# Optimal filtering in closed manifolds- a deterministic perspective

Gaël Le Ruz, Philippe Moireau

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Gaël Le Ruz, Philippe Moireau. Optimal filtering in closed manifolds- a deterministic perspective. 2025. hal-05357602v2

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Distributed under a Creative Commons Attribution 4.0 International License
# OPTIMAL FILTERING IN CLOSED MANIFOLDS - A DETERMINISTIC PERSPECTIVE

GAEL LE RUZ<fnref n="1" /> AND PHILIPPE MOIREAU<fnref n="2" />

**Abstract.** In this paper, we adopt an optimal control viewpoint to formulate a rigorous deterministic filtering theory when the dynamics and the observations are defined on manifolds. Therefore, our result extends the Mortensen observer to closed manifolds, namely a compact manifold without boundary, in both continuous and discrete time, where the second ultimately yields a convergent time discretization of the first. The resulting observer requires the computation of the viscosity solution of a Hamilton-Jacobi-Bellman equation on the state manifold, which we illustrate on the sphere.

**2020 Mathematics Subject Classification.** 49J15, 49L20, 49L25, 49Q20.

...

## 1. INTRODUCTION

In this work, we aim at formulating a rigorous filtering approach for nonlinear dynamics defined on manifolds. In the last decade, it has become increasingly important to propose adapted filtering strategies on manifolds, since many dynamical systems involve a state space defined as a manifold. Examples are systems on Lie groups in aerospace engineering [4] or more general geodesically complete Riemannian manifolds in shape and pattern recognition [22]. Another configuration of growing interest is when the dynamics is defined by a model reduction procedure and belongs to an invariant manifold derived from data [21].

From an engineering perspective, the strategy of proposing nonlinear filtering methods for manifolds has been to heuristically adapt the different variants of the Kalman filter to be compatible with computation valid in manifolds. This has led to the recent proposals around the Extended Kalman Filter and the extension of the Unscented Kalman Filter to manifolds [10, 22, 31]. Even though the proposed algorithms are ultimately computationally effective, they are derived from the EKF or the UKF formulations in Euclidean spaces, which are already heuristic formulations from Bayesian filter theory or deterministic filter theory. Therefore, we believe that such approaches may lack a reference formulation with which they can be compared.

Here, we propose an exact formulation of the deterministic filtering problem in a closed manifold. Our work follows the path of the seminal work of [34], then popularized in [19] and recently revisited in [27, 32], where the filtering problem is defined as a minimum-energy estimator. Under suitable regularity assumptions, this leads to an observer –the so-called Mortensen observer – whose feedback dynamics are defined from a value function solution –using the dynamic programming principle– to a Hamilton-Jacobi-Bellman equation.

<footnote>Keywords and phrases: Mortensen observer, Hamilton-Jacobi-Bellman equation, Optimal control</footnote>

<footnote n="1">¹ LJLL, CNRS, Inria, Sorbonne Université, 75005 Paris ; e-mail: gael.le-ruz@inria.fr ; site web: www.lrzgael.github.io</footnote>

<footnote n="2">² CMAP, CNRS, Inria, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau ; e-mail: philippe.moireau@polytechnique.edu ; site web: www.cmap.polytechnique.fr/philippe.moireau</footnote>
In fact, our starting point is the recent reformulation of discrete-time dynamics adapted to engineering configurations in which measurements are always sampled in time [28]. Our resulting discrete-time estimator, with its typical forecast-analysis – or prediction-correction– steps [32], is then directly comparable to other filtering approaches that have been proposed in manifolds. Moreover, this will help us to link the discrete-time formulation with a Bayesian filtering approach that is easier to formulate in discrete-time than in continuous-time [24]. Here, we justify that the deterministic filter formulation can be viewed as a small noise asymptotic limit of the Bayesian formulation [23], where the value-function – up to a Hopf-Cole transform – is a reminiscence of the estimator distribution.

In a second stage, we will consider the continuous-time version when the measurement sampling period goes to 0. This will allow us to generalize the Mortensen observer initially introduced in [34] to the context of manifolds. Moreover, our procedure from the discrete-time to the continuous-time formulation will be proved to be convergent, hence justifying the final choice of numerical approximation to effectively compute the estimator. The resulting estimator remains heavy to compute and faces a curse of dimensionality, so we limit our numerical illustration to surface manifolds in 3D. However, we believe that our formulation can serve the community as a reference estimator when proposing a Gaussian filter approximation, for instance.

The paper is organized as follows. We introduce the dynamics considered in the manifolds of interest and formulate the estimation problem in section 2. In section 3, the discrete-time Mortensen estimator in manifolds is formulated in a typical two steps splitting scheme involving a forecast step where we follow the dynamics and an analysis step where we incorporate the data. There, we connect the Mortensen estimator to a Bayesian filtering approach through small-noise asymptotic analysis. The section 4 deals with the limit when the sampling period goes to 0, allowing us to recover a continuous-time formulation. This procedure will be proved to be convergent and paving the way to justified numerical methods for solving estimation problems in section 5 with numerical illustrations on the sphere.

## 2. PROBLEM SETTING

Let $\mathcal{X}$ be a complete, simply connected, closed – namely a compact manifold without boundary – smooth Riemannian manifold of finite dimension $d$, with positive injectivity radius. We consider an intrinsic distance $\text{dist}_{\mathcal{X}}$ and we denote $L_{\mathcal{X}} = \sup_{(x,z) \in \mathcal{X}^2} \text{dist}_{\mathcal{X}}(x, z)$. For all $x \in \mathcal{X}$, the tangent space of $x$ in $\mathcal{X}$ is denoted $\mathcal{T}_x\mathcal{X}$ and is equipped with the norm $\|\cdot\|_x$ that takes into account the metric tensor associated with the distance considered. The exponential map is defined for all $x \in \mathcal{X}$ and denoted $\text{Exp}_x(\nu) \in \mathcal{X}$ for $\nu \in T_x\mathcal{X}$. The logarithm map is the inverse map and denoted $\text{Log}_x(z) \in T_x\mathcal{X}$ for $z \in \mathcal{X}$. Finally, for all $(x, z) \in \mathcal{X}^2$, $\Pi_{xz}$ or $\Pi_x^z$ is the parallel transport operator along the geodesic taking any vector $v$ from $\mathcal{T}_x\mathcal{X}$ to $\mathcal{T}_z\mathcal{X}$.

We consider a time-dependent vector field $f: \mathbb{R}^+ \times \mathcal{X} \to T\mathcal{X}$ such that the physical system can be modeled by a dynamical system defined on $\mathcal{X}$ of the form

$$ \left\{ \begin{array}{l} \frac{dx}{dt} = f(t, x) + \nu(t, x), \quad t \ge 0 \\ x(0) = x_0 \end{array} \right. \qquad (1) $$

and, in the rest of this paper, the time derivation will be denoted by $\dot{x}$. Here, $f$ is assumed to be known whereas $\nu$ is considered as an unknown model perturbation. We assumed that $f$ is globally Lipschitz continuous via parallel transport with respect to its second variable in the sense of [17, Definition 2.1], namely there exists a constant $L > 0$ such that, for any $\mathcal{U}$ totally normal neighbourhood of $\mathcal{X}$, it holds

$$ \forall (t, x, z) \in \mathbb{R}_+^* \times \mathcal{U}^2, \quad \|f(t, x) - \Pi_{zx} f(t, z)\|_x \le L \text{dist}_{\mathcal{X}}(x, z), $$

and we note $\|f(t, \cdot)\|_{\text{Lip}(\mathcal{X})}$ the smallest such $L$ and $L_f = \sup_{t \ge 0} \|f(t, \cdot)\|_{\text{Lip}(\mathcal{X})}$. We also assume that $\nu$ is globally Lipschitz continuous and so is $f + \nu$. Therefore, from [17, Theorem 2.5], the Local Cauchy-Lipschitz Theorem implies the existence of a unique maximal flow map for the dynamics for any initial condition $x_0 \in \mathcal{X}$.
Moreover, the Escape Lemma (see [29], for instance), together with the compactness of $\mathcal{X}$, implies that this flow map must be global. Consequently, the solution exists and is unique on the maximal time interval $I = \mathbb{R}^+$. For all $(s, t) \in \mathbb{R}^+$ and $z \in \mathcal{X}$, we denote by $\phi_{f+\nu}(t, s, z)$ the flow-map associated with the solution to (1) at time $t$ given $x(s) = z$. We consider that the true system of interest is modeled by a trajectory $\tilde{x}$ corresponding to a particular instance of the initial condition $\tilde{x}(0) = \tilde{x}_0$, and a particular instance of the model disturbance $\tilde{\nu}$. To justify that $\tilde{\nu}$ is a perturbation, we consider that

$$ \sup_{t \in \mathbb{R}^+} \left[ \|\tilde{\nu}(t, \cdot)\|_{L^\infty(\mathcal{X})} + \|\tilde{\nu}(t, \cdot)\|_{\text{Lip}(\mathcal{X})} \right] \le \alpha, $$

where $\alpha \in \mathbb{R}^+$ will be a small parameter in $[0, 1)$.

We now assume that we have at our disposal some measurements or observations of such a trajectory of the physical system of interest. These measurements are sampled over the time window at times $(t_n)_{n \ge 0}$ and correspond to partial observations $(y^\delta(t_n))_{n \ge 0}$ of the state of the true trajectory at these times. These partial observations belong to an observation space $\mathcal{Y}$ that is considered as a complete, simply connected, smooth, and closed Riemannian manifold of finite dimension $m$, with positive injectivity radius. Its intrinsic distance is denoted disty. We introduce the observation map that models the measurement procedure $h: \mathbb{R}^+ \times \mathcal{X} \to \mathcal{Y}$, which is considered to be a Lipschitz continuous function with constant $L_h$, namely

$$ \forall (t, x, z) \in \mathbb{R}_+^* \times \mathcal{X}^2, \quad \text{dist}_y(h(t, x), h(t, z)) \le L_h \text{dist}_\mathcal{X}(x, z). $$

In practice, we have $y^\delta(t_n) \simeq h(t_n, \tilde{x}(t_n))$ as the measurement procedure is not error free. As we consider an observation manifold, we propose to model the measurement error as

$$ y^\delta(t_n) = \text{Exp}^\mathcal{Y}_{h(t_n, \tilde{x}(t_n))}(\eta_n). $$

where $(\eta_n)_n$ is a sequence of measurement unknown error in $\mathcal{T}\mathcal{X}$ with for all $n \in \mathbb{N}$, $\eta_n \in \mathcal{T}_{h(\tilde{x}(t_n))}\mathcal{X}$ and such that,

$$ \forall n \in \mathbb{N}, \quad \|\eta_n\|_{h(\tilde{x}(t_n))} = \text{dist}_y(y^\delta(t_n), h(t_n, \tilde{x}(t_n))) \le \delta. $$

with $\delta$ a small parameter in $[0, 1)$. To simplify, we may assume a constant time-step $\tau > 0$ such that for all $n \in \mathbb{N}$, $t_{n+1} - t_n = \tau$.

