Legendre’s Necessary Condition for Fractional Bolza Functionals with Mixed Initial/Final Constraints

Abstract

The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore, we were eager to elaborate a correct proof and it turns out that this goal is highly nontrivial, especially when considering final constraints. This paper is the result of our reflections on this subject. Precisely, we consider here a constrained minimization problem of a general Bolza functional that depends on a Caputo fractional derivative of order $0 < \alpha \le 1$ and on a Riemann–Liouville fractional integral of order $\beta > 0$, the constraint set describing general mixed initial/final constraints. The main contribution of our work is to derive corresponding first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the transversality conditions and, of course, the Legendre condition. A detailed discussion is provided on the obstructions encountered with the classical strategy, while the new proof that we propose here is based on the Ekeland variational principle. Furthermore, we underline that some subsidiary contributions are provided all along the paper. In particular, we prove an independent and intrinsic result of fractional calculus stating that it does not exist a nontrivial function which is, together with its Caputo fractional derivative of order $0< \alpha <1$, compactly supported. Moreover, we also discuss some evidences claiming that Riemann–Liouville fractional integrals should be considered in the formulation of fractional calculus of variations problems in order to preserve the existence of solutions.

Appendices

A Proof of Proposition 3.3

In this section we provide a detailed proof of Proposition 3.3. In order to accomplish it, recall that for all $\rho \in {\mathbb {R}}$, there exists a real sequence $(\rho _k)_{k \in {\mathbb {N}}}$ such that

$$\begin{aligned} (1-\xi )^\rho = \sum _{k \in {\mathbb {N}}} \rho _k \xi ^k, \end{aligned}$$

(9)

for all $\xi \in (-1,1)$. We also recall that the series convergence is uniform on all compact subsets included in $(-1,1)$. Moreover, if $\rho \in {\mathbb {R}}\backslash {\mathbb {N}}$, the terms $\rho _k$ are all different from zero.

Lemma A.1

Let $c > a$ be a real number and $u \in \mathrm {L}^1([a,c],{\mathbb {R}})$. If

$$\begin{aligned} \int _a^c (s-a)^k u(s) \, ds = 0, \end{aligned}$$

for all $k \in {\mathbb {N}}$, then $u=0$.

Proof

This result easily follows from the density of polynomial functions in $\mathrm {C}([a,c],{\mathbb {R}})$ and from [22, Corollary 4.24 p.110]. $\square$

Lemma A.2

Let $c > a$ be a real number and $u \in \mathrm {L}^1([a,c],{\mathbb {R}})$. Let us consider the function

$$\begin{aligned}{}\begin{array}[t]{lrcl}\varPsi :&{}(c,+\infty ) &{}\longrightarrow &{}{\mathbb {R}}\\ {} &{}t&{} \longmapsto &{}\varPsi (t) := \displaystyle \int _a^c (t-s)^\mu u(s) \, ds , \end{array} \end{aligned}$$

where $\mu \in {\mathbb {R}}\backslash {\mathbb {N}}$. If the function $\varPsi$ is polynomial over a subinterval $I \subset (c,+\infty )$ with a nonempty interior, then $u = 0$.

Proof

Without loss of generality, we can assume that $I = [c_1,c_2]$ is compact with $c< c_1 < c_2$. From the LDC theorem, one can easily see that $\varPsi$ is of class $\mathrm {C}^\infty$ with

$$\begin{aligned} \varPsi ^{(r)}(t) = \mu (\mu - 1) \ldots (\mu - r+1) \displaystyle \int _a^c (t-s)^{\mu -r} u(s) \, ds, \end{aligned}$$

for all $t > c$ and all $r \ge 1$. In the sequel we fix some $r \ge 1$ larger than the degree of $\varPsi$ (polynomial over I) plus one. Since $\mu \in {\mathbb {R}}\backslash {\mathbb {N}}$, we get that

$$\begin{aligned} \displaystyle \int _a^c (t-s)^{\mu -r} u(s) \, ds = 0, \end{aligned}$$

and thus

$$\begin{aligned} \displaystyle \int _a^c \left( 1 - \dfrac{s-a}{t-a} \right) ^{\mu -r} u(s) \, ds = 0, \end{aligned}$$

for all $t \in I$. From Equality (9) (with $\rho := \mu -r \in {\mathbb {R}}\backslash {\mathbb {N}}$) and the uniform convergence of the power series (since $0 \le \frac{s-a}{t-a} \le \frac{c-a}{c_1 - c } <1$ for all $(t,s) \in I \times [a,c]$), we get that

$$\begin{aligned} \displaystyle \sum _{k \in {\mathbb {N}}} \dfrac{\rho _k}{(t-a)^k} \int _a^c (s-a)^k u(s) \, ds = 0, \end{aligned}$$

for all $t \in I$. Finally, using the change of variable $T=\frac{1}{t-a}$, we can write that $\sum _{k \in {\mathbb {N}}} \lambda _k T^k = 0$ for all $T \in [\frac{1}{c_2 - a},\frac{1}{c_1-a}]$, where $\lambda _k := \rho _k \int _a^c (s-a)^k u(s) \, ds$ for all $k \in {\mathbb {N}}$. Since the zeros of a nonzero power series are isolated, we deduce that $\lambda _k = 0$ for all $k \in {\mathbb {N}}$. Since all $\rho _k$ are different from zero, Lemma A.1 concludes the proof. $\square$

We are now in a position to prove Proposition 3.3. Actually we can even prove the more general following statement.

Proposition A.1

Let $0< \alpha <1$ and $x \in {}_{\mathrm {c}} \mathrm {AC}^\alpha _{a+}$. If there exist two real numbers $a \le c < d \le b$ such that:

• (i) x is polynomial over [c, d];

• (ii) ${}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x](t) = 0_{{\mathbb {R}}^n}$ for almost every $t \in [c,d]$;

then x is constant over [a, d].

