Convergence of the solutions of the discounted Hamilton–Jacobi equation

Abstract

We consider a continuous coercive Hamiltonian H on the cotangent bundle of the compact connected manifold M which is convex in the momentum. If $u_\lambda :M\rightarrow \mathbb {R}$ is the viscosity solution of the discounted equation

$$\begin{aligned} \lambda u_\lambda (x)+H(x,\mathrm{d}_x u_\lambda )=c(H), \end{aligned}$$

where c(H) is the critical value, we prove that $u_\lambda$ converges uniformly, as $\lambda \rightarrow 0$, to a specific solution $u_0:M\rightarrow \mathbb {R}$ of the critical equation

$$\begin{aligned} H(x,\mathrm{d}_x u)=c(H). \end{aligned}$$

We characterize $u_0$ in terms of Peierls barrier and projected Mather measures. As a corollary, we infer that the ergodic approximation, as introduced by Lions, Papanicolaou and Varadhan in 1987 in their seminal paper on homogenization of Hamilton–Jacobi equations, selects a specific corrector in the limit.

Appendices

Appendix 1: Aubry–Mather Theory for non–smooth Lagrangians

In this appendix, we present the main results of weak KAM Theory we use. This material is well known in the case of a Tonelli Hamiltonian, see [4, 7, 11]. The lack of Hamiltonian and Lagrangian flows requires some different arguments, see [8–10, 14]. Although given specifically for the torus, the results of [8, 9, 14] can be easily rephrased in our setting and proved along the same lines.

1.1 Weak KAM Theory

Let H be a continuous Hamiltonian satisfying (H1)–($\hbox {H2}'$). We can associate with it a Lagrangian $L:TM\rightarrow \mathbb {R}$ through the Fenchel transform by setting

$$\begin{aligned} L(x,v):=\sup _{p\in T_x^*M} p(v) - H(x,p)\qquad \hbox {for every }(x,v)\in TM. \end{aligned}$$

(5.1)

The function L is continuous on TM and satisfies properties analogous to (H1) and ($\hbox {H2}'$), see Appendix 1 in [6]. In particular, $L(x,\cdot )$ is superlinear in $T_x M$ for every fixed $x\in M$. The following facts are well known, see [21, Theorem 23.5].

Proposition 5.1

Let H and L be as above. The following inequality, called Fenchel inequality, holds

$$\begin{aligned} L(x,v)+H(x,p)\ge p(v), \quad {\text { for every }}(x,v)\in TM{\text { and }}(x,p)\in T^*M,\nonumber \\ \end{aligned}$$

(5.2)

and

$$\begin{aligned} H(x,p)=\sup _{v\in T_x M}\left\{ p(v)-L(x,v)\,\right\} \qquad \hbox {for every }(x,p)\in T^*M. \end{aligned}$$

For every $t>0$, we define a function $h_t:M\times M\rightarrow \mathbb {R}$ by setting

$$\begin{aligned}&h_t(x,y)\\&\quad :=\inf \left\{ \int _{-t}^0 \big [ L(\gamma ,\dot{\gamma })+c(H)\big ]\,\mathrm{d}s\ \Big |\ \gamma \in \mathrm{AC}([-t,0];\,M),\,\gamma (-t)=x,\,\gamma (0)=y \right\} , \end{aligned}$$

where we have denoted by $\mathrm{AC}([-t,0];\,M)$ the family of absolutely continuous curves from $[-t,0]$ to M.

The following characterization holds, see [9, 11, 14]:

Proposition 5.2

Let $w\in \mathrm{C}(M)$. Then w is a critical subsolution if and only if

$$\begin{aligned} w(x)-w(y)\le h_t(y,x)\qquad \hbox {for every}\quad x,y\in M\hbox { and } t>0. \end{aligned}$$

The Peierls barrier is the function $h:M\times M\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} h(x,y):=\liminf _{t\rightarrow +\infty } h_t(x,y). \end{aligned}$$

(5.3)

It satisfies the following properties, see for instance [9]:

Proposition 5.3

• (a) The Peierls barrier h is finite valued and Lipschitz continuous.