As our observations are sampled, we propose to first consider a discretized version of the dynamics (1) on which we are going to introduce the filtering problem at the discrete-time level. The continuous-time version of the problem will be recovered in a second step by considering that the time-step $\tau$ tends to 0. To this end, we propose a particular splitting time-scheme between the flow map of $f$ and the perturbation induced by the model disturbance $\nu$. As $f$ is known, we consider that, we have a time scheme approximating $\phi_f(t_{n+1}, t_n, x)$ denoted $\phi_n^\tau(x)$. The time-discretization of (1) is then defined from $x_0$ as

$$ x_{n+1} = \text{Exp}_{\phi_n^\tau(x_n)}(\tau\nu_{n+1}), \qquad (2) $$

where $(\nu_n)_{n \ge 1}$ is a sequence of elements of $\mathcal{T}\mathcal{X}$ defined sequentially with $\nu_n \in \mathcal{T}_{\phi_n^\tau(x_n)}\mathcal{X}$. Such discretization is a consistent discretization of (1), assuming that the sequence of disturbance is recursively defined by $\nu_{n+1} = \nu(t_{n+1}, \phi_n^\tau(x_n))$. We can easily show the following proposition.

**Proposition 2.1.** The sequence $(x_n)_{n \in \mathbb{N}}$ as a function of its input parameter $(x_0, (\nu_n)_{n \in \mathbb{N}})$ is Lipschitz continuous with respect to its initial condition and Lipschitz continuous via parallel transport with respect to its disturbance input, namely, there exists a constant $C^{st}$ such that

$$ \forall n \in \mathbb{N}, \quad \text{dist}(x_n, x'_n) \le C^{st} \left[ \sup_{1 \le k \le n} \|\nu'_k\|_{\phi_k^\tau(x_k)} \text{dist}(x_0, x'_0) + \sum_{k=1}^n \tau \|\nu_k - \Pi_{x'_k}^{x_k} \nu'_k\|_{x_k} \right]. $$
For the observed trajectory, the corresponding $(\tilde{\nu}_n)_{n \in \mathbb{N}}$ are small but unknown with

$$ \forall n \in \mathbb{N}, \quad \|\tilde{\nu}_{n+1}\|_{\tilde{x}(t_n)} = \frac{1}{\tau} \mathrm{dist}_{\mathcal{X}}(\tilde{x}(t_{n+1}), \phi(\tilde{x}(t_n))) \le \alpha. $$

So at the discrete-time level, the deterministic filtering problem consists of sequentially estimating the trajectory $(\tilde{x}_n)_{n \in \mathbb{N}}$ in $\mathcal{X}$ using the measurements at our disposal $(y^\delta(t_n))_{n \in \mathbb{N}}$ in $\mathcal{Y}$. When $\mathcal{X}$ and $\mathcal{Y}$ are Euclidean spaces, [27, 32] have introduced an energy-optimal sequential estimator at the discrete that generalizes the discrete-time Kalman filter with its prediction-correction sub-steps – also called forward-analysis steps or a priori-a posteriori steps in the data assimilation community [1, 30]. This estimator corresponds to a particular discretized version of the Mortensen estimator introduced by [34] using a splitting time-scheme combining the prediction and correction steps. Our objective in this work is to generalize such a Mortensen estimator for simply connected, closed, smooth Riemannian manifolds of finite dimension for the state and observation spaces.

# 3. DISCRETE-TIME FORMULATION

## 3.1. The Mortensen cost-to-come and estimator

To balance the various sources of uncertainties in our problem, we naturally introduce the following least-square functional, also called a disturbance energy functional

$$ \mathcal{J}_n(x_0, (\nu_k)_{k=1}^n, (\eta_k)_{k=1}^n) = \frac{1}{2} \mathcal{Y}_0(x_0) + \sum_{k=1}^n \left[ \frac{\tau}{2\alpha^2} \|\nu_k\|_{x_{k-1}}^2 + \frac{\tau}{2\delta^2} \|\eta_k\|_{h_{k-1}(x_{k-1})}^2 \right]. \quad (3) $$

with $h_k(x_k) = h(t_k, x_k)$, and under the discrete-time dynamics (2) constraint. Here, $\mathcal{Y}_0$ is a Lipschitz and positive function which is assumed to be $C^2(\mathcal{X})$. With $\mathcal{Y}_0$ assumed to have only one strict minimum, denoted $\hat{x}_0$. We typically penalize the initial condition for being away from $\hat{x}_0$.

Such functional can be parametrized only with respect to a trajectory sequence $(x_k)_{k=0}^n$ by reformulating

$$ \mathcal{J}_n((x_k)_{k=0}^n) = \frac{1}{2} \mathcal{Y}_0(x_0) + \sum_{k=1}^n \left[ \frac{1}{2\tau\alpha^2} \mathrm{dist}_{\mathcal{X}}(x_k, \phi_k^\tau(x_{k-1}))^2 + \frac{\tau}{2\delta^2} \mathrm{dist}_{\mathcal{Y}}(y_k^\delta, h_k(x_k))^2 \right]. \quad (4) $$

**Remark 3.1.** We remark that up to a redefinition of the Riemannian distances, typically by replacing $\mathrm{dist}_{\mathcal{X}}$ by $\mathrm{dist}_{\alpha, \mathcal{X}}(\cdot, \cdot) = \alpha^{-1}\mathrm{dist}_{\mathcal{X}}$ and $\mathrm{dist}_{\mathcal{Y}}$ by $\mathrm{dist}_{\delta, \mathcal{Y}}(\cdot, \cdot) = \delta^{-1}\mathrm{dist}_{\mathcal{Y}}$, we can consider that $\alpha = \delta = 1$ in the rest of the paper until the numerical illustrations.

In the spirit of [34], albeit at the discrete-time level here as in [27, 32], we associate to this disturbance energy functional a value function – referred to as cost-to-come in robust control theory, see for instance [5] –

$$ \mathcal{Y}_n(x) = \inf_{\substack{(x_k)_{k=0}^n \in \mathcal{X}^n \\ x_n=x}} \mathcal{J}_n((x_k)_{k=0}^n). $$

Note that the infimum is clearly a minimum here as $\mathcal{X}$ is compact and $\mathcal{J}_n$ is continuous.

**Proposition 3.2.** For all $n \ge 0$, the cost-to-come $\mathcal{Y}_n$ is a Lipschitz function.

*Proof.* Let us consider two points $x, x' \in \mathcal{X}$. Without loss of generality, we can assume that $\mathcal{Y}_n(x') \ge \mathcal{Y}_n(x)$. Let $(x_k)_{k=0}^n \in \mathrm{argmin} \mathcal{J}_n((x_k)_{k=0}^n, x)$. We have, for all $(x'_k)_{k=0}^n$ with $x'_n = x'$,

$$ \mathcal{Y}_n(x') - \mathcal{Y}_n(x) \le \mathcal{J}((x'_k)_{k=0}^n) - \mathcal{J}((x_k)_{k=0}^n). $$
Let us choose $x'_k = x_k$ and for all $0 \le k \le n-1$, then we have

$$ \mathcal{Y}_n(x') - \mathcal{Y}_n(x) \le \frac{\tau}{2} \left[ \operatorname{dist}_Y(y_n^\delta, h_n(x'))^2 - \operatorname{dist}_Y(y_n^\delta, h_n(x))^2 \right] \\ + \frac{1}{2\tau} \left[ \operatorname{dist}_X(x', \phi_n^\tau(x_{n-1}))^2 - \operatorname{dist}_X(x, \phi_n^\tau(x_{n-1}))^2 \right]. $$

Then factorizing the square, bounding the distances and using the triangular inequality, we get

$$ \begin{aligned} \mathcal{Y}_n(x') - \mathcal{Y}_n(x) &\le \tau L_Y \left[ \operatorname{dist}_Y(y_n^\delta, h_n(x')) - \operatorname{dist}_Y(y_n^\delta, h_n(x)) \right] \\ &\quad + \frac{L_X}{\tau} \left[ \operatorname{dist}_X(x', \phi_n^\tau(x_{n-1})) - \operatorname{dist}_X(x, \phi_n^\tau(x_{n-1})) \right] \\ &\le \tau L_Y \operatorname{dist}_Y(h_n(x'), h_n(x)) + \frac{L_X}{\tau} \operatorname{dist}_X(x', x) \\ &\le \left[ \tau L_Y L_{h_n} + \frac{L_X}{\tau} \right] \operatorname{dist}_X(x', x). \end{aligned} $$

Therefore $\mathcal{Y}_n$ is a Lipschitz function. $\square$

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As the cost-to-come is continuous on the compact manifold $\mathcal{X}$, it has a minimum and this minimum is attained. Hence, we can define the discrete-time Mortensen estimator of the trajectory state $\hat{x}_n \in \mathcal{X}$ as the state $\hat{x}_n \in \mathcal{X}$ associated with the minimum energy up to time $n$, so that

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126

$$ \hat{x}_n \in \underset{x \in \mathcal{X}}{\operatorname{argmin}} \mathcal{Y}_n(x). \qquad (5) $$

**Remark 3.3.** In this work, when considering the state estimator, we are going to assume that for all $n \in \mathcal{N}$ (or at least up to $\mathcal{N} \in \mathbb{N}$) there exists only one global minimum for the value function, typically enforced by an appropriate choice of the initial value function $\mathcal{Y}_0$ and small enough parameters $\alpha$ and $\delta$.

127

128

129

## 3.2. Dynamic programming

130

Our objective is now to derive a recursive dynamics for the cost-to-come $\mathcal{Y}_n$ and the estimator $\hat{x}_n$. In this respect, we first prove a Bellman principle.

131

132

**Theorem 3.4 (Bellman principle).** For all $n \in \mathbb{N}^*$ and for all $x \in \mathcal{X}$, we have

133

$$ \mathcal{Y}_n(x) = \min_{z \in \mathcal{X}} \left\{ \mathcal{Y}_{n-1}(z) + \frac{1}{2\tau} \operatorname{dist}_{\mathcal{X}}(x, \phi_{n-1}^{\tau}(z))^2 \right\} + \frac{\tau}{2} \operatorname{dist}_Y(y^{\delta}, h_n(x))^2. \qquad (6) $$

*Proof.* We easily justify that

$$ \begin{aligned} \mathcal{Y}_n(x) &= \min_{(x_k)_{k=0}^{n-1} \in \mathcal{X}^{n-1}} \mathcal{J}_n(((x_k)_{k=0}^{n-1}, x)) \\ &= \min_{(x_k)_{k=0}^{n-1} \in \mathcal{X}^{n-1}} \left\{ \mathcal{J}_{n-1}((x_k)_{k=0}^{n-1}) + \frac{1}{2\tau} \operatorname{dist}_{\mathcal{X}}(x, \phi_{n-1}^{\tau}(x_{n-1}))^2 \right\} + \frac{\tau}{2} \operatorname{dist}_Y(y_n^{\delta}, h_n(x))^2. \end{aligned} $$

Therefore, we get directly the desired result for $n = 1$ and for $n \ge 2$, by minimizing separately with respect to $x_{n-1}$, we have

$$ \mathcal{Y}_n(x) = \min_{x_{n-1} \in \mathcal{X}} \left\{ \min_{(x_k)_{k=0}^{n-2} \in \mathcal{X}^{n-2}} \left\{ \mathcal{J}_{n-1}((x_k)_{k=0}^{n-1}) \right\} + \frac{1}{2\tau} \operatorname{dist}_{\mathcal{X}}(x, \phi_{n-1}^{\tau}(x_{n-1}))^2 \right\} + \frac{\tau}{2} \operatorname{dist}_Y(y_n^{\delta}, h_n(x))^2, $$

