Let $L:\mathbb{T}^d\times\mathbb{R}^d\mapsto\mathbb{R}$ be a Lagrangian over the $d$-dimentional standard torus times $\mathbb{R}^d$. Let $\alpha >0$, the infinite horizon optimal control problem consist in minimizing $$ u_{\alpha}(x) = \inf_{\textbf{x}:\textbf{x}(0)=x}\int_{-\infty}^{0}e^{\alpha s}L(\textbf{x},\dot{\textbf{x}})\text{ d} s, $$ amang all globaly Lipschitz trajectories $\mathbf{x}$ with initial condition $\mathbf{x}(0)=x$. Let $T \in \mathbb{R}$, the initial value problem, consist in minimizing $$ V(x,t) = \inf_{\textbf{x}:\textbf{x}(t)=x}\int_{-T}^{t}L(\textbf{x},\dot{\textbf{x}}) \text{ d} s + \psi(\textbf{x}(-T)), $$ for $t\geq -T$, amang all globaly Lipschitz trajectories $\mathbf{x}$ with initial condition $\mathbf{x}(t)=x$.
This functions satisfies the dynamic programming principle, that is, for any $T>0$,
$$ u_{\alpha}(x) = \inf_{\textbf{x}:\textbf{x}(0)=x} \left( \int_{-T}^{0}e^{\alpha s}L(\textbf{x},\dot{\textbf{x}}) \text{ d} s + e^{-\alpha T} u_{\alpha}(\textbf{x}(-T))\right). $$ Similarly: $$ V(x,t) = \inf_{\textbf{x}:\textbf{x}(t)=x} \left( \int_{-\tilde{t}}^{t}L(\textbf{x},\dot{\textbf{x}}) \text{ d} s + V(\textbf{x}(-\tilde{t}),-\tilde{t})\right), $$ for all $-T\leq -\tilde{t} \leq t.$
I need to prove that both infima are in fact minima, and I am trying to use the compactness of the torus and some argument of calculus of variation, but I don't achieve anything.
Thanks!
Basically it is enough to note that the functions $u_{\alpha}$ and $V$ are continuous and then use the extreme value theorem thanks to the compactness of the torus.