• (b) If w is a critical subsolution, then

  $$\begin{aligned} w(x)-w(y)\le h(y,x)\quad {\text { for every }}x,y\in M. \end{aligned}$$

• (c) For every fixed $y\in M$, the function $h(y,\cdot )$ is a critical solution.

• (d) For every fixed $y\in M$, the function $-h(\cdot ,y)$ is a critical subsolution.

The projected Aubry set $\mathcal {A}$ is the closed set defined by

$$\begin{aligned} \mathcal {A}:=\{y\in M\,|\,h(y,y)=0\,\}. \end{aligned}$$

The following holds, see [10, 14]:

Theorem 5.4

There exists a critical subsolution w which is both strict and of class $\hbox {C}^1$ in $M\setminus \mathcal {A}$, i.e. satisfies

$$\begin{aligned} H(x,\mathrm{d}_x w)<c(H)\quad {\text { for every }}x\in M\setminus \mathcal {A}. \end{aligned}$$

In particular, the projected Aubry set $\mathcal {A}$ is nonempty.

The last assertion directly follows from the definition of c(H), see (2.4). A consequence of Theorem 5.4 is that $\mathcal {A}$ is a uniqueness set for the critical equation. In fact, we have, see [10, 14]:

Theorem 5.5

Let w, u be a critical sub and supersolution, respectively. If $w\le u$ on $\mathcal {A}$, then $w\le u$ on M. In particular, if two critical solutions coincide on the projected Aubry set $\mathcal {A}$, then they coincide on the whole manifold M.

1.2 Mather measures

Let X be a metric separable space. A probability measure on X is a nonnegative, countably additive set function $\mu$ defined on the $\sigma$–algebra $\mathscr {B}(X)$ of Borel subsets of X such that $\mu (X)=1$. In this paper, we deal with probability measures defined either on the compact manifold M or on its tangent bundle TM. A measure on TM is denoted by $\tilde{\mu }$, where the tilde on the top is to keep track of the fact that the measure is on the space TM. We say that a sequence $(\tilde{\mu }_n)_{n}$ of probability measures on TM (weakly) converges to a probability measure $\tilde{\mu }$ on TM if

$$\begin{aligned}&\lim _{n\rightarrow +\infty }\int _{TM} f(x,v)\,\mathrm{d}\tilde{\mu }_n(x,v)\nonumber \\&\quad =\int _{TM} f(x,v)\,\mathrm{d}\tilde{\mu }(x,v)\quad \hbox {for every }f\in \mathrm{C}_c(TM), \end{aligned}$$

(5.4)

where $\mathrm{C}_c(TM)$ denotes the family of continuous real functions with compact support on TM. If $\tilde{\mu }$ is a probability measure on TM, we denote by $\mu$ its projection $\pi _\#\tilde{\mu }$ on M, i.e. the probability measure on M defined as

$$\begin{aligned} \pi _{\#} \tilde{\mu }(B):=\tilde{\mu }\big (\pi ^{-1}(B)\big )\qquad \hbox {for every }B\in \mathscr {B}(M). \end{aligned}$$

Note that

$$\begin{aligned} \int _{M} f(x)\,\pi _{\#}\tilde{\mu }(x) = \int _{TM} \left( f\circ \pi \right) (x,v)\,\mathrm{d}\tilde{\mu }(x,v), \end{aligned}$$

for every $f\in \mathrm{C}(M)$.

Let H be a continuous Hamiltonian satisfying (H1)–($\hbox {H2}'$) and let us denote by L the associated Lagrangian. Mather theory states that the constant $-c(H)$, where c(H) is the critical value, can be also obtained by minimizing the integral of the Lagrangian over TM with respect to a suitable family of probability measures on TM. In the case of a Tonelli Hamiltonian, it is customary to choose this family as the one made up by probability measures on TM that are invariant by the Euler–Lagrange flow, see [20]. It was shown that this minimization problem yields the same result if it is done on the set of closed measures [1, 13, 14, 19] and the minimizing measures are the same. This is a set that does not depend on the Hamiltonian, and therefore this is the approach that can be adapted to the case when a Hamiltonian flow cannot be defined. The definition of closed measure is the following:

Definition 5.6

A probability measure $\tilde{\mu }$ on TM is called closed if it satisfies the following properties:

• (a)    $\displaystyle {\int _{TM} \Vert v\Vert _x\,\mathrm{d}\tilde{\mu }(x,v)<+\infty }$;

• (b)    $\displaystyle {\int _{TM} \mathrm{d}_x\varphi (v)\, \mathrm{d}\tilde{\mu } (x,v) =0}$   for every $\varphi \in \mathrm{C}^1(M)$.

