20 Summary
Notation \color{blue}\frac{\mathrm{d}}{\mathrm{d}\alpha}\bigg|_{\alpha=0}=\delta
Problem: minimize \color{blue}\int_a^b L(x,x^\prime,t)\,\mathrm{d}t \;+\; \color{SeaGreen}\Phi(x(a),x(b))
Boundary conditions: \color{OrangeRed} x(a) = x_a, \quad x(b)=\text{free}, \quad b_k(x(a),x(b)) = 0, \cdots
Integral constraints: \color{Brown} \int_a^b g_k(x,x^\prime,t)\,\mathrm{d}t \;+\; \Psi_k( x(a),x(b)) = 0
Higher order problem: minimize
\color{blue}\int_a^b L(x,x^\prime,x^{\prime\prime},\ldots,x^{(n)},t)\,\mathrm{d}t \;+\;\color{SeaGreen}\Phi(x(a),x(b))
20.1 Admissible Spaces
- \mathbb{V} = \Big\{x\in C^m([a,b]),\ldots\Big\}
- \mathbb{D} = \Big\{\delta x\in C^\infty([a,b]),\delta x(a) = \delta x(b) = 0, \delta x^\prime(a) = \delta x^\prime(b) = 0, \ldots \Big\}
Endpoint Costs
\begin{aligned} \mathcal{F}(x) &= \int_a^b L(x,x^\prime,t)\,\mathrm{d}t + \Phi(x(a),x(b)) \\ \delta\mathcal{F}(x) &= \delta\int_a^b L(x,x^\prime,t)\,\mathrm{d}t + \delta \Phi(x(a),x(b),x^\prime(a),x^{\prime}(b)) \\ &= \int_a^b \delta L(x,x^\prime,t)\,\mathrm{d}t + \delta \Phi(x(a),x(b)) \\ &= \int_a^b L_x(x,x^\prime,t)\delta x+ L_{x^\prime}(x,x^\prime,t)\delta x^\prime\,\mathrm{d}t \\ & \quad + \Phi_{x_a}(x(a),x(b)){\color{OrangeRed}\delta x(a)} + \Phi_{x_b}(x(a),x(b)){\color{OrangeRed}\delta x(b)} \\[0.5em] &\color{red}\text{integration by part} \\[0.5em] &= \int_a^b \left[L_x(x,x^\prime,t)- \frac{\mathrm{d}}{\mathrm{d}t}L_{x^\prime}(x,x^\prime,t)\right]{\color{OrangeRed}\delta x}\,\mathrm{d}t \\[0.5em] & \quad+ \left(\Phi_{x_a}(x(a),x(b))- L_{x^\prime}(x(a),x^\prime(a),a)\right){\color{OrangeRed}\delta x(a)} \\[0.5em] & \quad+ \left(\Phi_{x_b}(x(a),x(b))+L_{x^\prime}(x(b),x^\prime(b),b)\right){\color{OrangeRed}\delta x(b)} \\[0.5em] \end{aligned}
Endpoint Constraints
Define \begin{aligned} \mathcal{F}(x) & = \int_a^b L(x,x^\prime,t)\,\mathrm{d}t- \sum_{k=1}^p \lambda_k b_k(x(a),x(b)) \\ \delta\mathcal{F}(x) &= \delta\int_a^b L(x,x^\prime,t)\,\mathrm{d}t - \delta\sum_{k=1}^p \lambda_k b_k(x(a),x(b)) \\ &= \int_a^b L_x(x,x^\prime,t)\delta x+ L_{x^\prime}(x,x^\prime,t)\delta x^\prime\,\mathrm{d}t - \sum_{k=1}^p b_k(x(a),x(b)){\color{OrangeRed}\delta\lambda_k} \\ & \quad - \sum_{k=1}^p \lambda_k \left( \frac{\partial b_k}{\partial x_a}(x(a),x(b)){\color{OrangeRed}\delta x(a)}+ \frac{\partial b_k}{\partial x_b}(x(a),x(b)){\color{OrangeRed}\delta x(b)} \right) \\[2em] &\color{red}\text{integration by part} \\[0.5em] &= \int_a^b \left[L_x(x,x^\prime,t) - \frac{\mathrm{d}}{\mathrm{d}t} L_{x^\prime}(x,x^\prime,t)\right]{\color{OrangeRed}\delta x}\,\mathrm{d}t - \sum_{k=1}^p b_k(x(a),x(b)){\color{OrangeRed}\delta\lambda_k} \\ & \quad - \left(\sum_{k=1}^p \lambda_k \frac{\partial b_k}{\partial x_a}(x(a),x(b)) +L_{x^\prime}(x(a),x^\prime(a),a)\right){\color{OrangeRed}\delta x(a)} \\ & \quad - \left( \sum_{k=1}^p \lambda_k \frac{\partial b_k}{\partial x_b}(x(a),x(b)) -L_{x^\prime}(x(b),x^\prime(b),b) \right) {\color{OrangeRed}\delta x(b)} \end{aligned}
