Nonuniqueness of Solutions in the Calculus of Variations: A Geometric Approach
Dominique Henri · SIAM Journal on Control and Optimization · 1980
Differential topology is the latest of the many mathematical tools which have been used in the study of optimization problems. In this paper we apply it to the fundamental problem of the calculus of variations in $\mathbb{R}^n $: \[\mathcal{P}_{\xi ,T} \left\{ \begin{gathered} {\operatorname{Inf}}\int_0^T {f(x(t),\dot x(t))dt,} \hfill \\ x(0) = \xi _0 ,\qquad x(T) = \xi . \hfill \\ \end{gathered} \right. \] We show that if f is smooth, coercive (i.e., grows quickly at infinity), and convex with respect to $\dot x$, this problem has exactly one solution for almost every end condition $(\xi ,T)$. Then, using Thorn’s transversality theorems, we classify those points $(\xi ,T)$ in $\mathbb{R}^n \times \mathbb{R}$ where there is more than one solution (the singularity set). If the dimension is low $(n \leqq 4)$, there are but a finite number of singularity types which will fit almost all functions f.