A way to construct a closed measure is the following: if $\gamma : [a,b]\rightarrow M$ is an absolutely continuous curve, define the probability measure $\tilde{\mu }_\gamma$ on TM, by

$$\begin{aligned} \int _{TM}f(x,v) \,d\tilde{\mu }_\gamma (x,v):=\frac{1}{b-a}\int _a^b f\big (\gamma (t),\dot{\gamma }(t)\big )\,dt, \end{aligned}$$

(5.5)

for every $f\in \mathrm{C}_{c}(TM)$. It is easily seen that $\tilde{\mu }_\gamma$ is a closed measure whenever $\gamma$ is a loop.

The relation linking closed probability measures to the critical value is clarified by the next theorem.

Theorem 5.7

The following holds:

$$\begin{aligned} \min _{\tilde{\mu }} \int _{TM} L(x,v)\, d \tilde{\mu }(x,v)=-c(H) \end{aligned}$$

(5.6)

where $\tilde{\mu }$ varies in the set of closed measures and c(H) is the critical value for H.

Proof

Let us first show that the integral appearing at the right–hand side of (5.6) is greater than or equal to $-c(H)$. Using Theorem 2.2, we can construct a sequence of C$^1$ function $u_n: M\rightarrow \mathbb {R}$ such that $H(x,\mathrm{d}_x u_n)\le c(H)+1/n$ for every $x\in M$. For every $(x,v)\in TM$, by Fenchel’s inequality, we have

$$\begin{aligned} \mathrm{d}_x u_n(v)\le L(x,v)+H(x,\mathrm{d}_x u_n)\le L(x,v)+c(H)+\frac{1}{n}. \end{aligned}$$

By integrating this inequality with respect to a closed measure $\tilde{\mu }$, we get

$$\begin{aligned} \int _{TM}L(x,v)\, \mathrm{d}\tilde{\mu }(x,v)\ge -c(H)-\frac{1}{n}. \end{aligned}$$

The asserted inequality follows by letting $n\rightarrow +\infty$. Let us show that the infimum in (5.6) is achieved by a closed measure. Let $(\tilde{\mu }_n)_n$ a minimizing sequence of closed measure for the problem (5.6). According to Theorem 2-4.1-(3) in [7], up to extraction of a subsequence, there exists a probability measure $\tilde{\mu }$ on TM satisfying item (a) in Definition 5.6 and such that (5.4) holds for every $f\in \mathrm{C}(TM)$ enjoying the following property:

$$\begin{aligned} \sup _{(x,v)\in TM}\frac{|f(x,v)|}{1+\Vert v\Vert _x}<+\infty . \end{aligned}$$

This readily implies that $\tilde{\mu }$ is closed. Moreover, according to the same Theorem 2-4.1-(3) in [7], we know that $\int _{TM} L\,\mathrm{d}\tilde{\mu }\le -c(H)$, showing that $\tilde{\mu }$ is a solution of the minimization problem (5.6). $\square$

Note also that Proposition 3.6 provides some examples of minimizing measures.

Definition 5.8

A Mather measure for the Lagrangian L is a closed probability measure $\tilde{\mu }$ on TM such that $\int _{TM} L(x,v)\,d \tilde{\mu }(x,v)=-c(H)$. The set of Mather measures will be denoted by $\tilde{\mathfrak {M}} (L)$. A projected Mather measure is a Borel probability measure in $\mu$ on M of the form $\mu =\pi _\#\tilde{\mu }$, where $\tilde{\mu }\in \tilde{\mathfrak {M}} (L)$. The set of projected Mather measures is denoted by $\mathfrak {M}(L)$.

