19 Path Constraints
We consider a variational problem in which we seek to minimize an objective functional with respect to a path constraint—an algebraic constraint that must hold continuously along the entire path of the function x(t).
19.1 Discrete Derivation
Problem Formulation
The problem can be stated as follows:
\begin{aligned} \text{minimize} \quad & \mathcal{F}(x) = \int_a^b L(x, x^\prime, t) \, \mathrm{d}t, \\ \text{subject to} \quad & g(x, x^\prime, t) = 0. \end{aligned}
where L(x, x^\prime, t) is the Lagrangian, which depends on x(t), its derivative x^\prime(t), and t; and g(x, x^\prime, t) = 0 represents the algebraic (path) constraint that must be satisfied at each point t \in [a, b].
Feasible Space
We define a feasible space \mathbb{W} for x, which includes all functions that satisfy both the path constraint g(x, x^\prime, t) = 0 and the boundary conditions of the problem:
\mathbb{W} = \left\{ x \in C^1([a, b]) \mid g(x, x^\prime, t) = 0, \; t \in [a, b] \right\}.
The goal, then, is to find an x \in \mathbb{W} that minimizes \mathcal{F}(x).
Feasible Directions
To analyze feasible directions, we introduce a perturbation of the form x + \alpha \delta x, where \delta x is a variation such that:
g(x(t), x^\prime(t), t) = 0, \quad \text{for all } t \in [a, b].
To obtain the variation of \mathcal{F}(x), we compute:
\begin{aligned} \delta \mathcal{F}(x; \delta x) &= \int_a^b \left( \frac{\partial L}{\partial x}(x, x^\prime, t) \delta x + \frac{\partial L}{\partial x^\prime}(x, x^\prime, t) \delta x^\prime \right) \, \mathrm{d}t \\ &= \int_a^b \left( \frac{\partial L}{\partial x}(x, x^\prime, t) - \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial L}{\partial x^\prime}(x, x^\prime, t) \right) \right) \delta x \, \mathrm{d}t \\ &\quad + \left[ \frac{\partial L}{\partial x^\prime}(x, x^\prime, t) \delta x \right]_{t = a}^{t = b}. \end{aligned}
However, for \delta x to be feasible, it must also satisfy the constraint:
\delta g(x, x^\prime, t) = 0, \quad \Rightarrow \quad \frac{\partial g}{\partial x}(x, x^\prime, t) \delta x + \frac{\partial g}{\partial x^\prime}(x, x^\prime, t) \delta x^\prime = 0.
this approach is impractical, thus, we try to use finite difference.
Finite Differences
To discretize the problem, we divide the interval [a, b] into n subintervals of width h = \frac{b - a}{n} and define points t_k = a + k h for k = 0, 1, \dots, n. Let x_k approximate the solution at each point t_k, so that x_k \approx x(t_k).
First Derivative Approximation: We approximate the first derivative x^\prime(t) at each point t_k using a finite difference formula:
x^\prime(t_{k+1/2}) = \frac{x(t_{k+1}) - x(t_k)}{h} + \mathcal{O}(h^2)
(check using Taylor series)
Midpoint Approximation: We approximate the midpoint x(t_{k+1/2}) using left and right values:
x(t_{k+1/2}) = \frac{x(t_{k+1})+x(t_k)}{2} + \mathcal{O}(h^2)
(check using Taylor series)
Applying the Constraint: We impose the path constraint g(x, x^\prime, t) = 0 at each (mid-)point t_{k+1/2} by substituting the finite difference approximation:
g\left( \frac{x_{k+1}+x_k}{2}, \frac{x_{k+1} - x_k}{h}, \frac{t_{k+1}+t_k}{2}\right) = 0.
This transforms the continuous path constraint into a set of discrete constraints, one at each point t_{k+1/2}.