134

which concludes the proof. $\square$

135
Let us now introduce the predicted cost-to-come

$$ \mathcal{Y}_{n-}(x) := \min_{z \in \mathcal{X}} \left\{ \mathcal{Y}_{n-1}(z) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(x, \phi_{n-1}^{\tau}(z))^2 \right\}. \quad (7) $$

We can easily show that $\mathcal{Y}_{n-}$ is also Lipschitz and can be characterized by

$$ \mathcal{Y}_{n-}(x) = \inf_{\substack{(x_k)_k=0 \\ \hat{x}_n=x}} \mathcal{J}_{n-}((x_k)_{k=0}^n), $$

where

$$ \mathcal{J}_{n-}((x_k)_{k=0}^n) = \frac{1}{2} \mathcal{Y}_0(x_0) + \sum_{k=1}^n \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(x_k, \phi_k^{\tau}(x_{k-1}))^2 + \sum_{k=1}^{n-1} \frac{\tau}{2} \mathrm{dist}_{\mathcal{Y}}(y_k^{\delta}, h_k(x_k))^2. $$

Moreover as we assume that $\phi_{n-1}^{\tau}$ is invertible, we can rewrite

$$ \mathcal{Y}_{n-}(x) = \min_{z \in \mathcal{X}} \left\{ \mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(z) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(x, z)^2 \right\}, \quad (8) $$

which is the proximal of $\mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}$ in the manifold $\mathcal{X}$ equipped with the distance $\mathrm{dist}_{\mathcal{X}}$. As $\mathcal{X}$ is compact and $\mathcal{Y}_{n-1}$ is continuous, the inf-convolution is well-defined and continuous. We then introduce

$$ \hat{x}_{n-} \in \underset{x \in \mathcal{X}}{\operatorname{argmin}} \mathcal{Y}_{n-}(x), \quad (9) $$

which existence and uniqueness is given by the following proposition.

**Proposition 3.5.** Let $n \in \mathbb{N}^*$ and assume that $\mathcal{Y}_{n-1}$ has a unique minimizer $\hat{x}_{n-1}$. Then, $\mathcal{Y}_{n-}$ has a unique minimizer $\hat{x}_{n-}$ given by

$$ \hat{x}_{n-} = \phi_{n-1}^{\tau}(\hat{x}_{n-1}). \quad (10) $$

*Proof.* This is a classical property of the inf-convolution. We recall that

$$ \forall (x, z) \in \mathcal{X}^2, \quad \mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(z) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(x, z)^2 \ge \mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(z) \ge \mathcal{Y}_{n-1}(\hat{x}_{n-1}), $$

so that, by taking the infimum with respect to $z$, we still have $\mathcal{Y}_{n-}(x) \ge \mathcal{Y}_{n-1}(\hat{x}_{n-1})$. Moreover, with $z = \hat{x}_{n-} = \phi_{n-1}^{\tau}(\hat{x}_{n-1})$ in the inf-convolution that defines $\mathcal{Y}_{n-}(\hat{x}_{n-})$, we have

$$ \mathcal{Y}_{n-}(\hat{x}_{n-}) \le \mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(\hat{x}_{n-}) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(\hat{x}_{n-}, \hat{x}_{n-})^2 = \mathcal{Y}_{n-1}(\hat{x}_{n-}). $$

Conversely, Let us consider $\tilde{x}$ such that $\mathcal{Y}_{n-}(\tilde{x}) = \mathcal{Y}_{n-}(\hat{x}_{n-})$. Then, as the infimum is attained, there exists $\tilde{z}$ such that

$$ \mathcal{Y}_{n-}(\tilde{x}) = \mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(\tilde{z}) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(\tilde{x}, \tilde{z})^2. $$

As $\mathcal{Y}_{n-}(\hat{x}_{n-}) = \mathcal{Y}_{n-1}(\hat{x}_{n-1})$, we get that

$$ \mathcal{Y}_{n-1}(\hat{x}_{n-1}) = \mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(\tilde{z}) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(\tilde{x}, \tilde{z})^2 \ge \mathcal{Y}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(\tilde{z}). $$

By uniqueness of the minimum of $\mathcal{Y}_{n-1}$, we have that $\tilde{z} = \phi_{n-1}^{\tau}(\hat{x}_{n-1}) = \hat{x}_{n-}$. Therefore,

$$ \mathcal{Y}_{n-}(\tilde{x}) = \mathcal{Y}_{n-}(\hat{x}_{n-}) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(\tilde{x}, \hat{x}_{n-})^2, $$

so that $\mathrm{dist}_{\mathcal{X}}(\tilde{x}, \hat{x}_{n-}) = 0$ and we get $\tilde{x} = \hat{x}_{n-}$. $\square$
Therefore, we can summarize the discrete-time Mortensen estimator in manifolds as the prediction-correction sequence defined by:

• Initialization Step:

$$ \hat{x}_0 = \underset{x \in \mathcal{X}}{\operatorname{argmin}} \mathcal{V}_0(x). \qquad (11a) $$

• Then for all $n \ge 1$:

– Prediction Step:

$$ \left\{ \begin{array}{l} \mathcal{V}_{n^{-}}(x) = \min_{z \in \mathcal{X}} \left\{ \mathcal{V}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(z) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(x, z)^2 \right\}, \\ \hat{x}_{n^{-}} = \phi_{n-1}^{\tau}(\hat{x}_{n-1}). \end{array} \right. \qquad (11b) $$

– Correction Step:

$$ \left\{ \begin{array}{l} \mathcal{V}_n(x) = \mathcal{V}_{n^{-}}(x) + \frac{\tau}{2} \mathrm{dist}_Y(y_n^\delta, h_n(x))^2, \\ \hat{x}_n = \underset{x \in \mathcal{X}}{\operatorname{argmin}} \mathcal{V}_n(x). \end{array} \right. \qquad (11c) $$

### 3.3. Hopf-Lax formula

One central tool to understand the dynamics of the cost-to-come is the Hopf-Lax formula, which links the inf-convolution (7) to a Hamilton-Jacobi-Bellman equation.

**Proposition 3.6.** The predicted cost-to-come $\mathcal{V}_{n^{-}}(x) = v(\tau, x)$, where $v : [0, \tau] \times \mathcal{X} \to \mathbb{R}$ is the unique viscosity solution to the HJB equation

$$ \left\{ \begin{array}{l} \partial_t v(s, x) + \frac{1}{2} \|\mathrm{grad}\,v(s, x)\|_x^2 = 0, \quad (s, x) \in (0, \tau) \times \mathcal{X}, \\ v(0, x) = \mathcal{V}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(x), \quad x \in \mathcal{X}. \end{array} \right. \qquad (12) $$

The proof of Proposition 3.6 can be found in appendix B. Moreover, we know that $v$ defined in Proposition 3.6 is also the vanishing viscosity limit of $v^\epsilon \in C^2((0, \tau]; \mathcal{X})$ solution to

$$ \left\{ \begin{array}{l} \partial_t v^\epsilon(s, x) + \frac{1}{2} \|\mathrm{grad}\,v^\epsilon(s, x)\|_x^2 = \frac{\epsilon}{2} \Delta v^\epsilon(s, x), \quad (s, x) \in (0, \tau) \times \mathcal{X}, \\ v^\epsilon(0, x) = \mathcal{V}_{n-1} \circ (\phi_{n-1}^\tau)^{-1}(x), \quad x \in \mathcal{X}. \end{array} \right. \qquad (13) $$

where $\Delta$ stands for the Laplace-Beltrami operator on $\mathcal{X}$. In fact from appendix D, there exists a constant $C^{\mathrm{st}}$ such that

$$ \forall \epsilon > 0, \quad \sup_{x \in \mathcal{X}} |v^\epsilon(\tau, x) - \mathcal{V}_{n^{-}}(x)| \le C^{\mathrm{st}} \sqrt{\epsilon}. \qquad (14) $$

Finally, we recall that $v^\epsilon$ can be computed by relying on the Hopf-Cole transformation

$$ p^{\epsilon}(s, x) = \exp\left(-\frac{v^{\epsilon}(s, x)}{\epsilon}\right), $$

so that $p^\epsilon \in C^2((0, \tau]; \mathcal{X})$ is the solution to

$$ \left\{ \begin{array}{l} \partial_t p^{\epsilon}(s, x) = \frac{\epsilon}{2} \Delta p^{\epsilon}(s, x), \quad (s, x) \in (0, \tau) \times \mathcal{X}, \\ p^{\epsilon}(0, x) = \exp\left(-\frac{\mathcal{V}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(x)}{\epsilon}\right), \quad x \in \mathcal{X}. \end{array} \right. \qquad (15) $$
These results are of central importance for linking the discrete-time Mortensen estimator to its stochastic counterpart in the next section. They will also help us to define its continuous counterpart when the observation sampling step $\tau$ tends to 0, or finally to define a convergent spatial discretization.

### 3.4. From discrete-time stochastic filtering to deterministic filtering

The deterministic point of view for defining observation errors and model disturbances may be considered too restrictive when compared to the stochastic formalism. However, this deterministic formalism can be considered as an asymptotic formalism of the stochastic formalism [23], and fortunately, we are able to extend this asymptotic limit in the context of filtering for manifolds, at least at the discrete-time level.

Let us then start by recalling that a Brownian motion $w_t^X$ on the manifold $\mathcal{X}$ has a transition density $k_t(x, z)$ given by the heat kernel on the compact manifold $\mathcal{X}$. The transition probability is therefore given by

$$ \mathbb{P}(w_t^X \in dx | w_0^X = z) = k_t^X(z, x) dx, $$

and $k_t^X$ is the heat kernel satisfying for all $x \in \mathcal{X}$,

$$ \left\{ \begin{array}{ll} \partial_t k_t^X(z, x) - \frac{1}{2} \Delta k_t^X(z, x) = 0, & z \in \mathcal{X}, t \ge 0, \\ \lim_{t \to 0} k_t^X(z, x) = \delta_x(z), & z \in \mathcal{X}. \end{array} \right. \qquad (16) $$

where $\Delta$ stands for the Laplace-Beltrami operator on $\mathcal{X}$.

Let us then consider a Markov chain $(\mathbf{x}_n^\varepsilon)_{n \in \mathbb{N}}$ defined by

$$ \mathbb{P}(\mathbf{x}_{n+1}^\varepsilon | \mathbf{x}_n^\varepsilon \cdots \mathbf{x}_0^\varepsilon) = \mathbb{P}(\mathbf{x}_{n+1}^\varepsilon | \mathbf{x}_n^\varepsilon), $$

and of transition kernel defined by

$$ \mathbb{P}(\mathbf{x}_{n+1}^\varepsilon \in dx | \mathbf{x}_n^\varepsilon = z) = k_{\varepsilon\tau}^\chi(\phi_n^\tau(z), x) dx, \qquad (17) $$

starting from the initial distribution

$$ \mathbb{P}(\mathbf{x}_0^\varepsilon \in dx) \propto \exp \left( -\frac{\gamma_0(x)}{\varepsilon} \right) dx. \qquad (18) $$

The Markov-chain $(\mathbf{x}_n^\varepsilon)_{n \in \mathbb{N}}$ is partially observed, and these observations are given as a sequence of random variable $(\mathbf{y}_n^\varepsilon)_{n \in \mathbb{N}}$ with value in the observation manifold $\mathcal{Y}$ such that

$$ \mathbb{P}(\mathbf{y}_n^\varepsilon \in dy | \mathbf{x}_n^\varepsilon = x) = k_\tau^\chi(h_n(x), y) dy, $$

where $k_t^\gamma$ is the heat kernel on $\mathcal{Y}$.