Sometimes the terminology Mather minimizing measure, rather than Mather measure, is used to emphasize that a Mather measure is solving the miminization problem (5.6).

Appendix 2: Representation formulae for the discounted equation

For every $\lambda >0$ and $x\in M$, we define the discounted value function as

$$\begin{aligned} \hat{u}_\lambda (x):=\inf _{\gamma (0)=x}\left\{ \int _{-\infty }^0 \mathrm{e}^{\lambda s}\big [ L\big (\gamma (s),\dot{\gamma }(s)\big )+c(H)\big ]\,\mathrm{d}s \,\Big |\,\gamma \in \mathrm{AC}\left( [-t,0];\,M\right) \,\right\} ,\nonumber \\ \end{aligned}$$

(6.1)

where we have denoted by $\mathrm{AC}\left( [-t,0];\,M\right)$ the family of absolutely continuous curves from $[-t,0]$ to M. The following holds:

Theorem 6.1

For every $\lambda >0$, the function $\hat{u}_\lambda$ given by (6.1) is the unique continuous viscosity solution of (2.6).

Therefore $\hat{u}_\lambda$ is equal to the function $u_\lambda$ used in Sect. 3. The uniqueness part in the above statement is a consequence of the Comparison Principle stated in Theorem 2.5. The fact that the discounted value function is a continuous viscosity solution of (2.6) is usually proved in Optimal Control Theory under the assumption that the speed of admissible curves is bounded by a constant independent of $x\in M$, see for instance [2, Chapter III] or [3, Chapter 3]. These bounds are not known a priori here, but are actually a consequence of the fact that the functions $\hat{u}_{\lambda }$ are equi–Lipschitz. Indeed, the following holds:

Proposition 6.2

Let $\lambda >0$ and $x\in M$. Then there exists a curve $\gamma ^\lambda _x:(-\infty ,0]\rightarrow M$ with $\gamma ^\lambda _x(0)=x$ such that

$$\begin{aligned} \hat{u}_\lambda (x)= & {} \mathrm{e}^{-\lambda t} \hat{u}_\lambda (\gamma ^\lambda _x(-t))+\int _{-t}^0 \mathrm{e}^{\lambda s} \big [L\big (\gamma ^\lambda _x(s),\dot{\gamma }^\lambda _x(s)\big )+c(H)\big ]\,\mathrm{d}s \end{aligned}$$

(6.2)

$\hbox {for every} \quad t>0.$ Moreover, there exists a constant $\alpha >0$, independent of $\lambda$ and x, such that $\Vert \dot{\gamma }^\lambda _x\Vert _\infty \le \alpha$. In particular

$$\begin{aligned} \hat{u}_\lambda (x)=\int _{-\infty }^0 \mathrm{e}^{\lambda s} \big [L\big (\gamma ^\lambda _x(s),\dot{\gamma }^\lambda _x(s)\big ) +c(H)\big ]\,\mathrm{d}s. \end{aligned}$$

(6.3)

A proof for Theorem 6.1 can be easily recovered from this by arguing, for instance, as in [2, Chapter III, Prop. 2.8], to which we refer for the details. The remainder of this appendix is therefore devoted to give a proof of Proposition 6.2, that we also need to prove Theorem 1.1.

We begin by deriving some crucial information for $\hat{u}_\lambda$.

Proposition 6.3

The function $\hat{u}_\lambda$ defined by (6.1) satisfies the following properties:

• (i) For every $\lambda >0$

  $$\begin{aligned} \frac{\min _{TM} L+c(H)}{\lambda } \le \hat{u}_\lambda (x) \le \frac{ L(x,0)+c(H)}{\lambda }\qquad \hbox {for every }x\in M. \end{aligned}$$

  In particular, $\Vert \lambda \hat{u}_\lambda \Vert _\infty \le C_0$ for some positive constant $C_0$ independent of $\lambda >0$.