Integral Constraints
Define \begin{aligned} M(x,x^\prime,\bm{\lambda},t) &= L(x,x^\prime,t)-\sum_{k=1}^m \lambda_k g_k(x,x^\prime,t) \\ N(x_a,x_b,\bm{\lambda}) &= \Phi(x_a,x_b)-\sum_{k=1}^m \lambda_k \Psi_k(x_a,x_b) \end{aligned}
\begin{aligned} \mathcal{F}(x,\bm{\lambda}) &= \int_a^b M(x,x^\prime,\bm{\lambda},t)\,\mathrm{d}t +N(x(a),x(b),\bm{\lambda})\\ \delta\mathcal{F}(x,\bm{\lambda}) &= \delta\int_a^b M(x,x^\prime,\bm{\lambda},t)\,\mathrm{d}t +\delta N(x(a),x(b),\bm{\lambda}) \\ &= \int_a^b M_x(x,x^\prime,\bm{\lambda},t){\color{OrangeRed}\delta x}+ M_{x^\prime}(x,x^\prime,\bm{\lambda},t){\color{OrangeRed}\delta x^\prime}+ M_{\bm{\lambda}}(x,x^\prime,\bm{\lambda},t)\cdot {\color{OrangeRed}\delta\bm{\lambda}} \,\mathrm{d}t\\ & \quad +N_{x_a}(x(a),x(b),\bm{\lambda}) {\color{OrangeRed}\delta x(a)} +N_{x_b}(x(a),x(b),\bm{\lambda}) {\color{OrangeRed}\delta x(b)} +N_{\bm{\lambda}}\cdot {\color{OrangeRed}\delta\bm{\lambda}} \\[0.5em] &\color{red}\text{integration by part} \\[0.5em] &= \int_a^b \left[M_x(x,x^\prime,\bm{\lambda},t)- \frac{\mathrm{d}}{\mathrm{d}t}M_{x^\prime}(x,x^\prime,t)\right]{\color{OrangeRed}\delta x}\,\mathrm{d}t\\ & \quad - \sum_{k=1}^m {\color{OrangeRed}\delta\lambda_k} \left[\int_a^b g_k(x,x^\prime,t)\,\mathrm{d}t+ \Psi_k(x(a),x(b))\right]\\ & \quad+\Big(N_{x_a}(x(a),x(b),\bm{\lambda})- M_{x^\prime}(x(a),x^\prime(a),\bm{\lambda},a)\Big){\color{OrangeRed}\delta x(a)} \\ & \quad+\Big(N_{x_b}(x(a),x(b),\bm{\lambda})+M_{x^\prime}(x(b),x^\prime(b),\bm{\lambda},b)\Big){\color{OrangeRed}\delta x(b)} \end{aligned}
Higher-Order Reduction
Transform to optimal control problem
\begin{aligned} \text{minimize:}\qquad & \color{blue}\int_a^b L(x_1,x_2,x_3,\ldots,x_n,u,t)\,\mathrm{d}t\\ \text{subject to:}\qquad & \left\{ \color{red} \begin{aligned} x_1^\prime & = x_2\\ x_2^\prime & = x_3\\ \quad & \vdots \\ x_{n-1}^\prime & = x_n \\ x_n^\prime & = u \end{aligned} \right. \end{aligned}
20.2 General Problem
\begin{aligned} \text{minimize:}\quad & \color{blue}\int_a^b L(x,x^\prime,t)\,\mathrm{d}t \;+\; \color{SeaGreen}\Phi(x(a),x(b)) \\[2em] \text{subject to:}\quad & \color{brown}b_k(x(a),x(b)) = 0, \quad & k&=1,2,\ldots,p \\[0.5em] & \color{purple}\int_a^b g_k(x,x^\prime,t)\,\mathrm{d}t \;+\; \color{RoyalBlue}\Psi_k( x(a),x(b)) = 0, \quad & k& =1,2,\ldots,m \end{aligned}