Discrete Objective
To approximate the integral \mathcal{F}(x), we use the midpoint rule:
\begin{aligned} \mathcal{F}_h(\bm{x}) &= h \sum_{k=0}^{n-1} L\left(x_{k+1/2},x^\prime_{k+1/2}, t_{k+1/2}\right), \\[0.5em] x_{k+1/2}&=\frac{x_{k+1}+x_k}{2},\quad x^\prime_{k+1/2}=\frac{x_{k+1}-x_k}{h},\quad t_{k+1/2}=\frac{t_{k+1}+t_k}{2}, \end{aligned}
Our goal is now to minimize \mathcal{F}_h(\bm{x}) with respect to \bm{x} = (x_0, x_1, \dots, x_n), subject to the discrete constraints on g(x_{k+1/2}, x^\prime_{k+1/2}, t_{k+1/2}) = 0 for each k.
Discrete Multipliers
We introduce Lagrange multipliers \lambda_k at each point t_k to incorporate the constraints. The discrete Lagrangian becomes:
\begin{aligned} \mathcal{L}_h(\bm{x}, \bm{\lambda}) &= \mathcal{F}_h(\bm{x}) - h\sum_{k=0}^{n-1} \lambda_{k+1/2} \, g\left(x_{k+1/2}, x^\prime_{k+1/2}, t_{k+1/2}\right). \\ & = h\sum_{k=0}^{n-1} L\left(x_{k+1/2}, x^\prime_{k+1/2}, t_{k+1/2}\right) - h\sum_{k=0}^{n-1} \lambda_{k+1/2} \, g\left(x_{k+1/2}, x'_{k+1/2}, t_{k+1/2}\right) \\ & = h\sum_{k=0}^{n-1} M\left(x_{k+1/2}, x^\prime_{k+1/2}, \lambda_{k+1/2}, t_{k+1/2}\right) \end{aligned}
where
M(x,x^\prime,\lambda,t) = L(x,x^\prime,t)-\lambda g(x,x^\prime,t)
Nonlinear System
To find stationary points of \mathcal{L}_h(\bm{x}, \bm{\lambda}), we differentiate with respect to each x_k and \lambda_k, leading to a system of equations:
Derivative with respect to x_k:
\frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_k} = 0,\qquad k=0,1,\ldots,n
Constraint equations for each k:
g\left(x_{k+1/2}, x^\prime_{k+1/2}, t_{k+1/2}\right) = 0,\qquad k=0,1,\ldots,n-1
in details:
\begin{aligned} \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_0} & = h\frac{\partial}{\partial x_0} M\left(x_{1/2}, x^\prime_{1/2}, \lambda_{1/2}, t_{1/2}\right) \\ & = \frac{h}{2}\frac{\partial M}{\partial x} \left(x_{1/2}, x^\prime_{1/2}, \lambda_{1/2}, t_{1/2}\right) -\frac{\partial M}{\partial x^\prime} \left(x_{1/2}, x^\prime_{1/2}, \lambda_{1/2}, t_{1/2}\right) \\ \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_n} & = h\frac{\partial}{\partial x_n} M\left(x_{n-1/2}, x^\prime_{n-1/2}, \lambda_{n-1/2}, t_{n-1/2}\right) \\ & = \frac{h}{2}\frac{\partial M}{\partial x} \left(x_{n-1/2}, x^\prime_{n-1/2}, \lambda_{n-1/2}, t_{n-1/2}\right) +\frac{\partial M}{\partial x^\prime} \left(x_{n-1/2}, x^\prime_{n-1/2}, \lambda_{n-1/2}, t_{n-1/2}\right) \end{aligned}
for k=1,2,\ldots n-1 \begin{aligned} \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_k} & = h\frac{\partial}{\partial x_k} M\left(x_{k-1/2}, x^\prime_{k-1/2}, \lambda_{k-1/2}, t_{k-1/2}\right) + h\frac{\partial}{\partial x_k} M\left(x_{k+1/2}, x^\prime_{k+1/2}, \lambda_{k+1/2}, t_{k+1/2}\right) \\ & = \frac{h}{2}\bigg[ \frac{\partial M}{\partial x} \left(x_{k-1/2}, x^\prime_{k-1/2}, \lambda_{k-1/2}, t_{k-1/2}\right)+ \frac{\partial M}{\partial x} \left(x_{k+1/2}, x^\prime_{k+1/2}, \lambda_{k+1/2}, t_{k+1/2}\right) \bigg] \\ & + \bigg[ \frac{\partial M}{\partial x^\prime} \left(x_{k-1/2}, x^\prime_{k-1/2}, \lambda_{k-1/2}, t_{k-1/2}\right)- \frac{\partial M}{\partial x^\prime} \left(x_{k+1/2}, x^\prime_{k+1/2}, \lambda_{k+1/2}, t_{k+1/2}\right) \bigg] \end{aligned}