Proof

Without loss of generality, we assume in this proof that $n=1$. From Proposition 2.5, it holds that $x(t) = x(a) + \mathrm {I}^\alpha _{a+}[ {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x] ](t)$ for all $t \in [a,b]$, and thus

$$\begin{aligned} x(t) = x(a) + \displaystyle \int _a^c \dfrac{(t-s)^{\alpha -1}}{\varGamma (\alpha )} {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x](s) \, ds, \end{aligned}$$

for all $t \in [c,d]$. Let us denote by $u \in \mathrm {L}^1([a,c],{\mathbb {R}})$ the restriction of ${}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]$ over [a, c]. From the hypothesis, one can easily deduce that the function

$$\begin{aligned}{}\begin{array}[t]{lrcl}\varPsi :&{}(c,+\infty ) &{}\longrightarrow &{}{\mathbb {R}}\\ {} &{}t&{} \longmapsto &{}\varPsi (t) := \displaystyle \int _{a}^{c} (t-s)^{\alpha -1} u(s) \, ds , \end{array} \end{aligned}$$

is polynomial over (c, d]. From Lemma A.2, we deduce that $u=0$ and thus ${}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x](t) = 0$ for almost every $t \in [a,d]$. We deduce that $x(t)=x(a)$ for all $t \in [a,d]$ which completes the proof. $\square$

B Proof of Theorem 3.2

Our strategy of proof is based on the Ekeland variational principle [29]. Let us enunciate hereafter a simplified version (but sufficient for our purposes).

Proposition B.1

(Ekeland variational principle) Let $(\mathrm {E},\mathrm {d}_\mathrm {E})$ be a complete metric space and $\mathrm {J}: \mathrm {E}\rightarrow {\mathbb {R}}^+$ be a continuous nonnegative map. Let $\varepsilon >0$ and $\lambda \in \mathrm {E}$ such that $\mathrm {J}(\lambda ) = \varepsilon$. Then there exists $\lambda _\varepsilon \in \mathrm {E}$ such that $\mathrm {d}_\mathrm {E}( \lambda _\varepsilon , \lambda ) \le \sqrt{\varepsilon }$ and $-\sqrt{\varepsilon } \; \mathrm {d}_\mathrm {E}( \lambda ' , \lambda _\varepsilon ) \le \mathrm {J}(\lambda ')-\mathrm {J}(\lambda _\varepsilon )$ for all $\lambda ' \in \mathrm {E}$.

In Section B.1 we give some recalls about convex analysis. In Section B.2 we investigate the sensitivity analysis of the Bolza functional ${\mathcal {L}}$. Finally the proof of Theorem 3.2 is detailed in Section B.3 by applying the Ekeland variational principle on a penalized functional.

1.1 B.1 Basics of Convex Analysis

Let $\mathrm {d}_\mathrm {S}: {\mathbb {R}}^j \rightarrow {\mathbb {R}}_{+}$ denote the standard distance function to the nonempty closed convex subset $\mathrm {S}\subset {\mathbb {R}}^j$ defined by $\mathrm {d}_\mathrm {S}(z):= \inf _{z' \in \mathrm {S}} \Vert z-z' \Vert _{{\mathbb {R}}^j}$ for all $z \in {\mathbb {R}}^j$. We recall that for all $z \in {\mathbb {R}}^j$, there exists a unique element $\mathrm {P}_\mathrm {S}(z) \in \mathrm {S}$ (called the projection of z onto $\mathrm {S}$) such that $\mathrm {d}_\mathrm {S}(z)=\Vert z-\mathrm {P}_\mathrm {S}(z) \Vert _{{\mathbb {R}}^j}$. It can easily be shown that the map $\mathrm {P}_\mathrm {S}:{\mathbb {R}}^j \rightarrow \mathrm {S}$ is 1-Lipschitz continuous. Moreover it holds that $( z- \mathrm {P}_\mathrm {S}(z)) \cdot (z'-\mathrm {P}_\mathrm {S}(z)) \le 0$ for all $z'\in \mathrm {S}$, that is, $z-\mathrm {P}_\mathrm {S}(z)\in \mathrm {N}_\mathrm {S}[\mathrm {P}_\mathrm {S}(z)]$ for all $z \in {\mathbb {R}}^j$. Let us recall the two following required lemmas, whose proofs are detailed for the reader’s convenience.

Lemma B.1

Let $(z_k)_{k\in {\mathbb {N}}}$ be a sequence in ${\mathbb {R}}^j$ converging to some point $z\in \mathrm {S}$ and let $(\zeta _k)_{k\in {\mathbb {N}}}$ be a positive real sequence. If $\zeta _k(z_k-\mathrm {P}_\mathrm {S}(z_k))$ converges to some ${\overline{z}} \in {\mathbb {R}}^j$, then ${\overline{z}} \in \mathrm {N}_\mathrm {S}[z]$.

Proof

Since $z_k - \mathrm {P}_\mathrm {S}(z_k) \in \mathrm {N}_\mathrm {S}[\mathrm {P}_\mathrm {S}(z_k)]$ and $\zeta _k > 0$ for all $k \in {\mathbb {N}}$, we obtain that $\zeta _k (z_k - \mathrm {P}_\mathrm {S}(z_k)) \cdot (z' - \mathrm {P}_\mathrm {S}(z_k)) \le 0$ for all $z' \in \mathrm {S}$ and all $k \in {\mathbb {N}}$. Passing to the limit $k \rightarrow \infty$, and since $z \in \mathrm {S}$, we obtain that ${\overline{z}} \cdot (z' - z) \le 0$ for all $z' \in \mathrm {S}$ which exactly means that ${\overline{z}} \in \mathrm {N}_\mathrm {S}[z]$. $\square$

Lemma B.2

The map

$$\begin{aligned}{}\begin{array}[t]{lrcl}\mathrm {d}^2_\mathrm {S} :&{}{\mathbb {R}}^j &{}\longrightarrow &{}{\mathbb {R}}_+\\ {} &{}z&{} \longmapsto &{}\mathrm {d}^2_\mathrm {S}(z) := \mathrm {d}_\mathrm {S}(z)^2 , \end{array} \end{aligned}$$

is Fréchet-differentiable on ${\mathbb {R}}^j$, and its differential ${\mathcal {D}}\mathrm {d}^2_\mathrm {S}(z)$ at every $z \in {\mathbb {R}}^j$ can be expressed as

$$\begin{aligned} {\mathcal {D}}\mathrm {d}^2_\mathrm {S}(z)(z') = 2 ( z-\mathrm {P}_\mathrm {S}(z) ) \cdot z' , \end{aligned}$$

for all $z' \in {\mathbb {R}}^j$.

