_2)$ with $y_1,y_2>0$, find the curve $y(x)$ connecting these points such that when rotated around the $x$-axis, it generates a surface with minimal area. This can be formulated as minimising the functional:

$\mathcal{L}(y)={\int }_{x_1}^{x_2}2\pi y\sqrt{1+(y^{\prime }{)}^2},dx.$

Derive the Euler-Lagrange equation and the associated boundary conditions.

Euler Lagrange:

$$\frac{\partial f}{\partial y}-\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y^{\prime }}\right)$$ $$2\pi \sqrt{1+(y^{\prime }{)}^2}+\frac{\partial }{\partial x}(2\pi y\frac{y^{\prime }}{\sqrt{1+(y^{\prime }{)}^2}})$$

Since $\frac{\partial f}{\partial y^{\prime }}$ doesnt directly depend on x, we can simplify:

$$f-y^{\prime }\frac{\partial f}{\partial y^{\prime }}=C$$

…

…

$$2\pi y\frac{1}{\sqrt{1+(y^{\prime }{)}^2}}=C$$ $$y=\frac{C}{2\pi }\sqrt{1+(y^{\prime }{)}^2}$$ $$y^{\prime }=\sqrt{\frac{y^2}{(\frac{C}{2\pi }{)}^2}-1}$$

2. Isoperimetric Problem

Find the closed plane curve of fixed perimeter $L$ that encloses the maximum area. This can be formulated as maximising the functional:

$\mathcal{A}(y)={\int }_{x_1}^{x_2}y(x),dx,$

subject to the constraint:

${\int }_{x_1}^{x_2}\sqrt{1+(y^{\prime }{)}^2},dx=L.$

Derive the Euler-Lagrange equation with the appropriate Lagrange multiplier and the associated boundary conditions.

3. Hanging Chain (Catenary) Problem

Find the shape of a flexible, inextensible chain of uniform density hanging under its own weight between two fixed points $(x_1,y_1)$ and $(x_2,y_2)$. This can be formulated as minimising the functional representing the potential energy:

$\mathcal{L}(y)={\int }_{x_1}^{x_2}y\sqrt{1+(y^{\prime }{)}^2},dx.$

Derive the Euler-Lagrange equation and the associated boundary conditions.

4. Fermat’s Principle of Least Time

Given two points $(x_1,y_1)$ and $(x_2,y_2)$ in a medium where the speed of light varies with height according to $v(y)=v_0/n(y)$, where $n(y)$ is the refractive index, find the path that light takes between these points. By Fermat’s principle, light follows the path that minimizes travel time, formulated as:

$\mathcal{L}(y)={\int }_{x_1}^{x_2}\frac{n(y)}{v_0}\sqrt{1+(y^{\prime }{)}^2},dx.$

Derive the Euler-Lagrange equation and the associated boundary conditions.

5. Geodesic on a Surface

Find the shortest path (geodesic) between two points on a surface $z=f(x,y)$. For a surface with metric $d{s}^2=E,d{x}^2+2F,dx,dy+G,d{y}^2$, this can be formulated as minimising the functional:

$\mathcal{L}(y)={\int }_{x_1}^{x_2}\sqrt{E+2F{y}^{\prime }+G(y^{\prime }{)}^2},dx,$

where $E$, $F$, and $G$ are functions of $x$ and $y(x)$.

Derive the Euler-Lagrange equation and the associated boundary conditions.