• (ii) For every absolutely continuous curve $\gamma :[a,b]\rightarrow M$, we have

  $$\begin{aligned} \mathrm{e}^{\lambda b}\hat{u}_\lambda \big (\gamma (b)\big )-\mathrm{e}^{\lambda a}\hat{u}_\lambda \big (\gamma (a)\big ) \le \int _{a}^b \mathrm{e}^{\lambda s} \big [L\big (\gamma (s),\dot{\gamma }(s)\big )+c(H)\big ]\,\mathrm{d}s. \end{aligned}$$

  (6.4)

• (iii) There exists a positive constant $\kappa$, independent of $\lambda >0$, such that

  $$\begin{aligned} \hat{u}_\lambda (x)-\hat{u}_\lambda (y) \le \kappa d(x,y)\qquad \hbox {for every }x,y\in M\hbox { and }\lambda >0, \end{aligned}$$

  that is, the functions $\{\hat{u}_\lambda \,|\,\lambda >0\,\}$ are equi–Lipschitz.

Proof

In (i), the first inequality comes from the fact that every absolutely continuous curve $\gamma :(-\infty ,0]\rightarrow M$ satisfies

$$\begin{aligned} \int _{-\infty }^0 \mathrm{e}^{\lambda s} \big [L\big (\gamma (s),\dot{\gamma }(s)\big )+c(H)\big ]\,\mathrm{d}s&\ge \left( \min _{TM} L+c(H)\right) \int _{-\infty }^0 \mathrm{e}^{\lambda s}\,\mathrm{d}s\\&= \frac{\min _{TM} L+c(H)}{\lambda }. \end{aligned}$$

The second inequality follows by choosing, as a competitor, the steady curve identically equal to the point x.

To prove (ii), we first note that we can assume $b=0$, since we can always reduce to this case by replacing $\gamma$ with the curve $\gamma _{-b}(\cdot ):=\gamma (\cdot +b)$ defined on the interval $[a-b,0]$ and by dividing (6.4) by $\mathrm{e}^{\lambda \,b}$. Note that a change of variables gives

$$\begin{aligned} \int _{a}^b \mathrm{e}^{\lambda s} L\big (\gamma (s),\dot{\gamma }(s)\big )\,\mathrm{d}s =\mathrm{e}^{\lambda b} \int _{a-b}^0 \mathrm{e}^{\lambda s} L\big (\gamma _{-b}(s),\dot{\gamma }_{-b}(s)\big )\,\mathrm{d}s. \end{aligned}$$

So, let $\gamma \in \mathrm{AC}\left( [a,0];\,M\right)$ be fixed. For every absolutely continuous curve $\xi :(-\infty ,0]\rightarrow M$ with $\xi (0)=\gamma (a)$, we define a curve $\xi _a:(-\infty ,a]\rightarrow M$ by setting $\xi _a(\cdot ):=\xi (\cdot -a)$ and a curve $\eta :=\xi _a\star \gamma :(-\infty ,0]\rightarrow M$ obtained by concatenation of $\xi _a$ and $\gamma$. By definition of $\hat{u}_\lambda$ and arguing as above we get:

$$\begin{aligned} \hat{u}_\lambda \big (\gamma (0)\big )&\le \int _{-\infty }^0 \mathrm{e}^{\lambda s} \big [L(\eta ,\dot{\eta })+c(H)\big ]\,\mathrm{d}s \\&= \int _{-\infty }^a \mathrm{e}^{\lambda s} \big [L(\xi _a,\dot{\xi }_a)+c(H)\big ]\,\mathrm{d}s + \int _{a}^0 \mathrm{e}^{\lambda s} \big [L(\gamma ,\dot{\gamma })+c(H)\big ]\,\mathrm{d}s\\&= \mathrm{e}^{\lambda a} \int _{-\infty }^0 \mathrm{e}^{\lambda s}\big [ L(\xi ,\dot{\xi })+c(H)\big ]\,\mathrm{d}s + \int _{a}^0 \mathrm{e}^{\lambda s} \big [L(\gamma ,\dot{\gamma })+c(H)\big ]\,\mathrm{d}s. \end{aligned}$$

By minimizing with respect to all $\xi \in \mathrm{AC}\big ((-\infty ,0];\,M\big )$ with $\xi (0)=\gamma (a)$ we get the assertion by definition of $\hat{u}_\lambda \big (\gamma (a)\big )$.