Proof

Let $z \in {\mathbb {R}}^j$ and let us prove that $\mathrm {d}^2_\mathrm {S}$ is Fréchet-differentiable at z with ${\mathcal {D}}\mathrm {d}^2_\mathrm {S}(z)(z') = 2 ( z-\mathrm {P}_\mathrm {S}(z) ) \cdot z'$. One has

$$\begin{aligned} \mathrm {d}^2_\mathrm {S}(z+z') - \mathrm {d}^2_\mathrm {S}(z) \le \Vert z+z' - \mathrm {P}_\mathrm {S}(z) \Vert ^2 - \Vert z - \mathrm {P}_\mathrm {S}(z) \Vert ^2 = 2 ( z-\mathrm {P}_\mathrm {S}(z) ) \cdot z' + \Vert z' \Vert ^2, \end{aligned}$$

and, from Cauchy–Schwarz inequality and 1-Lipschitz continuity of $\mathrm {P}_\mathrm {S}$, one gets

$$\begin{aligned}&\mathrm {d}^2_\mathrm {S}(z) - \mathrm {d}^2_\mathrm {S}(z+z') \le \Vert z - \mathrm {P}_\mathrm {S}(z+z') \Vert ^2 - \Vert z+z' - \mathrm {P}_\mathrm {S}(z+z') \Vert ^2 \\&\quad = -2 ( z-\mathrm {P}_\mathrm {S}(z+z') ) \cdot z' - \Vert z' \Vert ^2 \\&\quad = -2 ( z-\mathrm {P}_\mathrm {S}(z) ) \cdot z' + 2 ( \mathrm {P}_\mathrm {S}(z+z')-\mathrm {P}_\mathrm {S}(z)) \cdot z' - \Vert z' \Vert ^2 \\&\quad \le -2 ( z-\mathrm {P}_\mathrm {S}(z) ) \cdot z' + \Vert z' \Vert ^2 , \end{aligned}$$

for all $z' \in {\mathbb {R}}^j$. Using both inequalities, the proof is complete. $\square$

1.2 B.2 Sensitivity Analysis of the Bolza Functional

In the proof of Theorem 3.2 (see Section B.3), we denote:

• by $r_\alpha$ some real number satisfying $r_\alpha > \frac{1}{\alpha }$ and by $r'_\alpha := \frac{r_\alpha }{r_\alpha - 1}$ the classical conjugate of $r_\alpha$ satisfying $\frac{1}{r_\alpha } + \frac{1}{r'_\alpha } = 1$;

• and, for all $(u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n$, by $x(\cdot ,u,y) \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}$ the function defined by

  $$\begin{aligned} x(t,u,y) := y + \mathrm {I}^\alpha _{a+}[u](t), \end{aligned}$$

  for all $t \in [a,b]$;

• and, for all $(u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n$, by ${\mathcal {P}}(u,y)$ the set of Lebesgue points $\tau \in (a,b)$ of both the functions u and $L(x(\cdot ,u,y),u,\cdot )$.

Remark B.1

Note that $r'_\alpha (\alpha -1) +1 > 0$.

Remark B.2

Let $x \in \mathrm {L}^1$. From Proposition 2.5 and Remark 2.5, note that $x \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}$ if and only if there exists $(u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n$ such that $x=x(\cdot ,u,y)$. In that case, the couple (u, y) is unique and is given by $u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]$ and $y=x(a)$.

Remark B.3

For all $(u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n$, note that the set ${\mathcal {P}}(u,y)$ is of full measure in [a, b].

We introduce the set

$$\begin{aligned} \mathrm {L}^\infty _R := \mathrm {L}^\infty \Big ( [a,b] , {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R ) \Big ), \end{aligned}$$

for all $R \ge 0$, where ${\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R )$ denotes the standard closed ball of ${\mathbb {R}}^n$ centered at the origin $0_{{\mathbb {R}}^n}$ with radius $R \ge 0$. We endow the set $\mathrm {L}^\infty _R \times {\mathbb {R}}^n$ with the distance

$$\begin{aligned} \mathrm {d}_{ \mathrm {L}^\infty _R \times {\mathbb {R}}^n } \Big ( (u_2,y_2) , (u_1 , y_1) \Big ) := \Vert u_2 - u_1 \Vert _{\mathrm {L}^1} + \Vert y_2 - y_1 \Vert _{{\mathbb {R}}^n}, \end{aligned}$$

for all $(u_1,y_1)$, $(u_2,y_2) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$.

Lemma B.3

Let $R \ge 0$. The following assertions are true:

• (i) The metric space $( \mathrm {L}^\infty _R \times {\mathbb {R}}^n , \mathrm {d}_{ \mathrm {L}^\infty _R \times {\mathbb {R}}^n } )$ is complete;

• (ii) The map $(u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n \longmapsto x(\cdot ,u,y) \in \mathrm {C}$ is continuous;

• (iii) The map

  $$\begin{aligned}{}\begin{array}[t]{lrcl}\varPhi _R :&{}\mathrm {L}^\infty _R \times {\mathbb {R}}^n &{}\longrightarrow &{}{\mathbb {R}}\\ {} &{}(u,y)&{} \longmapsto &{}\varPhi _R (u,y) := {\mathcal {L}}(x(\cdot ,u,y)), \end{array} \end{aligned}$$

  is continuous.

Proof

The first item can be easily derived from the PCLDC theorem. Secondly, it can be proved from the classical Hölder inequality that

$$\begin{aligned}&\Vert x(t,u_2,y_2) - x(t,u_1,y_1) \Vert _{{\mathbb {R}}^n} \\&\quad \le \Vert y_2 - y_1 \Vert _{{\mathbb {R}}^n} +\dfrac{1}{\varGamma (\alpha )} \left( \dfrac{ R (b-a)^{r'_\alpha (\alpha -1) +1}}{r'_\alpha (\alpha -1) +1 } \right) ^{1/r'_\alpha } \Vert u_2 - u_1 \Vert ^{1/r_\alpha }_{\mathrm {L}^1} , \end{aligned}$$

for all $t \in [a,b]$ and all $(u_1,y_1)$, $(u_2,y_2) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$, which concludes the proof of the second item. The third item can be derived by contradiction and by using the LDC and PCLDC theorems. $\square$

The rest of this section is devoted to the sensitivity analysis of the Bolza functional $\varPhi _R$ under perturbations of the couple (u, y) (see Propositions B.2 and B.3). Before coming to these points, we first introduce the following notion of needle-perturbation of u.