consider now the discrete nonlinear system
\left\{ \begin{aligned} \frac{1}{h} \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_k}&=0,\qquad &k&=1,2,\ldots,n-1 \\ \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_0}&=0 \\ \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_n}&=0 \\ g\left(x_{k+1/2}, x^\prime_{k+1/2}, t_{k+1/2}\right) &= 0,\qquad & k&=0,1,\ldots,n-1 \end{aligned} \right.
As h \to 0, this finite difference approximation corresponds to the following approximations
\begin{aligned} \frac{1}{h} \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_k} & \approx \left[ \frac{\partial M(x,x^\prime,\lambda,t)}{\partial x} -\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial M(x,x^\prime,\lambda,t)}{\partial x^\prime} \right]_{t=t_k} \\ \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_0} & \approx -\frac{\partial M}{\partial x^\prime} \left(x(a), x^\prime(a), \lambda(a), a\right) \\ \frac{\partial \mathcal{L}_h(\bm{x}, \bm{\lambda})}{\partial x_n} & \approx \frac{\partial M}{\partial x^\prime} \left(x(b), x^\prime(b), \lambda(b), b\right) \\ g\left(x_{k+1/2}, x^\prime_{k+1/2}, t_{k+1/2}\right) &\approx g\left(x, x^\prime, t\right)\Big|_{t=t_k} \end{aligned}
thus the finite difference approximation can be viewed as the discretization of the following BVP
\left\{ \begin{aligned} \frac{\partial M(x,x^\prime,\lambda,t)}{\partial x} -\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial M(x,x^\prime,\lambda,t)}{\partial x^\prime} &=0 \\ -\frac{\partial M}{\partial x^\prime} \left(x(a), x^\prime(a), \lambda(a), a\right) &=0\\ \frac{\partial M}{\partial x^\prime} \left(x(b), x^\prime(b), \lambda(b), b\right) &=0 \\ g\left(x, x^\prime, t\right) &=0 \end{aligned} \right.
19.2 Continuous Multipliers
In this section, we develop an analogy for handling constrained variational problems, where we aim to minimize an objective functional subject to an algebraic (or path) constraint. This technique introduces an augmented functional by incorporating a Lagrange multiplier to enforce the constraint continuously over the interval.
Problem
We start with the following constrained minimization problem: \begin{aligned} \text{minimize} \quad & \mathcal{F}(x) = \int_a^b L(x, x^\prime, t) \, \mathrm{d}t, \\ \text{subject to} \quad & g(x, x^\prime, t) = 0. \end{aligned} Where:
\mathcal{F}(x) is the objective functional, with L(x,x',t) as the Lagrangian depending on the function x(t), its derivative x^\prime(t), and the independent variable t.
The path constraint g(x,x',t) = 0 must hold continuously over the interval [a,b].