To prove (iii), pick $x,\,y\in M$ and let $\gamma :[-d(x,y),0]\rightarrow M$ be the geodesic joining y to x parameterized by the arc–length. According to item (ii), we have

$$\begin{aligned}&\hat{u}_\lambda (x)-\hat{u}_\lambda (y) \le -\hat{u}_\lambda (y)\big (1-\mathrm{e}^{-\lambda d(x,y)}\big )\\&\quad +\int ^{0}_{-d(x,y)} \mathrm{e}^{\lambda s}\big [ L\big (\gamma (s),\dot{\gamma }(s)\big )+c(H)\big ]\,\mathrm{d}s. \end{aligned}$$

Let $C_1:=\max \left\{ L(z,v)\,:\,z\in M,\,\Vert v\Vert _z\le 1\,\right\}$ and $C_0$ the constant given by item (i). We get

$$\begin{aligned} \hat{u}_\lambda (x)-\hat{u}_\lambda (y)\le & {} \big (\Vert \lambda \hat{u}_\lambda \Vert _\infty +C_1+c(H)\big )\,\frac{1-\mathrm{e}^{-\lambda d(x,y)}}{\lambda }\\\le & {} \big (C_0+C_1+c(H)\big )d(x,y), \end{aligned}$$

where, for the last inequality, we have used the fact that, by concavity, $1-\mathrm{e}^{-h}\le h$ for every $h\in \mathbb {R}$. $\square$

In the sequel, we will use the following result:

Theorem 6.4

Let [a, b] be a compact interval in $\mathbb {R}$ and $\lambda >0$. Let $(\gamma _n)_n$ be a sequence in $\mathrm{AC}([a,b];\,M)$ such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\ \int _a^b \mathrm{e}^{\lambda s} L\big (\gamma _n(s),\dot{\gamma }_{n}(s)\big )\,\mathrm{d}s<+\infty . \end{aligned}$$

Then there exists a subsequence $\left( \gamma _{n_k}\right) _k$ uniformly converging to a curve $\gamma \in \mathrm{AC}([a,b];\,M)$. Moreover

$$\begin{aligned} \int _a^b \mathrm{e}^{\lambda s} L\big (\gamma (s),\dot{\gamma }(s)\big )\,\mathrm{d}s \le \liminf _{k\rightarrow +\infty }\int _a^b \mathrm{e}^{\lambda s} L\big (\gamma _{n_k}(s),\dot{\gamma }_{n_k}(s)\big )\,\mathrm{d}s. \end{aligned}$$

When M is contained in $\mathbb {R}^k$, the above theorem follows by making use of the Dunford–Pettis theorem, see [5, Theorems 2.11 and 2.12], and of standard semicontinuity results in the Calculus of Variations, see [5, Theorem 3.6]. To get the result on an abstract compact manifold, it suffices to show that we can always reduce to this case by localizing the argument and by reasoning in local charts, see for instance [11].

The following holds:

Proposition 6.5

Let $\lambda >0$. For every $x\in M$ and $t>0$

$$\begin{aligned} \hat{u}_\lambda (x) =\inf _{\gamma (0)=x}\left\{ \mathrm{e}^{-\lambda t}\,\hat{u}_\lambda \big (\gamma (-t)\big )+ \int _{-t}^0 \mathrm{e}^{\lambda s} \big [L(\gamma ,\dot{\gamma })+c(H)\big ]\,\mathrm{d}s\,\Big |\,\gamma \in \mathrm{AC}\left( [-t,0];\,M\right) \,\right\} .\nonumber \\ \end{aligned}$$

(6.5)

Moreover, the above infimum is attained.