Definition B.1

(Needle-perturbation of u) Let $R \ge 0$ and let $(u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$. A needle-perturbation of u associated with $(\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$ and $0 < h \le b-\tau$ is the function $u^{(\tau ,v)}(\cdot ,h) \in \mathrm {L}^\infty _R$ defined by

$$\begin{aligned} u^{(\tau ,v)}(t,h) := \left\{ \begin{array}{lcl} v &{} \text {if} &{} t \in [\tau ,\tau +h) , \\ u(t) &{} \text {if} &{} t \notin [\tau ,\tau +h) , \end{array} \right. \end{aligned}$$

for almost every $t \in [a,b]$.

Lemma B.4

Let $R \ge 0$ and let $(u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$. It holds that

$$\begin{aligned} \left\| x \Big ( t ,u^{(\tau ,v)}(\cdot ,h),y \Big )- x(t,u,y) \right\| _{{\mathbb {R}}^n} \le \dfrac{2R}{\varGamma (\alpha +1)} h^\alpha , \end{aligned}$$

(10)

for all $t \in [a,b]$ and

$$\begin{aligned}&\left\| \dfrac{ x \Big ( t , u^{(\tau ,v)}(\cdot ,h),y \Big )- x(t,u,y)}{h} - \dfrac{(t-\tau )^{\alpha -1}}{\varGamma (\alpha )} (v-u(\tau )) \right\| _{{\mathbb {R}}^n} \nonumber \\&\quad \le \dfrac{( t - \tau )^{\alpha -1}}{\varGamma (\alpha )} \left\| \dfrac{1}{h} \int _\tau ^{\tau +h} u(s) \; ds - u(\tau ) \right\| _{{\mathbb {R}}^n} \nonumber \\&\qquad +\dfrac{2R}{\varGamma (\alpha )} \Bigg ( \Big ( t - (\tau +h) \Big )^{\alpha -1} - (t-\tau )^{\alpha -1} \Bigg ) , \end{aligned}$$

(11)

for all $t \in (\tau +h,b]$, all $(\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$ and all $0 < h \le b-\tau$.

Proof

Let $(\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$ and $0 < h \le b-\tau$ being fixed for the whole proof. It holds that

$$\begin{aligned}&x \Big ( t ,u^{(\tau ,v)}(\cdot ,h),y \Big )- x(t,u,y) = \mathrm {I}^\alpha _{a+} [ u^{(\tau ,v)}(\cdot ,h) -u ](t) \\&\quad = \int _a^t \dfrac{(t-s)^{\alpha - 1}}{\varGamma (\alpha )} \Big ( u^{(\tau ,v)}(s,h) -u(s) \Big ) \; ds \\&\quad = \left\{ \begin{array}{lcl} 0_{{\mathbb {R}}^n} &{} \text {if} &{} a \le t < \tau , \\ \displaystyle \int _{\tau }^t \dfrac{(t-s)^{\alpha - 1}}{\varGamma (\alpha )} ( v -u(s) ) \; ds &{} \text {if} &{} \tau \le t \le \tau + h , \\ \displaystyle \int _{\tau }^{\tau + h} \dfrac{(t-s)^{\alpha - 1}}{\varGamma (\alpha )} ( v -u(s) ) \; ds &{} \text {if} &{} t > \tau + h , \end{array} \right. \end{aligned}$$

for all $t \in [a,b]$. Since $u \in \mathrm {L}^\infty _R$ and $v \in {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$, we obtain that

$$\begin{aligned}&\left\| x \Big ( t ,u^{(\tau ,v)}(\cdot ,h),y \Big )- x(t,u,y) \right\| _{{\mathbb {R}}^n} \le \left\{ \begin{array}{lcl} 0 &{} \text {if} &{} a \le t< \tau , \\ 2 R \displaystyle \int _{\tau }^t \dfrac{(t-s)^{\alpha - 1}}{\varGamma (\alpha )} \; ds &{} \text {if} &{} \tau \le t \le \tau + h , \\ 2R \displaystyle \int _{\tau }^{\tau + h} \dfrac{(t-s)^{\alpha - 1}}{\varGamma (\alpha )} \; ds &{} \text {if} &{} t> \tau + h , \end{array} \right. \\&\quad = \left\{ \begin{array}{lcl} 0 &{} \text {if} &{} a \le t < \tau , \\ 2 R \dfrac{(t-\tau )^\alpha }{\varGamma (\alpha +1)} &{} \text {if} &{} \tau \le t \le \tau + h , \\ 2R \dfrac{(t-\tau )^\alpha - (t-(\tau +h))^\alpha }{\varGamma (\alpha +1)} &{} \text {if} &{} t > \tau + h , \end{array} \right. \end{aligned}$$

for all $t \in [a,b]$. To prove Inequality (10), one has just to see that in all of the three above cases, the right-hand side term is less than $\frac{2R}{\varGamma (\alpha +1)}h^\alpha$. For the last case, one has just to invoke the basic inequality $\chi _2^\alpha - \chi _1^\alpha \le (\chi _2 - \chi _1 )^\alpha$ which is satisfied for all $0 \le \chi _1 \le \chi _2$ (see the proof of Lemma 3.3 for some details). Now let us prove Inequality (11). Using similar arguments, one has