Augmented Functional
To incorporate the constraint into the minimization process, we construct an augmented functional L(x, \lambda) by introducing a Lagrange multiplier function \lambda(t) that enforces the constraint g(x, x', t) = 0 at every point in the interval:
\mathcal{L}(x,\lambda) = %\int_a^b L(x, x^\prime, t)-\lambda g(x, x^\prime, t) \, \mathrm{d}t = \int_a^b M(x, x^\prime, \lambda, t)\, \mathrm{d}t
where we define the augmented Lagrangian (or the modified integrand) as:
M(x, x', \lambda, t) = L(x, x', t) - \lambda g(x, x', t).
First Variation
To find stationary points of \mathcal{L}(x, \lambda), we calculate the first variation \delta \mathcal{L}(x, \lambda; \delta x, \delta \lambda) with respect to x and \lambda:
\delta\mathcal{L}(x,\lambda;\delta x,\delta\lambda) = \int_a^b \frac{\partial M}{\partial x}\delta x+ \frac{\partial M}{\partial x^\prime}\delta x^\prime + \frac{\partial M}{\partial\lambda}\delta \lambda \, \mathrm{d}t
To further simplify, we integrate by parts on the term involving \delta x^\prime:
\delta\mathcal{L}(x,\lambda;\delta x,\delta\lambda) = \int_a^b \left[ \frac{\partial M}{\partial x} -\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial M}{\partial x^\prime} \right]\delta x+ \frac{\partial M}{\partial\lambda}\delta \lambda \, \mathrm{d}t + \frac{\partial M}{\partial x^\prime}\delta x\Big|_{t=a}^{t=b}
Necessary Conditions
For free endpoints, stationarity with respect to arbitrary variations \delta x and \delta\lambda gives the Euler–Lagrange equation, two natural boundary conditions, and the path constraint:
\left\{ \begin{aligned} \frac{\partial M}{\partial x}(x, x^\prime, \lambda,t) -\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial M}{\partial x^\prime}(x, x^\prime, \lambda, t) &=0 \\ \frac{\partial M}{\partial x^\prime}(x(b), x^\prime(b), \lambda(b), b) &=0 \\ -\frac{\partial M}{\partial x^\prime}(x(a), x^\prime(a), \lambda(a), a) &=0 \\ -\frac{\partial M}{\partial\lambda}(x, x^\prime, \lambda, t) = g(x, x^\prime, t) &= 0 \end{aligned} \right.
If an endpoint value is fixed, the corresponding variation vanishes and its natural boundary condition is omitted. The multiplier \lambda(t) is a function because the constraint is imposed at every t.
19.3 Inequality Path Constraints
For a pointwise inequality
c(x,x',t)\le0,
introduce a multiplier function \mu(t) and the augmented integrand
M(x,x',\mu,t)=L(x,x',t)+\mu(t)c(x,x',t).
Under an appropriate constraint qualification, the first-order KKT conditions are
\begin{aligned} M_x-\frac{\mathrm d}{\mathrm dt}M_{x'}&=0,\\ c(x,x',t)&\le0,\\ \mu(t)&\ge0,\\ \mu(t)c(x,x',t)&=0. \end{aligned}
Thus \mu(t)=0 wherever the constraint is inactive and may be positive only on active arcs. Junctions between active and inactive arcs require matching conditions; nonsmooth multipliers may arise when the active set changes.
19.4 Regularity of Path-Constraint Multipliers
The notation \lambda(t) or \mu(t) suggests an ordinary function, but this regularity is not automatic. For smooth equality constraints with constant rank, a sufficiently regular multiplier function is often obtained locally. For state inequalities and changing active sets, the natural multiplier can be less regular and, in more general formulations, may be a measure. Therefore the pointwise KKT equations above should be read under the stated smoothness and constraint-qualification assumptions.
For a constraint involving x' the augmented Euler–Lagrange equation is valid on smooth arcs. At junctions, one must additionally enforce the appropriate Weierstrass–Erdmann matching conditions for the augmented integrand. These junction conditions are part of the necessary system and cannot in general be recovered by solving the differential equation separately on each arc.