Bolza Problem and Transversality Condition

Formulations with Free Endpoints

In the simplest formulation of the calculus of variations, both endpoints of the curve are fixed: $y(x_0) = y_0$, $y(x_1) = y_1$. However, in many real problems, this is not the case. For example, one may need to find the shortest path from point A to a certain curve C (the endpoint is not fixed, but only has to lie on C). Or, the functional contains terms that depend on the endpoints. The Bolza problem is the most general formulation that includes all such cases.

Three Classical Formulations

Lagrange problem: $\min J = \int F, dx$ subject to differential constraints $G(x, y, y') = 0$. Motion along a curve with additional constraints.

Mayer problem: $\min J = g(x_0, y(x_0), x_1, y(x_1))$—we minimize only a boundary function, without an integral. This is a problem with free endpoints.

Bolza problem (general form): $\min J = \int F, dx + g(x_0, y_0, x_1, y_1)$ under additional constraints. It unifies the Lagrange and Mayer problems.

All three formulations are equivalent (can be reduced to each other), but each is convenient for a certain type of problem.

Transversality Condition

When the endpoint is not fixed and should lie on a curve C: $\varphi(x_1, y_1) = 0$, the boundary conditions change.

Derivation: when varying $\delta J = 0$ with an unfixed right endpoint, a “remainder” arises from integration by parts:

$[F - y' F_{y'}]\ \delta x_1 + F_{y'}\ \delta y_1 = 0$

Since the endpoint moves along C: $\varphi_x\ \delta x_1 + \varphi_y\ \delta y_1 = 0$. From these two conditions:

Transversality condition: $F_{y'} \cdot \varphi_x = (F - y' F_{y'}) \cdot \varphi_y$

Geometric meaning for the geodesic problem: if we seek the shortest distance from point A to a curve C, the transversality condition means that the extremal is perpendicular to C at the endpoint. This is intuitively evident: the shortest path from a point to a curve is the perpendicular to that curve.

Multidimensional Problems: Functionals of $u(x, y)$

For a functional of two variables:

$J[u] = \iint_{\Omega} F(x, y, u, u_x, u_y)\ dx, dy$

Euler–Lagrange equation: $F_u - \frac{\partial}{\partial x} F_{u_x} - \frac{\partial}{\partial y} F_{u_y} = 0$

This is a partial differential equation (PDE)!

Example: soap film (Plateau's problem). Find a surface $u(x, y)$ of minimal area stretched over a closed wire frame.

$F = \sqrt{1 + u_x^2 + u_y^2}$—element of area. The Euler–Lagrange equation:

$\frac{\partial}{\partial x} \left( \frac{u_x}{\sqrt{1 + u_x^2 + u_y^2}} \right) + \frac{\partial}{\partial y} \left( \frac{u_y}{\sqrt{1 + u_x^2 + u_y^2}} \right) = 0$

This is the condition of zero mean curvature: $H = (\kappa_1 + \kappa_2)/2 = 0$—a minimal surface. Physically, at each point of the soap film, the sum of the two principal curvatures equals zero, which corresponds to the balance of pressures on both sides.

Full Analysis: Shortest Distance from a Point to a Parabola

Problem: find the shortest curve from point $A = (0, 2)$ to the parabola $C: y = x^2/2$.

Length functional: $J[y] = \int_{0}^{x_1} \sqrt{1 + y'^2}\ dx$.

Euler–Lagrange equation: extremals are straight lines $y = a x + b$.

From the initial condition $y(0) = 2$: $b = 2$. Thus, the extremal: $y = a x + 2$.

Transversality condition: the endpoint lies on C: $y_1 = x_1^2/2$. Curve C is given by $\varphi(x, y) = y - x^2/2 = 0$, $\varphi_x = -x$, $\varphi_y = 1$.

For the straight line: $F_{y'} = \frac{y'}{\sqrt{1 + y'^2}} = \frac{a}{\sqrt{1 + a^2}}$, $F - y'F_{y'} = \frac{1}{\sqrt{1 + a^2}}$.

Transversality condition: $\frac{a}{(1 + a^2)^{1/2}} \cdot (-x_1) = (1 + a^2)^{-1/2} \cdot 1 \Rightarrow -a x_1 = 1 \Rightarrow x_1 = -1/a$.

Also, $y_1 = a x_1 + 2 = -1 + 2 = 1$, and $y_1 = x_1^2/2 = 1/(2 a^2)$. From $y_1 = 1$: $1 = 1/(2 a^2) \Rightarrow a^2 = 1/2 \Rightarrow a = \pm 1/\sqrt{2}$.

Take $a = -1/\sqrt{2}$ (the downward path): $x_1 = \sqrt{2}$, and the line is $y = -x/\sqrt{2} + 2$. Length: $\sqrt{1 + 1/2} \cdot \sqrt{2} = \sqrt{3}$. This is the shortest distance, and the straight line is perpendicular to the parabola at the point $(\sqrt{2}, 1)$.

Applications of the Bolza Problem

The Bolza problem covers most practical optimal control problems:

• Rocket control: the flight time $T$ is variable, the final position is the target point (Mayer). Fuel is minimized (Lagrange).
• Economic growth: Ramsey’s problem—a consumer maximizes $\int_{0}^{\infty} e^{-\rho t} u(c(t)),dt$ under capital dynamics $\dot{k} = f(k) - c$. Infinite horizon, a terminal condition on the limit $k(t)$.
• Robotics: manipulator trajectory—minimizing energy plus a penalty for final orientation.

Numerical Methods

For Bolza problems with moving boundaries, the following are used:

• Direct methods: discretization of state and controls, reduction to nonlinear programming (NLP). Solvers: IPOPT, SNOPT, Knitro.
• Shooting method: integrating the system with adjustment of initial conditions via Newton’s method.
• Pseudospectral methods (Radau, Chebyshev): representing the trajectory as a high-order polynomial, with exponential accuracy for smooth solutions. Used in GPOPS-II, DIDO.

In the aerospace industry, pseudospectral methods are the standard for designing interplanetary mission trajectories.