$$\begin{aligned}&\dfrac{ x \Big ( t , u^{(\tau ,v)}(\cdot ,h),y \Big )- x(t,u,y)}{h} - \dfrac{(t-\tau )^{\alpha -1}}{\varGamma (\alpha )} (v-u(\tau )) \\&\quad = \dfrac{1}{h} \int _\tau ^{\tau +h} \dfrac{(t-s)^{\alpha -1}}{\varGamma (\alpha )} (v-u(s)) \; ds - \dfrac{(t-\tau )^{\alpha -1}}{\varGamma (\alpha )} (v-u(\tau )) \\&\quad = \dfrac{1}{h} \int _\tau ^{\tau +h} \dfrac{(t-s)^{\alpha -1} -(t-\tau )^{\alpha -1} }{\varGamma (\alpha )} (v-u(s)) \; ds + \dfrac{(t-\tau )^{\alpha -1}}{\varGamma (\alpha )} \left( u(\tau ) - \dfrac{1}{h} \int _\tau ^{\tau +h} u(s) \; ds \right) , \end{aligned}$$

for all $t \in (\tau +h,b]$. Since $u \in \mathrm {L}^\infty _R$ and $v \in {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$, we obtain that

$$\begin{aligned}&\left\| \dfrac{ x \Big ( t , u^{(\tau ,v)}(\cdot ,h),y \Big )- x(t,u,y)}{h} - \dfrac{(t-\tau )^{\alpha -1}}{\varGamma (\alpha )} (v-u(\tau )) \right\| _{{\mathbb {R}}^n} \\&\quad \le \dfrac{2R}{\varGamma (\alpha )} \dfrac{1}{h} \int _\tau ^{\tau +h} (t-s)^{\alpha -1} -(t-\tau )^{\alpha -1} \; ds + \dfrac{( t - \tau )^{\alpha -1}}{\varGamma (\alpha )} \left\| \dfrac{1}{h} \int _\tau ^{\tau +h} u(s) \; ds - u(\tau ) \right\| _{{\mathbb {R}}^n} , \end{aligned}$$

for all $t \in (\tau +h,b]$. Noting that $(t-s)^{\alpha -1} \le (t-(\tau +h))^{\alpha -1}$ for all $s \in [\tau ,\tau +h]$ and all $t \in (\tau +h,b]$, the proof of Inequality (11) is complete. $\square$

Proposition B.2

(Sensitivity analysis under needle-perturbation of u) Let $R \ge 0$ and let $(u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$. It holds that

$$\begin{aligned}&\lim \limits _{h \rightarrow 0^+} \dfrac{\varPhi _R \Big ( u^{(\tau ,v)}(\cdot ,h),y \Big )-\varPhi _R(u,y)}{h} = \dfrac{(b-\tau )^{\beta -1}}{\varGamma (\beta )} \Big ( L(x(\tau ),v,\tau )-L(x(\tau ),u(\tau ),\tau ) \Big ) \\&\quad +\, \left( \dfrac{ (b-\tau )^{\alpha -1}}{\varGamma (\alpha )} \partial _2 \varphi (x(a),x(b)) + \mathrm {I}^\alpha _{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x,u,\cdot ) \right] (\tau ) \right) \cdot (v-u(\tau )) , \end{aligned}$$

for all $(\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$, where $x = x(\cdot ,u,y) \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}$.

Proof

Let $(\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$. For simplicity of notations in this proof, we denote by

$$\begin{aligned} u_h := u^{(\tau ,v)}(\cdot ,h) \quad \text {and} \quad x_h := x \Big ( \cdot , u^{(\tau ,v)}(\cdot ,h),y \Big ), \end{aligned}$$

for all $0 < h \le b-\tau$. From Inequality (11) and since $\tau$ is a Lebesgue point of u, it is clear that

$$\begin{aligned} \lim \limits _{h \rightarrow 0^+} \dfrac{\varphi (x_h(a),x_h(b))-\varphi (x(a),x(b))}{h} = \dfrac{ (b-\tau )^{\alpha -1}}{\varGamma (\alpha )} \partial _2 \varphi (x(a),x(b)) \cdot (v-u(\tau )). \end{aligned}$$

Moreover the term

$$\begin{aligned} \dfrac{\mathrm {I}^{\beta }_{a+} [ L(x_h,u_h,\cdot )](b) - \mathrm {I}^{\beta }_{a+} [ L(x,u,\cdot ) ](b) }{h}, \end{aligned}$$

can be decomposed as:

$$\begin{aligned}&\dfrac{1}{h} \int _\tau ^{\tau +h} \dfrac{(b-s)^{\beta -1}}{\varGamma (\beta )} \Big ( L(x_h(s),v,s) - L(x(s),v,s) \Big ) \; ds \\&\quad +\, \dfrac{1}{h} \int _\tau ^{\tau +h} \dfrac{(b-s)^{\beta -1}}{\varGamma (\beta )} \Big ( L(x(s),v,s) - L(x(s),u(s),s) \Big ) \; ds \\&\quad +\, \dfrac{1}{h} \int _{\tau +h}^b \dfrac{(b-s)^{\beta -1}}{\varGamma (\beta )} \Big ( L(x_h(s),u(s),s) - L(x(s),u(s),s) \Big ) \; ds, \end{aligned}$$

for all $0 < h \le b-\tau$. From the uniform convergence obtained in Inequality (10), the first term tends to zero when $h \rightarrow 0^+$. Since $\tau$ is a Lebesgue point of the function $L(x,u,\cdot )$, it is clear that the second term tends to

$$\begin{aligned} \dfrac{(b-\tau )^{\beta -1}}{\varGamma (\beta )} \Big ( L(x(\tau ),v,\tau )-L(x(\tau ),u(\tau ),\tau ) \Big ), \end{aligned}$$

when $h \rightarrow 0^+$. From a Taylor expansion with integral rest, the last term can be decomposed as:

$$\begin{aligned}&\int _{\tau +h}^b \int _0^1 \dfrac{(b-s)^{\beta -1}}{\varGamma (\beta )} \partial _1 L (\star _\theta ) \cdot \left( \dfrac{x_h(s)-x(s)}{h} - \dfrac{(s-\tau )^{\alpha -1}}{\varGamma (\alpha )} (v-u(\tau )) \right) \; d\theta ds \\&\quad +\, \int _{\tau +h}^b \int _0^1 \dfrac{(b-s)^{\beta -1}}{\varGamma (\beta )} \dfrac{(s-\tau )^{\alpha -1}}{\varGamma (\alpha )} ( \partial _1 L (\star _\theta ) - \partial _1 L (\star _0) ) \cdot (v-u(\tau )) \; d\theta ds \\&\quad +\, \int _{\tau +h}^b \dfrac{(b-s)^{\beta -1}}{\varGamma (\beta )} \dfrac{(s-\tau )^{\alpha -1}}{\varGamma (\alpha )} \partial _1 L (\star _0) \cdot (v-u(\tau )) \; ds \end{aligned}$$