In stochastic filtering the objective is then to compute the conditional probabilities of $\mathbf{x}_n^\varepsilon$ knowing $\mathbf{y}_{1:n} = (\mathbf{y}_k)_{k=1}^n$, namely $\mathbb{P}(\mathbf{x}_n^\varepsilon \in dx | \mathbf{y}_{1:n}^\varepsilon = y_{1:n}^\delta)$, and potentially the conditional probabilities of predicting $\mathbf{x}_{n+1}^\varepsilon$ knowing $\mathbf{y}_{1:n} = (\mathbf{y}_k)_{k=1}^n$, namely $\mathbb{P}(\mathbf{x}_{n+1}^\varepsilon \in dx | \mathbf{y}_{1:n}^\varepsilon = y_{1:n}^\delta)$.

Let us now recursively define the following unnormalized densities :

* Initialization Step:

$$ \pi_0^\varepsilon(x) = \exp \left( -\frac{\gamma_0(x)}{\varepsilon} \right), \qquad (19a) $$

* Then for all $n \ge 1$:

- Prediction Step:

$$ \pi_{n-}^\varepsilon(x) = \int_{\mathcal{X}} \pi_{n-1}^\varepsilon(z) k_{\varepsilon\tau}^\chi(\phi_n^\tau(z), x) dz, \qquad (19b) $$
– Correction Step:

$$ \pi_n^\epsilon(x) = \pi_{n-}^\epsilon(x) k_{\frac{\epsilon}{\tau}}^\mathcal{Y}(h_n(x), y_n^\delta). \quad (19c) $$

From the traditional Bayes formulas, we easily verify the following proposition.

**Proposition 3.7.** We have

$$ \mathbb{P}(\mathbf{x}_0^\epsilon \in dx) = \frac{\pi_0^\epsilon(x) dx}{\int_X \pi_0^\epsilon(x) dx}, \quad (20) $$

and then for all $n \in \mathbb{N}^*$,

$$ \mathbb{P}(\mathbf{x}_n^\epsilon \in dx | \mathbf{y}_{1:n-1}^\delta = \mathbf{y}_{1:n-1}^\delta) = \frac{\pi_{n-}^\epsilon(x) dx}{\int_X \pi_{n-}^\epsilon(x) dx}, \quad (21) $$

while

$$ \mathbb{P}(\mathbf{x}_n^\epsilon \in dx | \mathbf{y}_{1:n}^\epsilon = \mathbf{y}_{1:n}^\delta) = \frac{\pi_n^\epsilon(x) dx}{\int_X \pi_n^\epsilon(x) dx}. \quad (22) $$

*Proof.* We proceed by induction. The result is true for $\mathbb{P}(\mathbf{x}_0^\epsilon \in dx)$. Then at iteration $n$ this is a direct application of Bayes theorem, we have – see for instance [24, p. 174] –

$$ \mathbb{P}(\mathbf{x}_n^\epsilon \in dx | \mathbf{y}_{1:n}^\epsilon \in dy_1 \times \cdots \times dy_n) = \frac{\mathbb{P}(\mathbf{y}_n \in dy_n | \mathbf{x}_n^\epsilon \in dx) \mathbb{P}(\mathbf{x}_n^\epsilon \in dx | \mathbf{y}_{1:n-1}^\epsilon \in dy_1 \times \cdots \times dy_{n-1})}{\mathbb{P}(\mathbf{y}_n \in dy_n | \mathbf{y}_{1:n-1}^\epsilon \in dy_1 \times \cdots \times dy_{n-1})}, $$

and

$$ \mathbb{P}(\mathbf{x}_{n+1}^\epsilon \in dx | \mathbf{y}_{1:n}^\epsilon = y_{1:n}) = \int_X \mathbb{P}(\mathbf{x}_{n+1}^\epsilon \in dx | \mathbf{x}_n^\epsilon \in dz) \mathbb{P}(\mathbf{x}_n^\epsilon \in dz | \mathbf{y}_{1:n}^\epsilon = y_{1:n}^\delta). $$

<figure><img src="image_2.png" /></figure>

Therefore, we can justify that our discrete-time Mortensen estimator is the asymptotic limit of the Minimum Mean Squared Error (MMSE) Estimator of our Hidden Markov Chain in the manifold $\mathcal{X}$ with observations in the manifold $\mathcal{Y}$.

**Theorem 3.8.** For all $n \in \mathbb{N}$, we have

$$ \lim_{\epsilon \to 0} -\epsilon \log \pi_{n-}^{\epsilon}(x) = \mathcal{Y}_{n-}(x) \quad \text{and} \quad \lim_{\epsilon \to 0} -\epsilon \log \pi_{n}^{\epsilon}(x) = \mathcal{Y}_{n}(x). \quad (23) $$

*Proof.* We proceed by induction. At $n = 0$, (23) is satisfied for $\pi_{0-}$ by construction. Then, using (19c), we get that

$$ -\epsilon \log \pi_n^\epsilon(x) = -\epsilon \log \pi_{n-}^\epsilon(x) - \epsilon \log k_{\frac{\epsilon}{\tau}}^\mathcal{Y}(y_n^\delta, h_n(x)) $$

from Varadhan's formula in Manifold [33], we have

$$ \lim_{\epsilon \to 0} -2\epsilon \log \left( k_{\frac{\epsilon}{\tau}}^{\mathcal{Y}}(y_n^{\delta}, h_n(x)) \right) = \tau \lim_{\epsilon \to 0} -2\frac{\epsilon}{\tau} \log \left( k_{\frac{\epsilon}{\tau}}^{\mathcal{Y}}(y_n^{\delta}, h_n(x)) \right) = \tau \mathrm{dist}(y_n^{\delta}, h_n(x))^2. \quad (24) $$

so that we get

$$ \lim_{\epsilon \to 0} -\epsilon \log \pi_n^\epsilon(x) = \mathcal{Y}_{n-}(x) + \frac{\tau}{2} \mathrm{dist}(z, x)^2. $$

If we now consider (19b) and proceed to the change of variable $\tilde{z} = \phi_n^\tau(z)$, we obtain

$$ \pi_{n+1-}^{\epsilon}(x) = \int_{\mathcal{X}} \pi_{n}^{\epsilon}((\phi_{n}^{\tau})^{-1}(\tilde{z})) k_{\epsilon\tau}^{\mathcal{X}}(x, \tilde{z}) |\det(D\phi_{n}^{\tau}((\phi_{n}^{\tau})^{-1}(\tilde{z})))|^{-1} d\tilde{z}. $$
210 As we recognized the heat equation kernel solution to (16) in the previous expression, we see that $\pi_{n+1-}^{\epsilon}(x) =$

211 $\pi^{\epsilon}(\tau, x)$ where $\pi^{\epsilon}(s, x)$ is solution to

$$ \left\{ \begin{array}{ll} \partial_t \pi^{\epsilon}(s, x) - \frac{\epsilon}{2} \Delta \pi^{\epsilon}(s, x) = 0, & s \in (0, \tau), x \in \mathcal{X}, \\ \\ \pi^{\epsilon}(0, x) = \pi_n^{\epsilon} \circ (\phi_n^{\tau})^{-1}(x) |\det(D\phi_n^{\tau}((\phi_n^{\tau})^{-1}(x)))|^{-1}, & x \in \mathcal{X}. \end{array} \right. \qquad (25) $$

Moreover from Hopf-Lax formula, $\tilde{v}^{\epsilon} = -\epsilon \log \pi^{\epsilon}$ is therefore the solution to

$$ \left\{ \begin{array}{ll} \partial_t \tilde{v}^{\epsilon}(s, x) + \frac{1}{2} \|\mathrm{grad} \, v^{\epsilon}(s, x)\|_x^2 - \frac{\epsilon}{2} \Delta \tilde{v}^{\epsilon}(s, x) = 0 & (s, x) \in (0, \tau) \times \mathcal{X}, \\ \\ \tilde{v}^{\epsilon}(0, x) = -\epsilon \log \pi_n^{\epsilon} \circ (\phi_n^{\tau})^{-1}(x) + \epsilon \log \left( |\det(D\phi_n^{\tau}((\phi_n^{\tau})^{-1}(x)))| \right) & x \in \mathcal{X}. \end{array} \right. $$

212 Then, we known that there exists constants $C_1^{\mathrm{st}}$ and $C_2^{\mathrm{st}}$ such that

$$ \sup_{x \in \mathcal{X}} |\tilde{v}^{\epsilon}(\tau, x) - v^{\epsilon}(\tau, x)| \leq C_1^{\mathrm{st}} \sup_{x \in \mathcal{X}} |\tilde{v}^{\epsilon}(0, x) - v^{\epsilon}(0, x)| \leq C_2^{\mathrm{st}} \sqrt{\epsilon}, $$

213 where $v^{\epsilon}$ is solution to (62) of which the solution $v$ of (12) is a vanishing viscosity limit. Therefore, we finally

214 get

$$ \lim_{\epsilon \to 0} -\epsilon \log \pi_{n+1-}^{\epsilon}(x) = \lim_{\epsilon \to 0} \tilde{v}^{\epsilon}(\tau, x) = \lim_{\epsilon \to 0} v^{\epsilon}(\tau, x) = v(\tau, x) = \mathcal{Y}_{n+1-}(x), $$

215 which concludes the proof.

<figure><img src="image_3.png" /></figure>

216 **Remark 3.9.** Note that we could have considered a transition kernel in the observation space of the form

$$ \tilde{k}_{\tau}^{\mathcal{Y}}(y^{\delta}, h(x)) \propto \exp \left( -\frac{\tau}{2\epsilon} \mathrm{dist}(y^{\delta}, h(x))^2 \right), \qquad (26) $$

217 for representing uncertainties in manifolds with generalized normal laws as in [35]. Then the asymptotic regime

218 when $\epsilon \to 0$ would have been the same as a consequence of Varadhan's formula. Moreover, if we want to use a

219 normal law transition kernel for the modeling error terms, our proof could be adapted by directly using Laplace's

220 method when looking for the asymptotics of (19b) instead of the previous link between the heat kernel and the

221 Hopf-Cole transform.

## 4. CONTINUOUS-TIME FORMULATION

222

223 Our objective in this section is to identify the limit of our problem when the measurement sampling tends

224 to 0. We have already seen that the discrete dynamics converge to the continuous-time dynamical system. The

225 question is then to establish the corresponding asymptotic dynamics for the cost-to-come and for the associated

226 estimator.