Proof

The fact that the discounted value function satisfies the Dynamical Programming Principle (6.5) is standard, see for instance [2, Chapter III, Prop. 2.5]. To prove that the infimum is actually a minimum, we take minimizing sequence $\gamma _n:[-t,0]\rightarrow M$ with $\gamma _n(0)=x$, i.e. such that

$$\begin{aligned} \lim _{n\rightarrow +\infty } \mathrm{e}^{-\lambda t}\,\hat{u}_\lambda \big (\gamma _n(-t)\big ) + \int _{-t}^0 \mathrm{e}^{\lambda s} \big [L(\gamma _n,\dot{\gamma }_n)+c(H)\big ]\,\mathrm{d}s = \hat{u}_\lambda (x). \end{aligned}$$

For n large enough, we have:

$$\begin{aligned} \int _{-t}^0 \mathrm{e}^{\lambda s} \big [L(\gamma _n,\dot{\gamma }_n)+c(H)\big ] \mathrm{d}s&\le 1+\hat{u}_\lambda \big (\gamma _n(0)\big )\\&\quad - \mathrm{e}^{-\lambda t} \hat{u}_\lambda \big (\gamma _n(-t)\big ) \le 1+2\Vert \hat{u}_\lambda \Vert _\infty . \end{aligned}$$

According to Theorem 6.4, the curves $\gamma _n$ uniformly converge, up to subsequences, to an absolutely continuous curve $\gamma : [-t,0]\rightarrow M$ with $\gamma (0)=x$ and satisfying

$$\begin{aligned} \int _{-t}^0 \mathrm{e}^{\lambda s} \big [L(\gamma ,\dot{\gamma })+c(H)\big ] \mathrm{d}s\le \liminf _{n\rightarrow +\infty } \int _{-t}^0 \mathrm{e}^{\lambda s} \big [L(\gamma _n,\dot{\gamma }_n)+c(H)\big ] \mathrm{d}s. \end{aligned}$$

This readily implies that $\gamma$ is a minimizer of (6.5). $\square$

Proof of Proposition 6.2

According to Proposition 6.5 we know that, for every $n\in \mathbb {N}$, there exists a curve $\xi _n:[-n,0]\rightarrow M$ with $\xi _n(0)=x$ such that

$$\begin{aligned} \hat{u}_\lambda (x)=\mathrm{e}^{-\lambda n}\hat{u}_\lambda \big (\xi _n(-n)\big )+\int _{-n}^0 \mathrm{e}^{\lambda s}\big [ L\big (\xi _n(s),\dot{\xi }_n(s)\big )+c(H) \big ]\,\mathrm{d}s. \end{aligned}$$

It is then standard that, for every $[a,b]\subset [-n,0]$,

$$\begin{aligned} \mathrm{e}^{\lambda b}\hat{u}_\lambda \big (\xi _n (b)\big )-\mathrm{e}^{\lambda a}\hat{u}_\lambda \big (\xi _n (a)\big ) = \int _{a}^b \mathrm{e}^{\lambda s}\big [ L\big (\xi _n(s),\dot{\xi }_n(s)\big )+c(H) \big ]\,\mathrm{d}s.\qquad \quad \end{aligned}$$

(6.6)

By reasoning as in the proof of Proposition 6.5 and using a diagonal argument, we derive from Theorem 6.4 that there exists an absolutely continuous curve $\gamma _x^\lambda :(-\infty ,0]\rightarrow M$ with $\gamma _x^\lambda (0)=x$ which is, up to extraction of a subsequence, the uniform limit of the curves $\xi _n$ over compact subsets of $(-\infty ,0]$. Such curve satisfies

$$\begin{aligned} \mathrm{e}^{\lambda b}\hat{u}_\lambda \big (\gamma _x^\lambda (b)\big )-\mathrm{e}^{\lambda a}\hat{u}_\lambda \big (\gamma _x^\lambda (a)\big ) = \int _{a}^b \mathrm{e}^{\lambda s}\big [ L\big (\gamma _x^\lambda (s),\dot{\gamma }_x^\lambda (s)\big )+c(H) \big ] \,\mathrm{d}s.\nonumber \\ \end{aligned}$$

(6.7)

for every $[a,b]\subset (-\infty ,0]$. To see this, it suffices to pass to the limit in (6.6). The equality holds also for the limit curve $\gamma _x^\lambda$ by the lower semicontinuity of the integral functional stated in Theorem 6.4 and by Proposition 6.3–(ii). In particular, this proves assertion (6.2).