where

$$\begin{aligned} \star _\theta := ( x(s)+\theta (x_h(s)-x(s)) , u(s) , s), \end{aligned}$$

for all $\theta \in [0,1]$ and all $0 < h \le b-\tau$. The last above term clearly tends to

$$\begin{aligned} \mathrm {I}^\alpha _{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x,u,\cdot ) \right] (\tau ) \cdot (v-u(\tau )), \end{aligned}$$

when $h \rightarrow 0^+$. From the LDC theorem, the second above term clearly tends to zero when $h \rightarrow 0^+$. Finally one can prove from Inequality (11) that the norm of the first above term can be bounded by

$$\begin{aligned}&\dfrac{\Big ( b-(\tau +h) \Big )^{\alpha +\beta -1}}{\varGamma (\alpha +\beta )} \left\| \dfrac{1}{h} \int _\tau ^{\tau +h} u(s) \; ds - u(\tau ) \right\| _{{\mathbb {R}}^n} \\&\quad +\, \dfrac{2R}{\varGamma (\alpha +\beta )} \left( \Big ( b-(\tau +h) \Big )^{\alpha +\beta -1} - ( b-\tau )^{\alpha +\beta -1} \right) \\&\quad +\, 2R \int _\tau ^{\tau +h} \dfrac{(b-s)^{\beta -1}}{\varGamma (\beta )} \dfrac{(s-\tau )^{\alpha -1}}{\varGamma (\alpha )} \; ds, \end{aligned}$$

which tends to zero when $h \rightarrow 0^+$ (in particular since $\tau$ is a Lebesgue point of the function u). The proof is complete. $\square$

Proposition B.3

(Sensitivity analysis under perturbation of y) Let $R \ge 0$ and let $(u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$. It holds that

$$\begin{aligned}&\lim \limits _{h \rightarrow 0^+} \dfrac{\varPhi _R(u,y+hy')-\varPhi _R(u,y)}{h} \\&\quad = \left( \partial _1 \varphi (x(a),x(b)) + \partial _2 \varphi (x(a),x(b)) + \mathrm {I}^\beta _{a+} \Big [ \partial _1 L (x, u, \cdot ) \Big ](b) \right) \cdot y', \end{aligned}$$

for all $y' \in {\mathbb {R}}^n$, where $x = x(\cdot ,u,y) \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}$.

Proof

We apply Proposition 3.1 with the constant variation $\eta = y' \in {}_{\mathrm {c}} \mathrm {AC}^{\alpha ,\infty }_{a+}$. $\square$

1.3 B.3 Proof of Theorem 3.2 by Applying the Ekeland Variational Principle

Let $x \in \mathrm {K}$ be a solution to Problem (P). Using the notations introduced in Section B.2, it holds that $x = x(\cdot ,u,y)$ where $u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]$ and $y=x(a)$. In particular note that $g(y,x(b,u,y))=g(x(a),x(b)) \in \mathrm {S}$. Let $R := \Vert u \Vert _{\mathrm {L}^\infty } + 1 \ge 0$ and let us consider a positive sequence $(\varepsilon _k)_{k \in {\mathbb {N}}}$ which tends to zero when $k \rightarrow \infty$. We introduce the penalized functional

$$\begin{aligned}{}\begin{array}[t]{lrcl}\mathrm {J}_k :&{}\mathrm {L}^\infty _R \times {\mathbb {R}}^n &{}\longrightarrow &{}{\mathbb {R}}_+\\ {} &{}(u',y')&{} \longmapsto &{} \sqrt{ \Bigg ( \Big ( \varPhi _R (u',y') - \varPhi _R (u,y) + \varepsilon _k \Big )^+ \Bigg )^2 + \mathrm {d}^2_\mathrm {S}\Big ( g(y',x(b,u',y')) \Big ) } , \end{array} \end{aligned}$$

for all $k \in {\mathbb {N}}$. From Lemma B.3 and the continuities of g and $\mathrm {d}^2_\mathrm {S}$, it is clear that $J_k$ is a continuous nonnegative map defined on a complete metric space for all $k \in {\mathbb {N}}$ . Since $\mathrm {J}_k (u,y) = \varepsilon _k$ for all $k \in {\mathbb {N}}$, we deduce from the Ekeland variational principle (see Proposition B.1) that there exists a sequence $(u_k,y_k)_{k \in {\mathbb {N}}} \subset \mathrm {L}^\infty _R \times {\mathbb {R}}^n$ such that

$$\begin{aligned} \mathrm {d}_{\mathrm {L}^\infty _R \times {\mathbb {R}}^n} \Big ( (u_k,y_k),(u,y) \Big ) \le \sqrt{\varepsilon _k}, \end{aligned}$$

and

$$\begin{aligned} - \sqrt{\varepsilon _k} \; \mathrm {d}_{\mathrm {L}^\infty _R \times {\mathbb {R}}^n} \Big ( (u',y') , (u_k,y_k) \Big ) \le \mathrm {J}_k (u',y') - \mathrm {J}_k(u,y), \end{aligned}$$

(12)

for all $(u',y') \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$ and all $k \in {\mathbb {N}}$. In the sequel, we denote by $x_k := x(\cdot ,u_k,y_k)$ for all $k \in {\mathbb {N}}$. Note that the sequence $(u_k,y_k)_{k \in {\mathbb {N}}}$ converges to (u, y) in $\mathrm {L}^\infty _R \times {\mathbb {R}}^n$, and thus the sequence $(x_k)_{k \in {\mathbb {N}}}$ converges to x in $\mathrm {C}$ (see Lemma B.3).