### 4.1. Asymptotic dynamics

228 Let us first proceed formally in order to identify the asymptotic dynamics for the cost-to-come. Recalling

229 the two substeps of the cost-to-come evolution

$$ \frac{\mathcal{Y}_n(x) - \mathcal{Y}_{n-}(x)}{\tau} - \frac{1}{2} \mathrm{dist}_{\mathcal{Y}}(y^{\delta}(t_n), h(t_n, x))^2 = 0, \qquad (27) $$

230 and

$$ \frac{\mathcal{Y}_{n-}(x) - \mathcal{Y}_{n-1}(\phi_{n-1}^{\tau})^{-1}(x)}{\tau} + \frac{1}{2\tau} \int_{0}^{\tau} \|\mathrm{grad} \, v(s, x)\|_{x}^{2} \, ds = 0. \qquad (28) $$
By adding these two equations, we get

$$ \frac{\mathcal{V}_n(x) - \mathcal{V}_{n-1} \circ (\phi_{n-1}^{\tau})^{-1}(x)}{\tau} + \frac{1}{2\tau} \int_0^{\tau} \|\text{grad } v(x, s)\|_x^2 \, ds - \frac{1}{2} \text{dist}(y^{\delta}, h(x))^2 = 0, \quad (29) $$

which formally gives, when $\tau \to 0$, the following Hamilton-Jacobi-Bellman (HJB) equation

$$ \partial_t \mathcal{V}(t, x) + \langle f(t, x), \text{grad } \mathcal{V}(t, x) \rangle_x + \frac{1}{2} \|\text{grad } \mathcal{V}(x, s)\|_x^2 - \frac{1}{2} \text{dist}_y(y^{\delta}(t), h(t, x))^2 = 0. \quad (30) $$

## 4.2. Viscosity solution to the HJB equation

Let us restart by considering a least squares criterion associated with the dynamical system (1) where this time the observations $(y^{\delta}(t))_{t \ge 0}$ are defined for almost all time $t$. They are generated from a target trajectory $\tilde{x}$, up to some measurements disturbance $\eta$, from the identity

$$ y^{\delta}(t) = \text{Exp}_{h(t, \tilde{x}(t))}^{\mathcal{Y}}(\eta(t)). \quad (31) $$

The least-squares functional is here defined for all $t \ge 0$ by

$$ \mathcal{J}_t: \quad \begin{array}{l} \text{AC}([0, t]; \mathcal{X}) \to \mathbb{R} \\ \gamma \mapsto \mathcal{V}_0(\gamma(0)) + \int_0^t \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds, \end{array} \quad (32) $$

where

$$ \ell: (x, v, t) \mapsto \frac{1}{2} \|v - f(t, x)\|_x^2 + \frac{1}{2} \text{dist}_y(y^{\delta}(t), h(t, x))^2, \quad (33) $$

and $\text{AC}([0, t]; \mathcal{X})$ is the space of absolutely continuous functions in time with values in $\mathcal{X}$. The least-squares functional $\mathcal{J}_t$ is the limit of (3) when $\tau$ goes to 0. The associated cost-to-come is then defined by

$$ \mathcal{V}(t, x) = \inf_{\substack{\gamma \in \text{AC}([0, t]; \mathcal{X}) \\ \gamma(t)=x}} \mathcal{J}_t(\gamma). \quad (34) $$

The following functional enters the framework of the Tonelli Lagrangian as described in appendix A, namely $\ell$ is (1) of class $C^2$, (2) nondegenerate and convex in the fibers, (3) is superlinear on compact subsets of $\mathcal{X}$. More precisely, we consider the case of a Mañé Lagrangian with the term $\frac{1}{2}\|v - f(t, x)\|_x^2$. In particular, [16] proves that there are minimizer trajectories in the space of absolutely continuous functions. Moreover, the resulting minimizer may have an additional $C^1$ regularity, which is important for the cost-to-come solution to be a viscosity supersolution of an HJB equation. Such regularity is typically achieved when $f$ and $h$ are time-independent and $y^{\delta}$ is a time-periodic trajectory. The very general case has not been treated yet in the literature, so in the following we will assume that the existing minimizers in (34) are indeed $C^1$. Note that when considering the discrete-time formulation, we avoid facing such difficulty with respect to the time-dependency of our formulation and cost-to-come.

As a similar result compared to the discrete-time case, we justify the following regularity for the cost-to-come

**Proposition 4.1.** The cost-to-come $\mathcal{V}$ is a Lipschitz function in $[0, T] \times \mathcal{X}$.

*Proof.* Let us fix the time $t$ and introduce $x$ and $\tilde{x}$. Without loss of generality, we can assume that $\mathcal{V}(t, x) \ge \mathcal{V}(t, \tilde{x})$. Let us define, in accordance with Theorem A.2, a minimizing trajectory

$$ \bar{\gamma} = \underset{\substack{\gamma \in \text{AC}([0, t]; \mathcal{X}) \\ \gamma(t)=x}}{\operatorname{argmin}} \mathcal{J}_t(\gamma). $$
Then, we define the vector field on the manifold $\mathcal{X} \times [0, t]$ by

$$ \forall (s, x) \in [0, t] \times \mathcal{X}, \quad \nu(s, x) = \Pi_{\gamma(s)}^{x}(\dot{\gamma}(s) - f(s, \gamma(s))). $$

By construction, $\nu$ is Lipschitz continuous via parallel transport, namely, there exists $L_\nu \in \mathbb{R}_+^*$ such that

$$ \forall (s, x, \tilde{x}) \in [0, t] \times \mathcal{X}^2, \quad |\nu(s, x) - \Pi_{\tilde{x}}^x \nu(\tilde{x}, s)| \le L_\nu \mathrm{dist}_\chi(x, \tilde{x}). $$

And we can define the trajectory $\tilde{\gamma}$ solution to (1), such that $\tilde{\gamma}(t) = \tilde{x}$. We get

$$ \begin{aligned} \mathcal{V}(t, x) - \mathcal{V}(t, \tilde{x}) &\le \mathcal{J}_t(\tilde{\gamma}) - \mathcal{J}_t(\gamma) \le L_0 \mathrm{dist}_\chi(\bar{x}(0), \tilde{x}(0)) \\ &\quad + \int_0^t \frac{1}{2} \left[ \mathrm{dist}_\gamma(y^\delta(t), h(s, \bar{\gamma}(s)))^2 - \mathrm{dist}_\gamma(y^\delta(t), h(s, \tilde{\gamma}(s)))^2 \right] \mathrm{d}s \\ &\quad + \int_0^t \frac{1}{2} \left[ \|\nu(s, \bar{\gamma}(s))\|_{\bar{\gamma}(s)}^2 - \|\nu(s, \tilde{\gamma}(s))\|_{\tilde{\gamma}(s)}^2 \right] \mathrm{d}s. \end{aligned} $$

Moreover, we have

$$ \begin{aligned} & \mathrm{dist}_\gamma(y^\delta(t), h(s, \bar{\gamma}(s)))^2 - \mathrm{dist}_\gamma(y^\delta(t), h(s, \tilde{\gamma}(s)))^2 \\ &= \left| \mathrm{dist}_\gamma(y^\delta(t), h(s, \bar{\gamma}(s))) - \mathrm{dist}_\gamma(y^\delta(t), h(s, \tilde{\gamma}(s))) \right| \times \\ & \qquad \left( \mathrm{dist}_\gamma(y^\delta(t), h(s, \bar{\gamma}(s))) + \mathrm{dist}_\gamma(y^\delta(t), h(s, \tilde{\gamma}(s))) \right) \\ &\le 2L_\gamma L_h \mathrm{dist}_\chi(\tilde{\gamma}(s), \bar{\gamma}(s)), \end{aligned} $$

where $L_\gamma = \sup_{(y, \tilde{y}) \in \mathcal{X}^2} \mathrm{dist}_\gamma(y, \tilde{y})$. Similarly, we find

$$ \begin{aligned} & \left| \|\nu(\bar{\gamma}(s), s)\|_{\bar{\gamma}(s)}^2 - \|\nu(\tilde{\gamma}(s), s)\|_{\tilde{\gamma}(s)}^2 \right| \\ &= \left| \|\nu(\bar{\gamma}(s), s)\|_{\bar{\gamma}(s)} - \|\nu(\tilde{\gamma}(s), s)\|_{\tilde{\gamma}(s)} \right| \times \left| \|\nu(\bar{\gamma}(s), s)\|_{\bar{\gamma}(s)} + \|\nu(\tilde{\gamma}(s), s)\|_{\tilde{\gamma}(s)} \right| \\ &\le 2 \|\nu\|_{L^\infty((0,t) \times \mathcal{X})} L_\nu \mathrm{dist}_\chi(\tilde{x}(s), \bar{x}(s)). \end{aligned} $$

Finally from our stability estimate on the trajectory, see [17, Lemma 4.1],

$$ \mathrm{dist}_\chi(\tilde{\gamma}(s), \bar{\gamma}(s)) \le L_{f+\nu} e^{t-s} \mathrm{dist}_\chi(\tilde{x}, \bar{x}), $$

hence, there exists $C_1^{\mathrm{st}} \in \mathbb{R}_+^*$ such that

$$ \mathcal{V}(t, x) - \mathcal{V}(t, \tilde{x}) \le C_1^{\mathrm{st}} \mathrm{dist}_\chi(\tilde{x}, \bar{x}). $$

Now, let us consider $\tilde{t} > 0$ and without loss of generality that $\mathcal{V}(t, x) \ge \mathcal{V}(\tilde{t}, x)$. First consider the case where $t \ge \tilde{t}$, then with $\bar{\gamma}$ as previously defined

$$ \mathcal{V}(t, x) - \mathcal{V}(\tilde{t}, x) \le \mathcal{J}_t(\bar{\gamma}) - \mathcal{J}_{\tilde{t}}(\bar{\gamma}) \le \int_{\tilde{t}}^t \left[ \frac{1}{2} \mathrm{dist}_\gamma(y^\delta, h(\bar{\gamma}(s)))^2 + \frac{1}{2} \|\dot{\bar{\gamma}}(s) - f(s, \bar{\gamma}(s))\|_{\bar{\gamma}(s)}^2 \right] \mathrm{d}s. $$

As the optimum being $C^1([0, t], \mathcal{X})$ we conclude that there exists a constant $C_2^{\mathrm{st}}$ independent of $\tilde{t}$ such that

$$ \mathcal{V}(t, x) - \mathcal{V}(\tilde{t}, x) \le C_2^{\mathrm{st}} (t - \tilde{t}). $$
Conversely when $\tilde{t} > t$, we can consider that $\tilde{\gamma}$ the minimizer of $\mathcal{J}_{\tilde{t}}$ with ending point $x$ to get a constant $C_3^{\text{st}}$ independent of $t$ such that finally

$$ \mathcal{V}(t, x) - \mathcal{V}(\tilde{t}, x) \leq C_3^{\text{st}} |t - \tilde{t}|. \quad \square $$

Then, we justify the following Bellman's principle

**Proposition 4.2 (Bellman Principle).** The cost-to-come satisfies

$$ \mathcal{V}(t, x) = \min_{\substack{\gamma \in \text{AC}((\tilde{t}, t); \mathcal{X}) \\ \gamma(t)=x}} \left\{ \mathcal{V}(\tilde{t}, \gamma(\tilde{t})) + \int_{\tilde{t}}^{t} \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds \right\}. \quad (35) $$

*Proof.* For any trajectory $\gamma$ such that $\gamma(t) = x$, we have

$$ \mathcal{V}(t, x) \leq \mathcal{J}_t(\gamma) = \mathcal{V}_0(\gamma(0)) + \int_0^{\tilde{t}} \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds + \int_{\tilde{t}}^{t} \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds. $$

Taking the infimum over the set of all the trajectories $x_t$ such that $x_t(t) = \gamma(\tilde{t})$, we get

$$ \mathcal{V}(t, x) \leq \mathcal{V}(\tilde{t}, \gamma(\tilde{t})) + \int_{\tilde{t}}^{t} \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds. $$

Then let us consider – see appendix A – a minimizing trajectory

$$ \bar{\gamma} = \underset{\substack{\gamma \in \text{AC}([0, t]; \mathcal{X}) \\ \gamma(t)=x}}{\operatorname{argmin}} \mathcal{J}_t(\gamma). $$

We have by definition

$$ \mathcal{V}(t, x) - \mathcal{V}_0(\bar{\gamma}(0)) = \int_0^t \ell(s, \bar{\gamma}(s), \dot{\bar{\gamma}}(s)) \, ds. $$

Moreover, we have just proved that for all $\tilde{t} > 0$ and $\tilde{x} \in \mathcal{X}$,

$$ \mathcal{V}(\tilde{t}, \bar{\gamma}(\tilde{t})) - \mathcal{V}(0, \bar{\gamma}(0)) \leq \int_{0}^{\tilde{t}} \ell(s, \bar{\gamma}(s), \dot{\bar{\gamma}}(s)) \, ds. $$

Therefore, by subtracting the two previous expressions, we get

$$ \mathcal{V}(t, x) - \mathcal{V}(\tilde{t}, \bar{\gamma}(\tilde{t})) \geq \int_{\tilde{t}}^{t} \ell(s, \bar{\gamma}(s), \dot{\bar{\gamma}}(s)) \, ds, $$

so that we have the equality for this trajectory.

