Let $\pi\in\Pi$ . The subset $\mathcal{D}_{\pi}$ of ${\widehat{W}}$ is the set of those elements such that for any $w\in{\widehat{W}}_{\pi}$ and $d\in\mathcal{D}_{\pi}$ ,

$$\ell(wd)=\ell(w)+\ell(d).$$

We call $\mathcal{D}_{\pi}$ the set of distinguished right coset representatives of ${\widehat{W}}_{\pi}$ in ${\widehat{W}}$ .

The subset $\mathcal{D}_{\pi}^{-1}$ is called the set of distinguished left coset representatives of ${\widehat{W}}_{\pi}$ in ${\widehat{W}}$ ; elements $d\in\mathcal{D}_{\pi}^{-1}$ have the property that $\ell(dw)=\ell(d)+\ell(w)$ for any $w\in{\widehat{W}}_{\pi}$ .

Functional ( 4 ) is said to be invariant under the one-parameter group of infinitesimal transformations

$$\begin{cases}\bar{t}=t+\varepsilon\tau(t,q)+o(\varepsilon)\,,\\ \bar{q}(\bar{t})=q(t)+\varepsilon\xi(t,q)+o(\varepsilon)\,,\\ \end{cases}$$

(8)

if, and only if,

$$\int_{t_{a}}^{t_{b}}L\left(t,q(t),{{}_{t_{a}}D_{t}^{\alpha}q(t)}\right)dt\\ =\int_{\bar{t}(t_{a})}^{\bar{t}(t_{b})}L\left(\bar{t},\bar{q}(\bar{t}),{{}_{\bar{t}_{a}}D_{\bar{t}}^{\alpha}\bar{q}(\bar{t})}\right)d\bar{t}$$

(9)

for any subinterval $[{t_{a}},{t_{b}}]\subseteq[a,b]$ .

A fractional optimal control problem ( 10 ) is said to be invariant under the $\varepsilon$ -parameter local group of transformations

$$\begin{cases}\bar{t}=t+\varepsilon\tau(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \bar{q}(\bar{t})=q(t)+\varepsilon\xi(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \bar{u}(\bar{t})=u(t)+\varepsilon\sigma(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \bar{p}(\bar{t})=p(t)+\varepsilon\zeta(t,q(t),u(t),p(t))+o(\varepsilon)\,,\\ \end{cases}$$

(13)

if, and only if,

$$\displaystyle\left[{\cal H}(\bar{t},\bar{q}(\bar{t}),\bar{u}(\bar{t}),\bar{p}(\bar{t}))-\bar{p}(\bar{t})\cdot{{}_{\bar{a}}D_{\bar{t}}}^{\alpha}\bar{q}(\bar{t})\right]d\bar{t}\\ \displaystyle=\left[{\cal H}(t,q(t),u(t),p(t))-p(t)\cdot{{}_{a}D_{t}}^{\alpha}q(t)\right]dt\,.$$

(14)

Write $s=Z(\gamma)$ . The $M$ - cycle $\mu(\gamma)$ associated to a $N$ -reduced geodesic $\gamma=\gamma_{Q}$ is the sum

$$\mu(\gamma)=\left\{i\infty,\widehat{\gamma}\right\}+\left\{\widehat{\gamma},i+s\right\}+\left\{i+s,\widehat{\gamma^{\prime}}\right\}+\left\{\widehat{\gamma^{\prime}},i\infty\right\},$$

where $\gamma^{\prime}=\gamma_{Q^{\prime}}$ , $Q^{\prime}=(T^{s}S^{\frac{a}{2}}T^{-s})\circ Q$ with $a=\textit{sign}(A)$ , if $s=s^{\prime}$ and $T^{-s}\gamma_{Q}$ contains $i$ , and

$$\mu(\gamma)=\left\{i\infty,\widehat{\gamma}\right\}+\left\{\widehat{\gamma},P\right\}+\left\{M_{s}P,\widehat{M_{s}\gamma}\right\}+\left\{\widehat{M_{s}\gamma},i\infty\right\}$$

where $P=\gamma_{Q}\cap\{s-1,s+1\}$ , otherwise.