defining
\begin{aligned} M(x,x^\prime,\bm{\lambda},t) &= L(x,x^\prime,t)-\sum_{k=1}^m \lambda_k g_k(x,x^\prime,t) \\ N(x_a,x_b,\bm{\lambda},\bm{\mu}) &= \Phi(x_a,x_b) -\sum_{k=1}^m\lambda_k \Psi_k(x_a,x_b) -\sum_{k=1}^p\mu_k b_k(x_a,x_b) \end{aligned}
the stationary points satisfy the BVP
\left\{\begin{aligned} M_x(x,x^\prime,\bm{\lambda},t)-\frac{\mathrm{d}}{\mathrm{d}t}M_{x^\prime}(x,x^\prime,\bm{\lambda},t) &= 0\quad && \text{\color{red} Euler Lagrange} \\[0.5em] N_{x_a}(x(a),x(b),\bm{\lambda},\bm{\mu})- M_{x^\prime}(x(a),x^\prime(a),\bm{\lambda},a)&=0 \quad && \text{\color{red} Adjoint BC} \\[0.5em] N_{x_b}(x(a),x(b),\bm{\lambda},\bm{\mu})+M_{x^\prime}(x(b),x^\prime(b),\bm{\lambda},b) &=0\quad && \text{\color{red} Adjoint BC} \\[0.5em] b_k(x(a),x(b)) &= 0, \quad & k&=1,2,\ldots,p \\[0.5em] \int_a^b g_k(x,x^\prime,t)\,\mathrm{d}t + \Psi_k( x(a),x(b)) &= 0, \quad & k& =1,2,\ldots,m \end{aligned}\right.
20.3 Path Constraints
For g(x,x',t)=0 imposed pointwise, the multiplier is a function \lambda(t) and the augmented integrand is M=L-\lambda(t)g. For an inequality c(x,x',t)\le0, use M=L+\mu(t)c together with
\mu(t)\ge0, \qquad c(x,x',t)\le0, \qquad \mu(t)c(x,x',t)=0.
20.4 Stationarity and Minima
The Euler–Lagrange and multiplier equations identify stationary candidates. A local minimizer also satisfies \delta^2\mathcal J(x;\eta)\ge0 for every admissible direction. Joint convexity of the integrand and convexity of the admissible set make stationarity sufficient for global optimality; strict convexity gives uniqueness.
20.5 Classification and Existence
The Euler–Lagrange equations and transversality conditions characterize stationary candidates; they do not by themselves prove minimality or existence. For a fixed-endpoint first-order problem,
\delta^2\mathcal F(x;\eta)=\int_a^b\left(L_{xx}\eta^2+2L_{xx'}\eta\eta'+L_{x'x'}(\eta')^2\right)\,\mathrm dt.
A weak local minimum must satisfy the Legendre condition L_{x'x'}\ge0. Strong local minima also satisfy the Weierstrass excess inequality. Jacobi’s equation and conjugate points refine the second-order test. Existence is normally established separately through the direct method: coercivity, compactness, closedness of the feasible set, and lower semicontinuity.
20.6 Constraint Qualifications
Integral and path constraints require a regularity or constraint-qualification hypothesis for the usual normalized multiplier rule. If it fails, abnormal multipliers may occur. For inequality constraints, primal feasibility, dual feasibility, complementarity, and stationarity must all be imposed; active-set junctions may require additional matching conditions.