<figure><img src="image_4.png" /></figure>

Therefore, it can be justified that the cost-to-come is the unique viscosity solution to the HJB equation (59) starting from $\mathcal{V}_0$. We first introduce the Hamiltonian $H: \mathbb{R}^+ \times \mathcal{X} \times \mathcal{TX} \ni (t, x, q) \to \mathbb{R}$ defined by

$$ H(t, x, q) = \sup_{v \in \mathcal{T}_x \mathcal{X}} \left\{ \langle v, q \rangle_x - \ell(t, x, v) \right\} = \langle f(t, x), q \rangle_x + \frac{1}{2} \|q\|_x^2 - \frac{1}{2} \operatorname{dist}_Y(y^\delta, h(t, x))^2. \quad (36) $$
272 **Remark 4.3.** Note that in [16], the Hamiltonian is defined as a function of a co-tangent variable $H : \mathbb{R}^+ \times \mathcal{X} \times \mathcal{T}^*\mathcal{X} \ni (t, x, q) \to \mathbb{R}$ with the Legendre transform

$$H(t, x, q) = \sup_{v \in \mathcal{T}_{\bar{x}}\mathcal{X}} \{q(v) - \ell(t, x, v)\}.$$

274 Both definition are equivalent from the musical isomorphism between the tangent bundle and the co-tangent bundle.

276 **Theorem 4.4.** The cost-to-come $\mathcal{V}$ is a viscosity solution to the HJB equation (59) with initial condition
277 $\mathcal{V}(0, x) = \mathcal{V}_0(x)$. Namely, $\mathcal{V}$ is (1) a subsolution in the sense that, for all $\psi \in C^1((0, T] \times \mathcal{X})$, with $\psi \ge \mathcal{V}$ and
278 $\psi(\bar{t}, \bar{x}) = \mathcal{V}(\bar{t}, \bar{x})$ for a given $(\bar{t}, \bar{x}) \in (0, T]$ then

$$\partial_t \psi(\bar{t}, \bar{x}) + H(\bar{t}, \bar{x}, \mathrm{grad} \, \psi(\bar{t}, \bar{x})) \le 0. \qquad (37)$$

279 And (2), it is a supersolution in the sense that for all for all $\psi \in C^1((0, T] \times \mathcal{X})$, with $\psi \le \mathcal{V}$ and $\psi(\bar{t}, \bar{x}) = \mathcal{V}(\bar{t}, \bar{x})$ for a given $(\bar{t}, \bar{x}) \in (0, T]$ then

$$\partial_t \psi(\bar{t}, \bar{x}) + H(\bar{t}, \bar{x}, \mathrm{grad} \, \psi(\bar{t}, \bar{x})) \ge 0. \qquad (38)$$

281 **Proof.** The proof is classical and there is no specificity due to the manifold configuration. It is here reproduced as it is the core of the Mortensen observer theory. Let us start by showing that $\mathcal{V}$ is a subsolution. Suppose
282 $\psi \ge \mathcal{V}$ and $\psi(\bar{t}, \bar{x}) = \mathcal{V}(\bar{t}, \bar{x})$ for a given $(\bar{t}, \bar{x})$, then pick $v \in \mathcal{T}_{\bar{x}}\mathcal{X}$ and a $C^1([0, T]; \mathcal{X})$ curve $\gamma$ such that
283 $(\gamma(\bar{t}), \dot{\gamma}(\bar{t})) = (\bar{x}, v)$. Let $0 < t < \bar{t}$, then from Bellman's principle (35) we have

$$\mathcal{V}(\bar{t}, \bar{x}) - \mathcal{V}(t, \gamma(t)) \le \int_t^{\bar{t}} \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds,$$

285 leading with $\psi \ge \mathcal{V}$ and $\psi(\bar{t}, \bar{x}) = \mathcal{V}(\bar{t}, \bar{x})$ to

$$\psi(\bar{t}, \bar{x}) - \psi(t, \gamma(t)) \le \int_t^{\bar{t}} \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds.$$

286 Therefore, we have

$$\int_t^{\bar{t}} \left[ \frac{\partial \psi}{\partial t}(s, \gamma(s)) + \langle \mathrm{grad} \, \psi(s), \dot{\gamma}(s, \gamma(s)) \rangle_{\gamma(s)} - \ell(s, \gamma(s), \dot{\gamma}(s)) \right] ds \le 0.$$

287 and by considering the limit $t \to \bar{t}$, we get by continuity

$$\frac{\partial \psi}{\partial t}(\bar{t}, \bar{x}) + \langle \mathrm{grad} \, \psi(\bar{t}, \bar{x}), v \rangle_{\bar{x}} - \ell(\bar{t}, v, \bar{x}) \le 0.$$

288 This is in particular the case for $v = \mathrm{grad} \, \psi(\bar{t}, \bar{x}) + f(\bar{t}, \bar{x})$ leading to (37).

289 Let us now show that $\mathcal{V}$ is a supersolution. Suppose $\psi \le \mathcal{V}$ and $\psi(\bar{t}, \bar{x}) = \mathcal{V}(\bar{t}, \bar{x})$ for a given $(\bar{t}, \bar{x})$. Let us then consider

$$\bar{\gamma} = \underset{\gamma(\bar{t})=x}{\operatorname{argmin}} \mathcal{C}((t, \bar{t}); \mathcal{X}) \left\{ \mathcal{V}(t, \gamma(t)) + \int_t^{\bar{t}} \ell(s, \gamma(s), \dot{\gamma}(s)) \, ds \right\}.$$

291 Moreover, we have assumed that the minimizer $\bar{\gamma} \in C^1([t, \bar{t}], \mathcal{X})$. We have

$$\mathcal{V}(\bar{t}, \bar{x}) - \mathcal{V}(t, \gamma(t)) = \int_t^{\bar{t}} \ell(\bar{\gamma}(s), \dot{\bar{\gamma}}(s), s) \, ds,$$
leading with $\psi \le \mathcal{V}$ and $\psi(\bar{t}, \bar{x}) = \mathcal{V}(\bar{t}, \bar{x})$ to

$$ \psi(\bar{t}, \bar{x}) - \psi(t, \gamma(t)) \geq \int_{t}^{\bar{t}} \ell(\bar{\gamma}(s), \dot{\bar{\gamma}}(s), s) \, ds. $$

Therefore, we have

$$ \int_{t}^{\bar{t}} \left[ \frac{\partial \psi}{\partial t}(s, \bar{\gamma}(s)) + \langle \operatorname{grad} \psi(s), \dot{\bar{\gamma}}(s, \bar{\gamma}(s)) \rangle_{x} - \ell(\bar{\gamma}(s), \dot{\bar{\gamma}}(s), s) \right] ds \geq 0, $$

and by considering the limit $t \to \bar{t}$, we get by continuity

$$ \frac{\partial \psi}{\partial t}(\bar{t}, \bar{x}) + \langle \operatorname{grad} \psi(\bar{t}, \bar{x}), \dot{\bar{\gamma}}(t, \bar{x}) \rangle_{\bar{x}} \geq \ell(\bar{t}, \dot{\bar{\gamma}}(t, \bar{x}), \bar{x}). $$

Then by definition of $H$, we have

$$ \ell(\bar{t}, \dot{\bar{\gamma}}(t, \bar{x}), \bar{x}) \geq \langle \operatorname{grad} \psi(\bar{t}, \bar{x}), \dot{\bar{\gamma}}(t, \bar{x}) \rangle_{\bar{x}} - H(\bar{t}, \bar{x}, \operatorname{grad} \psi(\bar{t}, \bar{x})), $$

so that we finally get

$$ \frac{\partial \psi}{\partial t}(\bar{t}, \bar{x}) + \langle \operatorname{grad} \psi(\bar{t}, \bar{x}), \dot{\bar{\gamma}}(t, \bar{x}) \rangle_{\bar{x}} \geq \langle \operatorname{grad} \psi(\bar{t}, \bar{x}), \dot{\bar{\gamma}}(t, \bar{x}) \rangle_{\bar{x}} - H(\bar{t}, \bar{x}, \operatorname{grad} \psi(\bar{t}, \bar{x})), $$

which, after simplification justifies (38).

□

## 4.3. Convergence of discrete-time formulation

We can now show that the discrete-time cost-to-come converges to the continuous-time cost-to-come in the sense that when considering $\mathcal{V}^{\tau}(t, x)$, defined from $\mathcal{V}^{\tau}(0, x) = \mathcal{V}_{0}(x)$ by

$$ \mathcal{V}^{\tau}(t_n, x) = S_n^{\tau}(\mathcal{V}^{\tau}(t_{n-1}, \cdot))(x), $$

where the discrete semigroup $S_n^\tau$ is defined by

$$ S_n^\tau : \mathcal{W} \mapsto \min_{z \in \mathcal{X}} \left\{ \mathcal{W} \circ (\phi_n^\tau)^{-1}(z) + \frac{1}{2\tau} \operatorname{dist}_{\mathcal{X}}(x, z)^2 \right\} + \frac{\tau}{2} \operatorname{dist}_{\mathcal{Y}}(y_n^\delta, h_n(x))^2, \quad (39) $$

then $\mathcal{V}^\tau$ converges to $\mathcal{V}$ as $\tau$ tends to 0.

We follow the classical results first introduced in [3], where they prove the convergence of Hamilton-Jacobi-Bellman discretizations under the conditions of consistency, stability, and monotonicity.

**Theorem 4.5 (Convergence).** The discrete-time scheme (39) satisfies the three following conditions

**Consistency:** For all $(t, x) \in \mathbb{R}^+ \times \mathcal{X}$, we have

$$ \lim_{\substack{t_n \to t \\ \tau \downarrow 0}} \tau^{-1}(\mathcal{V}^{\tau}(t_n, x) - \mathcal{V}^{\tau}(t_{n-1}, x)) = \partial_t \mathcal{V}(t, x) + H(t, x, \operatorname{grad} \mathcal{V}). $$

**Stability:** for every constant $c$ and for any $\mathcal{W}$, we have

$$ S_n^\tau(\mathcal{W} + c)(x) = S_n^\tau(\mathcal{W})(x) + c. $$

**Monotonicity:** For any $\mathcal{U}$ and $\mathcal{W}$, we have

$$ \mathcal{U} \leq \mathcal{W} \Rightarrow S_n^\tau(\mathcal{U}) \leq S_n^\tau(\mathcal{W}). $$
Therefore we have

$$ \forall (t, x) \in \mathbb{R}^{+} \times \mathcal{X}, \quad \lim_{\substack{t_n \to t \\ \tau \downarrow 0}} \mathcal{V}^{\tau}(t_n, x) = \mathcal{V}(t, x). $$

Proof. The justification that consistency, stability, and monotonicity imply convergence follows the line of [20, Theorem 9.2]. Note that here we consider only the time discretization, whereas [20, Theorem 9.2] considers directly the space and time discretization, but the proof is identical.