From the optimality of x, one can easily see that $\mathrm {J}_k (u',y') > 0$ for all $(u',y') \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n$ and all $k \in {\mathbb {N}}$. As a consequence, we can correctly define

$$\begin{aligned} \psi ^0_k := - \dfrac{1}{\mathrm {J}_k (u_k,y_k)} \Big ( \varPhi _R (u_k,y_k) - \varPhi _R (u,y) + \varepsilon _k \Big )^+ \le 0, \end{aligned}$$

and

$$\begin{aligned} \psi _k := - \dfrac{1}{\mathrm {J}_k (u_k,y_k)} \Bigg ( g(x_k(a),x_k(b)) - \mathrm {P}_\mathrm {S}\Big ( g(x_k(a),x_k(b)) \Big ) \Bigg ) \in {\mathbb {R}}^j, \end{aligned}$$

which satisfy $\vert \psi ^0_k \vert ^2 + \Vert \psi _k \Vert _{{\mathbb {R}}^j}^2 = 1$ for all $k \in {\mathbb {N}}$. From a standard compactness argument and from the PCLDC theorem, we can extract subsequences (that we do not relabel) such that $(\psi ^0_k)_{k \in {\mathbb {N}}}$ converges to some $\psi ^0 \le 0$, $(\psi _k)_{k \in {\mathbb {N}}}$ converges to some $\psi \in {\mathbb {R}}^j$ satisfying $-\psi \in \mathrm {N}_\mathrm {S}[ g(x(a),x(b)) ]$ (see Lemma B.1) and $(u_k)_{k \in {\mathbb {N}}}$ converges to u pointwisely almost everywhere on [a, b]. Moreover, note that $\vert \psi ^0 \vert ^2 + \Vert \psi \Vert _{{\mathbb {R}}^j}^2 = 1$ and thus the couple $(\psi ^0,\psi )$ is not trivial.

Perturbation of $y_k$. Let $y' \in {\mathbb {R}}^n$ and let us fix some $k \in {\mathbb {N}}$. From Inequality (12), it holds that

$$\begin{aligned} - \sqrt{\varepsilon _k} \; \Vert y' \Vert _{{\mathbb {R}}^n} \le \dfrac{1}{\mathrm {J}_k (u_k,y_k+hy') + \mathrm {J}_k (u_k,y_k)} \times \dfrac{\mathrm {J}^2_k ( u_k,y_k+hy' ) - \mathrm {J}^2_k (u_k,y_k)}{h}, \end{aligned}$$

for all $h > 0$. Letting $h \rightarrow 0^+$, we exactly get from Proposition B.3 that

$$\begin{aligned}&\Bigg ( \psi ^0_k \left( \partial _1 \varphi (x_k(a),x_k(b)) + \partial _2 \varphi (x_k(a),x_k(b)) + \mathrm {I}^\beta _{a+} \Big [ \partial _1 L (x_k, u_k, \cdot ) \Big ](b) \right) \\&\quad +\, \Big ( \partial _1 g ( x_k(a) , x_k(b) )^\top + \partial _2 g ( x_k(a) , x_k(b) )^\top \Big ) \times \psi _k \Bigg ) \cdot y' \le \sqrt{\varepsilon _k} \Vert y' \Vert _{{\mathbb {R}}^n}. \end{aligned}$$

Finally, letting $k \rightarrow \infty$, we obtain (using in particular the LDC theorem) that

$$\begin{aligned}&\Bigg ( \psi ^0 \Big ( \partial _1 \varphi (x(a),x(b)) + \partial _2 \varphi (x(a),x(b)) + \mathrm {I}^\beta _{a+} \Big [ \partial _1 L (x, u, \cdot ) \Big ](b) \Big ) \\&\quad + \Big ( \partial _1 g ( x(a) , x(b) )^\top + \partial _2 g ( x(a) , x(b) )^\top \Big ) \times \psi \Bigg ) \cdot y' \le 0. \end{aligned}$$

Since the above inequality is satisfied for all $y' \in {\mathbb {R}}^n$ and we can write

$$\begin{aligned} \mathrm {I}^\beta _{a+} \Big [ \partial _1 L (x, u, \cdot ) \Big ](b) = \mathrm {I}^1_{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x,u,\cdot ) \right] (a), \end{aligned}$$

we deduce the crucial equality given by

$$\begin{aligned}&\Big ( \psi ^0 \partial _2 \varphi (x(a),x(b)) + \partial _2 g ( x(a) , x(b) )^\top \times \psi \Big ) + \psi ^0 \mathrm {I}^1_{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x,u,\cdot ) \right] (a) \nonumber \\&\quad = - \psi ^0 \partial _1 \varphi (x(a),x(b))- \partial _1 g ( x(a) , x(b) )^\top \times \psi . \end{aligned}$$

(13)

Needle-perturbation of $u_k$. Let $(\tau ,v) \in {\mathcal {P}}\times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$ where

$$\begin{aligned} {\mathcal {P}}:= \Big \lbrace \tau \in (a,b) \mid (u_k(\tau ))_{k \in {\mathbb {N}}} \text { converges to } u(\tau ) \Big \rbrace \cap \bigcap _{k \in {\mathbb {N}}} {\mathcal {P}}(u_k,y_k). \end{aligned}$$

Note that ${\mathcal {P}}$ is of full measure in [a, b]. Let us fix some $k \in {\mathbb {N}}$. From Inequality (12), it holds that

$$\begin{aligned} - 2R \sqrt{\varepsilon _k} \le \dfrac{1}{\mathrm {J}_k \Big ( u_k^{(\tau ,v)}(\cdot ,h),y_k \Big ) + \mathrm {J}_k (u_k,y_k)} \times \dfrac{\mathrm {J}^2_k \Big ( u_k^{(\tau ,v)}(\cdot ,h),y_k \Big ) - \mathrm {J}^2_k (u_k,y_k)}{h}, \end{aligned}$$

for all $0 < h \le b - \tau$. Letting $h \rightarrow 0^+$, we exactly get from Lemma B.4 and Proposition B.2 that

$$\begin{aligned}&\Bigg ( \dfrac{(b-\tau )^{\alpha -1}}{\varGamma (\alpha )} \Big ( \psi ^0_k \partial _2 \varphi (x_k(a),x_k(b)) + \partial _2 g ( x_k(a) , x_k(b) )^\top \times \psi _k \Big ) \\&\quad +\, \psi ^0_k \mathrm {I}^\alpha _{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x_k,u_k,\cdot ) \right] (\tau ) \Bigg ) \cdot (v-u_k(\tau )) \\&\quad +\, \psi ^0_k \dfrac{(b-\tau )^{\beta -1}}{\varGamma (\beta )} \Big ( L(x_k(\tau ),v,\tau )-L(x_k(\tau ),u_k(\tau ),\tau ) \Big ) \le 2R \sqrt{\varepsilon _k}. \end{aligned}$$