The fact that the curves $\gamma ^\lambda _x$ are equi–Lipschitz is a consequence of the fact that the functions $\hat{u}_\lambda$ are equi–Lipschitz, say $\kappa$–Lipschitz, according to Proposition 6.3. Indeed, by superlinearity of L, there exists a constant $A_\kappa$, depending on $\kappa$, such that

$$\begin{aligned} L(x,v)+c(H)\ge (\kappa +1)\Vert v\Vert _x-{A_\kappa }\qquad \hbox {for every } (x,v)\in TM. \end{aligned}$$

For every $a\in (-\infty ,0)$ and $h>0$ small enough, from (6.7) we get

$$\begin{aligned}&\mathrm{e}^{\lambda (a+h)}\hat{u}_\lambda \big (\gamma _x^\lambda (a+h)\big )-\mathrm{e}^{\lambda a}\hat{u}_\lambda \big (\gamma _x^\lambda (a)\big )\nonumber \\&\quad = \int _{a}^{a+h} \mathrm{e}^{\lambda s} \big [ L\big (\gamma _x^\lambda (s),\dot{\gamma }_x^\lambda (s)\big )+c(H) \big ] \,\mathrm{d}s\nonumber \\&\quad \ge \mathrm{e}^{\lambda a}\,(\kappa +1) \int _{a}^{a+h} \Vert \dot{\gamma }_x^\lambda (s)\Vert _{\gamma _x^\lambda (s)}\,\mathrm{d}s - {A_\kappa }\int _{a}^{a+h} \mathrm{e}^{\lambda s}\,\mathrm{d}s\nonumber \\&\quad \ge \mathrm{e}^{\lambda a}\,\left( \kappa \,d\big (\gamma _x^\lambda (a),\gamma _x^\lambda (a+h)\big ) + \int _{a}^{a+h} \Vert \dot{\gamma }_x^\lambda (s)\Vert _{\gamma _x^\lambda (s)}\,\mathrm{d}s - {A_\kappa }\,\frac{\mathrm{e}^{\lambda h}-1}{\lambda }\right) \end{aligned}$$

(6.8)

On the other hand

$$\begin{aligned}&\mathrm{e}^{\lambda (a+h)}\hat{u}_\lambda \big (\gamma _x^\lambda (a+h)\big )-\mathrm{e}^{\lambda a}\hat{u}_\lambda \big (\gamma _x^\lambda (a)\big )\nonumber \\&\quad \le (\mathrm{e}^{\lambda (a+h)}-\mathrm{e}^{\lambda a})\hat{u}_\lambda \big (\gamma _x^\lambda (a+h)\big ) + \mathrm{e}^{\lambda a}\,\kappa \,d\big (\gamma _x^\lambda (a),\gamma _x^\lambda (a+h)\big )\nonumber \\&\quad \le \mathrm{e}^{\lambda a}\,\left( C_0\,\frac{\mathrm{e}^{\lambda h}-1}{\lambda } + \kappa \,d\big (\gamma _x^\lambda (a),\gamma _x^\lambda (a+h)\big ) \right) , \end{aligned}$$

(6.9)

where $C_0$ is the constant given by Proposition 6.3–(i). Plugging (6.9) into (6.8) and dividing by $h\,\mathrm{e}^{\lambda a}$ we end up with

$$\begin{aligned} \frac{1}{h}\,\int _{a}^{a+h} \Vert \dot{\gamma }_x^\lambda (s)\Vert _{\gamma _x^\lambda (s)}\,\mathrm{d}s \le ({A_\kappa }+C_0)\,\frac{\mathrm{e}^{\lambda h}-1}{\lambda \,h}. \end{aligned}$$

Sending $h\rightarrow 0$ we infer

$$\begin{aligned} \Vert \dot{\gamma }_x^\lambda (a)\Vert _{\gamma _x^\lambda (a)}\le \alpha := ({A_\kappa }+C_0)\qquad \hbox {for a.e. }a\in (-\infty ,0], \end{aligned}$$

as it was to be shown. In particular, by sending $t\rightarrow +\infty$ in (6.2) we get (6.3) by the Dominated Convergence Theorem. $\square$