Therefore, we only need to verify that the proposed scheme satisfies the 3 required conditions, namely consistency, stability, and monotonicity. Concerning the consistency, this is a direct consequence of (29). Let us then consider a constant c, we directly get

$$ \min_{z \in \mathcal{X}} \left\{ \mathcal{W} \circ (\phi_n^\tau)^{-1}(z) + \frac{1}{2\tau} \mathrm{dist}_\mathcal{X}(x, z)^2 + c \right\} + \frac{\tau}{2} \mathrm{dist}_\mathcal{Y}(y_n^\delta, h_n(x))^2 = S_n^\tau(\mathcal{W}) + c. $$

hence $S_n^\tau(\mathcal{W} + c)(x) = S_n^\tau(\mathcal{W})(x) + c$ as required for the stability since [15].

Finally, the monotonicity is a direct consequence of the link between the inf-convolution

$$ \mathcal{W}^{-}(x) = \min_{z \in \mathcal{X}} \left\{ \mathcal{W} \circ (\phi_{n}^{\tau})^{-1}(z) + \frac{1}{2\tau} \mathrm{dist}_{\mathcal{X}}(x, z)^{2} \right\} $$

and the HJB equation

$$ \left\{ \begin{array}{ll} \partial_t v(t, x) + \frac{1}{2} \|\mathrm{grad} \, v(t, x)\|_x^2 = 0, & (t, x) \in (0, \tau) \times \mathcal{X}, \\ v(0, x) = \mathcal{W} \circ (\phi_{n-1}^{\tau})^{-1}(x), & x \in \mathcal{X}. \end{array} \right. \qquad (40) $$

in the sense that $\mathcal{W}^{-}(x) = v(\tau, x)$. Then we know that from the maximum principle that when $v$ and $\tilde{v}$ are two solutions of (40) for two different initial conditions, we have

$$ v(0, x) \leq \tilde{v}(0, x) \Rightarrow v(\tau, x) \leq \tilde{v}(\tau, x). $$

This concludes the proof.

□

## 4.4. Observer feedback

In fact, we know that for all $t \geq 0$, $\mathcal{V}(t, \cdot)$ admits a minimum, and we further assume in this work that such a minimum is unique for all time $t \geq 0$. As done in the discrete-time setting, we can define the continuous-time Mortensen estimator by

$$ \hat{x}(t) = \underset{x \in \mathcal{X}}{\operatorname{argmin}} \mathcal{V}(t, x). \qquad (41) $$

One question discussed by [19,34] is the definition of the estimator (41) as the solution to a dynamical system combining the initial model $f$ perturbed by a feedback law incorporating the data. To obtain such dynamics, let us assume that for all $t \geq 0$, $\mathcal{V}(t, \cdot)$ is regular enough, namely $C^2$ at its minimum $\hat{x}(t)$.

We have by the chain rule for the covariant derivative along curves

$$ \begin{aligned} \frac{D}{dt}(\mathrm{grad} \, \mathcal{V}(t, \hat{x}(t))) = 0 &\Rightarrow \frac{\partial}{\partial t} \mathrm{grad} \, \mathcal{V}(t, \hat{x}(t)) + \nabla_{\dot{\hat{x}}(t)}^{\mathcal{X}} \mathrm{grad} \, \mathcal{V}(t, \hat{x}(t)) = 0, \\ &\Rightarrow \frac{\partial}{\partial t} \mathrm{grad} \, \mathcal{V}(t, \hat{x}(t)) + \mathrm{Hess} \, \mathcal{V}(\hat{x}(t), t) \dot{\hat{x}}(t) = 0, \end{aligned} $$

where $\frac{D}{dt}$ is in fact the covariant derivative along the curve $(t, \hat{x}(t))$ in the product manifold $\mathbb{R} \times \mathcal{X}$, which reduces to the usual derivation $\partial/\partial t$ in $\mathbb{R}$ combined with the Levi-Civita connection $\nabla^{\mathcal{X}}$ on $\mathcal{X}$. By switching
the order of derivations, we get

$$
\begin{align*}
\frac{\partial}{\partial t} \mathrm{grad} \mathcal{V}(t, \hat{x}) &= \mathrm{grad} \frac{\partial}{\partial t} \mathcal{V}(t, \hat{x}), \\
&= -\mathrm{grad} \left[ \langle \mathrm{grad} \mathcal{V}(t, \hat{x}), f(\hat{x}) \rangle_{\hat{x}} \right] + \frac{1}{2} \mathrm{grad} \left[ \mathrm{dist}_y(y^\delta, h(t, \hat{x}))^2 \right] \\
&\quad - \frac{1}{2} \mathrm{grad} \langle \mathrm{grad} \mathcal{V}(t, \hat{x}), \mathrm{grad} \mathcal{V}(t, \hat{x}) \rangle_{\hat{x}}.
\end{align*}
$$

On the one hand, we know from [36, Supplementary Material] that the gradient of the square distance is

$$
\frac{1}{2} \mathrm{grad} \left[ \mathrm{dist}_y(y^\delta, h(t, \hat{x}))^2 \right] = -Dh(t, \hat{x})^* \mathrm{Log}_{h(t, \hat{x})} y^\delta.
$$

On the other hand, we recall from [8, Theorem 5.6] that considering three vector fields $u, v, w$ of $\mathcal{X}$, we have

$$
\langle \mathrm{grad} \langle u, v \rangle_x, w \rangle_x = \langle \nabla_w^\chi u, v \rangle + \langle u, \nabla_w^\chi v \rangle.
$$

Taking for $t$ fixed, $u : x \mapsto \mathrm{grad} \mathcal{V}(t, x)$, $v : x \mapsto f(t, x)$ and evaluating at $\hat{x}(t)$, it yields

$$
\langle \mathrm{grad} \langle \mathrm{grad} \mathcal{V}(t, x), f(t, x) \rangle_x, w \rangle_x \Big|_{x=\hat{x}(t)} \\
= \langle \nabla_w^\chi \mathrm{grad} \mathcal{V}(t, \hat{x}(t)), f(t, \hat{x}(t)) \rangle_{\hat{x}(t)} + \langle \mathrm{grad} \mathcal{V}(t, \hat{x}(t)), \nabla_w^\chi f(t, \hat{x}(t)) \rangle_{\hat{x}(t)}.
$$

Since grad $\mathcal{V}(t, \hat{x}(t)) = 0$ we have

$$
\begin{align*}
\langle \mathrm{grad} \langle \mathrm{grad} \mathcal{V}(t, x), f(t, x) \rangle_x, w \rangle_x \Big|_{x=\hat{x}(t)} &= \langle \nabla_w \mathrm{grad} \mathcal{V}(t, \hat{x}(t)), f(t, \hat{x}(t)) \rangle_{\hat{x}(t)} \\
&= \mathrm{Hess} \mathcal{V}(t, \hat{x}(t)) w,
\end{align*}
$$

Finally, with $w = f(t, \hat{x}(t))$, we get

$$
\langle \mathrm{grad} \langle \mathrm{grad} \mathcal{V}(t, x), f(t, x) \rangle_x, w \rangle_x \Big|_{x=\hat{x}(t)} = \mathrm{Hess} \mathcal{V}(t, \hat{x}(t)) f(t, \hat{x}(t)).
$$

329

Finally, we obtain

$$
\mathrm{Hess} \mathcal{V}(\hat{x}(t), t) \dot{\hat{x}}(t) - \mathrm{Hess} \mathcal{V}(\hat{x}(t), t) f(t, \hat{x}(t)) - Dh(t, \hat{x}(t))^* \mathrm{Log}_{h(t, \hat{x}(t))} y^\delta(t) = 0.
$$

330

leading, if Hess $\mathcal{V}(t, \hat{x}(t))$ is invertible, to the feedback dynamics

$$
\dot{\hat{x}} = f(t, \hat{x}(t)) + (\mathrm{Hess} \mathcal{V}(t, \hat{x}(t)))^{-1} Dh(t, \hat{x}(t))^* \mathrm{Log}_{h(t, \hat{x}(t))} y^\delta(t), \quad t \in \mathbb{R}_+. \quad (42)
$$

331

This is the counterpart of the Mortensen feedback dynamics given in [19, 34] in Manifolds. Note that this
332
dynamics is justified only when $\mathcal{V}$ is regular enough. Proving such regularity is out of the scope of the present
333
article. It was recently justified in Euclidean spaces for bilinear dynamics in [9], and we believe that this could
334
be an inspiring starting point for future extension in the Riemannian case.
335

332

333

334

Finally, let us discuss the fact that (42) is an observer in the sense of [26, Definition 2]. In essence, this means
335
that in the absence of measurement and model disturbances, the trajectory of $\hat{x}$ should converge asymptotically
336
in time to $\tilde{x}$. Moreover, $\hat{x}$ should depend continuously on the measurement error and the model disturbance
337
$\tilde{v}$. Verifying the observer properties for the manifold observer in full detail is beyond the scope of the present
338

336

337

338
339 article. However, we propose a starting point for future analysis. From the cost-to-come $\mathcal{V}$ and assuming
340 enough regularity, we can define the function

$$
d_{\nu} : \mathbb{R}_{+} \times \mathcal{X}^2 \ni (t, x, z) \mapsto \mathcal{V}(t, x) - \mathcal{V}(t, z) - \langle \operatorname{grad} \mathcal{V}(t, z), \operatorname{Log}_z(x) \rangle_z. \quad (43)
$$

In the case where $\mathcal{V}$ can be proved to be geodesic-convex, then $d_{\mathcal{V}}(t, \cdot, \cdot)$ can be seen as a time-dependent Bregmann divergence. Then, one could look at this divergence between the Mortensen estimator and the target trajectory, we get in the absence of noise

$$
\begin{align*}
\frac{d}{dt} d_{\mathcal{V}}(t, \tilde{x}(t), \hat{x}(t)) &= \frac{\partial \mathcal{V}}{\partial t}(t, \tilde{x}(t)) + \langle \operatorname{grad} \mathcal{V}(t, \tilde{x}(t)), \dot{\tilde{x}} \rangle_{\hat{x}(t)} \\
&\quad - \frac{\partial \mathcal{V}}{\partial t}(t, \hat{x}(t)) - \langle \operatorname{grad} \mathcal{V}(t, \hat{x}(t)), \dot{\hat{x}} \rangle_{\hat{x}(t)} \\
&= -\frac{1}{2} \| \operatorname{grad} \mathcal{V}(t, \tilde{x}(t)) \|_{\hat{x}(t)}^2 - \frac{1}{2} \operatorname{dist}_{\mathcal{V}}(h(t, \tilde{x}(t)), h(t, \hat{x}(t))^2.
\end{align*}
$$

So that, we could imagine that the asymptotic convergence of the proposed divergence goes to 0 when we have
some observability conditions of the form

$$
\exists T_0, \kappa > 0 \mid \forall (x_0, \tilde{x}_0) \in \mathcal{X}^2, \forall t \in \mathbb{R}_+, \int_t^{t+T_0} \mathrm{dist}_\mathcal{Y}(h(x(s)), h(\tilde{x}(s))^2 \mathrm{d}s \ge \kappa \mathrm{dist}_\mathcal{X}(x(t), \tilde{x}(t))^2,
$$

341 for any solution to (1). Then, if we are able to additionally justify a uniform control of the form: there exist
342 $C_1^{\mathrm{st}}$ and $C_2^{\mathrm{st}}$ such that for all $t \ge 0$

$$
C_1^{\mathrm{st}} d_{\mathcal{X}}(x, z) \le d_{\mathcal{V}}(t, x, z) \le C_2^{\mathrm{st}} d_{\mathcal{X}}(x, z),
$$

343 then we could envision the asymptotic convergence of the observer to the target trajectory in the absence of
344 noise with respect to the proposed divergence going to 0 by relying on Liapunov's theory [26].