Finally, letting $k \rightarrow \infty$, we obtain (using in particular the LDC theorem and the fact that $(u_k(\tau ))_{k \in {\mathbb {N}}}$ converges to $u(\tau )$) that

$$\begin{aligned}&\Bigg ( \dfrac{(b-\tau )^{\alpha -1}}{\varGamma (\alpha )} \Big ( \psi ^0 \partial _2 \varphi (x(a),x(b)) + \partial _2 g ( x(a) , x(b) )^\top \times \psi \Big ) \nonumber \\&\quad +\, \psi ^0 \mathrm {I}^\alpha _{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x,u,\cdot ) \right] (\tau ) \Bigg ) \cdot (v-u(\tau )) \nonumber \\&\quad +\, \psi ^0 \dfrac{(b-\tau )^{\beta -1}}{\varGamma (\beta )} \Big ( L(x(\tau ),v,\tau )-L(x(\tau ),u(\tau ),\tau ) \Big ) \le 0. \end{aligned}$$

(14)

Note that the above crucial inequality is satisfied for almost all $\tau \in [a,b]$ and all $v \in {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)$.

Introduction of an adjoint vector. Let us introduce the adjoint vector p defined by

$$\begin{aligned}&p(t) := \dfrac{(b-t)^{\alpha -1} }{\varGamma (\alpha )} \Big ( \psi ^0 \partial _2 \varphi (x(a),x(b)) + \partial _2 g ( x(a) , x(b) )^\top \times \psi \Big ) \\&\quad +\, \psi ^0 \mathrm {I}^\alpha _{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x,u,\cdot ) \right] (t), \end{aligned}$$

for almost every $t \in [a,b]$. In particular it holds that

$$\begin{aligned}&\mathrm {I}^{1-\alpha }_{b-}[p](t) = \Big ( \psi ^0 \partial _2 \varphi (x(a),x(b)) + \partial _2 g ( x(a) , x(b) )^\top \times \psi \Big ) \\&\quad +\, \psi ^0 \mathrm {I}^1_{b-} \left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x,u,\cdot ) \right] (t), \end{aligned}$$

for all $t \in [a,b]$. We deduce that $p \in \mathrm {AC}^\alpha _{b-}$ with

$$\begin{aligned} \mathrm {D}^\alpha _{b-}[p] (t) = \psi ^0 \dfrac{(b-t)^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x(t),u(t),t), \end{aligned}$$

for almost every $t \in [a,b]$. Moreover, in particular from Equality (13), it holds that

$$\begin{aligned} - \mathrm {I}^{1-\alpha }_{b-}[p](a)= & {} \psi ^0 \partial _1 \varphi (x(a),x(b)) + \partial _1 g ( x(a) , x(b) )^\top \times \psi , \\ \mathrm {I}^{1-\alpha }_{b-}[p](b)= & {} \psi ^0 \partial _2 \varphi (x(a),x(b)) + \partial _2 g ( x(a) , x(b) )^\top \times \psi . \end{aligned}$$

From Inequality (14), it is clear that

$$\begin{aligned} u(t) \in \mathop {\mathrm{arg max}}\limits _{v \in {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R) } \Big \lbrace p(t) \cdot v +\psi ^0 \dfrac{(b-t)^{\beta -1}}{\varGamma (\beta )} L(x(t),v,t) \Big \rbrace , \end{aligned}$$

(15)

for almost every $t \in [a,b]$. One can easily deduce that

$$\begin{aligned} p(t) = - \psi ^0 \dfrac{ (b-t)^{\beta -1}}{\varGamma (\beta )} \partial _2 L (x(t),u(t),t), \end{aligned}$$

for almost every $t \in [a,b]$.

Normalization. By contradiction, let us assume that $\psi ^0 = 0$. In that case, one can easily deduce from the above equalities that $\partial _1 g(x(a),x(b))^\top \times \psi = \partial _2 g(x(a),x(b)) )^\top \times \psi = 0_{{\mathbb {R}}^n}$. Since g is assumed to be regular at (x(a), x(b)), we deduce that $\psi = 0_{{\mathbb {R}}^j}$ which raises a contradiction with the nontriviality of the couple $(\psi ^0,\psi )$. We deduce that $\psi ^0 < 0$. Moreover, since the couple $(\psi ^0,\psi )$ is defined up to a positive multiplicative constant, we now normalize the couple $(\psi ^0,\psi )$ such that $\psi ^0 = -1$.

End of the proof. We deduce from the previous paragraphs that $p = \frac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _2 L (x,u,\cdot ) \in \mathrm {AC}^\alpha _{b-}$ with

$$\begin{aligned} \mathrm {D}^\alpha _{b-}\left[ \dfrac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _2 L (x,u,\cdot ) \right] (t) = - \dfrac{(b-t)^{\beta -1}}{\varGamma (\beta )} \partial _1 L (x(t),u(t),t), \end{aligned}$$

for almost every $t \in [a,b]$, which exactly corresponds to the Euler–Lagrange equation stated in Theorem 3.2 since $u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]$. We also deduce from the equalities on $\mathrm {I}^{1-\alpha }_{b-} [p](a)$ and $\mathrm {I}^{1-\alpha }_{b-} [p](b)$ in a previous paragraph that the transversality conditions given in Theorem 3.2 are satisfied. Finally, from the maximization condition (15), it is clear that the matrix $\frac{(b-t)^{\beta -1}}{\varGamma (\beta )} \partial ^2_{22} L(x(t),u(t),t)$ is positive semi-definite for almost all $t \in [a,b]$, which exactly corresponds to the Legendre condition given in Theorem 3.2 since $u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]$.