5. SPATIAL DISCRETIZATION AND NUMERICAL ILLUSTRATIONS

345

346 5.1. Spatial discretization

347 We can now move forward to the spatial discretization strategy used for the numerical illustrations. From
348 now on, we assume that the manifold $\mathcal{X}$ is a sub-manifold of $\mathbb{R}^3$ which is identified to a triangular mesh
349 $\mathcal{X} = \mathcal{X}_\hbar = \cup_{j=1}^J K_j$, where $\hbar$ is the discretization step-size$^1$. The mesh nodes are denoted $(x_i^\hbar)_{1\le i\le I}$. Moreover,
350 we assume that the triangulation is a Delaunay triangulation [11]. Again, in order to ensure that a proposed
351 discretization strategy is convergent, we must rely on a version of Theorem 4.5 in $\mathcal{X}_\hbar$. To do so, we must propose
352 an adequate discretization of the inf-convolution step.
353

354 Here considering a subsampling of $[0, \tau]$ into $M \in \mathbb{N}$ substeps, and denote the space of positive continuous
355 functions $C_+^0(\mathcal{X})$. we introduce the fully-discrete semigroup

$$
\tilde{S}_{n-}^{\tau, \hbar} : C_+^0(\mathcal{X}) \ni \mathcal{W} \mapsto (\mathcal{X} \ni x \mapsto -\hbar \log(p_M^h(x))),
$$

355 where $p_M^h(x)$ is the $\mathbb{P}_1$ solution from $p_0^h(x) = \exp\left(-\frac{I^h(\mathcal{W}(x))}{\hbar}\right)$ – with $I_\hbar$ is the $\mathbb{P}_1$ interpolator – of the following
356 heat equation defined in weak form by

$$
\forall q^\hbar \in \mathcal{V}_h, \quad \int_{\mathcal{X}} \left( \frac{p_{m+1}^\hbar - p_m^\hbar}{\tau} \right) q^\hbar \mathrm{d}x + \frac{1}{2} \int_{\mathcal{X}} \langle \operatorname{grad} p_{m+1}^\hbar, \operatorname{grad} q^\hbar \rangle_{g(x)} \mathrm{d}x = 0, \quad 0 \le k < M, \quad (44)
$$

<footnote n="1">¹we use the notation $\hbar$ to avoid any confusion with the observation map $h$</footnote>
357 where

$$
\mathcal{V}_h = \left\{ q \in C^0(\mathcal{X}) \text{ s.t. } q^h_{|K_j} \in \mathbb{P}_1(K_j), 1 \le j \le J \right\}. \quad (45)
$$

358 The operator $\tilde{S}_{n-}^{\tau, \hbar}$ is well defined from the following discrete maximum principle.

359 Theorem 5.1. Assuming that the triangulation is Delaunay, then for all $0 \le m \le M$, we have

$$
\forall x \in \mathcal{X}, \quad \inf_{x \in \mathcal{X}} p_0^h(x) \le p_m^h(x) \le \sup_{x \in \mathcal{X}} p_0^h(x).
$$

360 Then the full space and time discretization is defined by the following three steps:

361 • Initialization Step:

$$
\mathcal{V}_0^{\tau, \hbar}(t_0, x) = I_{\hbar}(\mathcal{V}_0^{\tau, \hbar}(x)). \quad (46a)
$$

362 • Then for all $n \ge 1$:

363 – Convection Step:

$$
\tilde{\mathcal{V}}_{n}^{\tau, \hbar}(t_n, x_i) = I_{\hbar}(\mathcal{V}_{n-1}^{\tau, \hbar}(t_0, \phi_{n-1}^{\tau}(\cdot))(x_i)), \quad \forall x_i \in \mathcal{X}_h.
$$

364 – Inf-convolution Step:

$$
\mathcal{V}_n^{\tau, \hbar}(t_n, x_i) = \tilde{S}_{n-}^{\tau, \hbar}(\tilde{\mathcal{V}}_n^{\tau, \hbar}(t_n, \cdot))(x_i), \quad \forall x_i \in \mathcal{X}_h.
$$

365 – Correction Step:

$$
\mathcal{V}_n(x) = \mathcal{V}_{n-}(x) + I_h \left( \frac{\tau}{2} \operatorname{dist}_Y (y_n^\delta, h_n(x))^2 \right),
$$

366 The procedure from one time-step to the sequel can be summarized using the following semigroup

$$
S_{n}^{\tau, \hbar} : \mathcal{W} \mapsto \left( \mathcal{X}_h \ni x_i \mapsto \tilde{S}_{n-}^{\tau, \hbar} \circ I_{\hbar}(\mathcal{W} \circ \phi_{n-1}^{\tau}(\cdot))(x_i) + \frac{\tau}{2} \mathrm{dist}_Y(y_n^{\delta}, h_n(x_i))^2 \right),
$$

367 to be compared with the time-discretized semigroup operator $S_n^\tau$.

368 To prove that we can apply Theorem 4.5 to $S_n^{\tau, \hbar}$ in the space $X_h$, we only need to remember that $I_\hbar$ is

369 monotone as a linear interpolation operator on the $X_h$. In particular, the convection step is solved by computing

$$
I_{\hbar}(\mathcal{V}_{n-1}^{\tau, \hbar}(t_0, \phi_{n-1}^{\tau}(\cdot))(x_i)) = \sum_{i=1}^{3} \lambda_i(\phi_{n-1}^{\tau}(x_i))\mathcal{V}_{n-1}^{\tau, \hbar}(t_0, z_{k(i)}),
$$

370 where $\phi_{n-1}^{\tau}(x_i)$ belongs to the triangle $(z_{k(1)}, z_{k(2)}, z_{k(3)})$ and $\lambda_1, \lambda_2, \lambda_3$ are the barycentric coordinates of

371 $\phi_{n-1}^{\tau}(x_i)$ in the triangle $(z_{k(1)}, z_{k(2)}, z_{k(3)})$. Moreover, $\tilde{\mathcal{V}}_n^{\tau, \hbar}$ is monotone from Theorem 5.1. Concerning the

372 stability, we easily justify that

$$
\tilde{S}_{n}^{\tau, \hbar}(\mathcal{W} + c) = \tilde{S}_{n}^{\tau, \hbar}(\mathcal{W}) + c.
$$

373 by linearity of the heat equation combined with $-\hbar \log(\exp(\frac{\mathcal{W}+c}{\hbar})) = -\hbar \log(\exp(\frac{\mathcal{W}}{\hbar})) + c$. The consistency is

374 given by the fact that when $\hbar$ goes to 0, we know that $\mathcal{V}_n^{\tau, \hbar}(t_n, x_i)$ converges to its vanishing viscosity limit

375 solution of the inf-convolution. Therefore, Theorem 4.5 adapted to be applied to $S_n^{\tau, \hbar}$ in the space $X_h$, we have

$$
\forall (t, x) \in \mathbb{R}^{+} \times \mathcal{X}, \quad \lim_{\substack{t_n \to t \\ x_i \to x \\ \tau, \hbar \downarrow 0}} \mathcal{V}^{\tau, \hbar}(t_n, x_i) = \mathcal{V}(t, x). \quad (47)
$$

376 **Remark 5.2.** Note that when considering with a fixed $\tau$, the convergence with respect to $\hbar$, we have that from

377 appendix D that

$$
\forall \hbar > 0, \quad \left| S_{n-}^{\tau, \hbar} \mathcal{W} - S_{n-}^{\tau} \mathcal{W} \right| \le C \sqrt{\hbar},
$$
which combined to the convergence in $O(\hbar)$ for the linear interpolator for $C^1$ functions gives a convergence in $O(\sqrt{\hbar})$ for the spatial discretization in a compact manifold without boundary.

## 5.2. Numerical illustration

We consider a linear dynamical system on $\mathbb{R}^3$ by $\dot{\tilde{x}} = A\tilde{x}$ where $A$ is a $3 \times 3$ matrix. Then $x = \frac{\tilde{x}}{\|\tilde{x}\|}$ evolves on the 2-sphere $S^2 = \{x \in \mathbb{R}^3 | \|x\|^2 = 1\}$ with dynamics given by

$$ \dot{x}(t) = Ax(t) - \langle x(t), Ax(t) \rangle x(t), \quad t \in \mathbb{R}^{+}, \qquad (48) $$

where $\langle \cdot, \cdot \rangle$ and $\| \cdot \|$ denote the scalar product and the norm in $\mathbb{R}^3$, respectively. Concerning the time discretization, we choose an implicit midpoint scheme or the form

$$ \frac{x_{n+1} - x_n}{\tau} = \left( A - \left\langle \frac{x_{n+\frac{1}{2}}}{\|x_{n+\frac{1}{2}}\|}, A \frac{x_{n+\frac{1}{2}}}{\|x_{n+\frac{1}{2}}\|} \right\rangle \right) x_{n+\frac{1}{2}}, \quad n \in \mathbb{N}, \qquad (49) $$

where $x_{n+\frac{1}{2}} = \frac{1}{2}(x_{n+1} + x_n)$. Given an initial condition $x_0 \in S^2$, it is easy to verify that the produced sequence stays on $S^2$. The numerical scheme (49) is a non-linear implicit scheme of the form $x_{n+1} = F(x_{n+1}, x_n)$ leading to an implicit definition of $\phi_n^\tau$ which here is independent of $n$. It can be reformulated as a minimization problem

$$ x_{n+1} = \underset{x \in \mathcal{X}}{\operatorname{argmin}} R(x, x_n), \quad \text{where } R(x, x_n) = \frac{1}{2} \operatorname{dist}_{\mathcal{X}}^2(x, F(x, x_n)). \qquad (50) $$

This can be solved with a gradient descent starting at $x_n$ with the gradient of $R$ with respect to its first component given by $\nabla_1 R(x, x_n) = -\log_x F(x) - DF(x)^* \log_F(x) x$. We solve this minimization problem with the Riemannian limited memory BFGS solver [8]$^2$. Computing the inverse $(\phi_n^\tau)^{-1}$ also leads to an implicit formulation $x_n = \tilde{F}(x_n, x_{n+1})$ solved as minimization problem of the form (50). In what follows, let $A$ denote the matrix with entries $A_{23} = -2$, $A_{32} = 2$, and $A_{ij} = 0$ otherwise.

Observer settings: The real trajectory is artificially created with the procedure described by (49) starting at $\tilde{x}(0) = (0.7061, 0.024, 0.7080)$. To generate a perturbation of the target trajectory, we use a normal distribution on a manifold [35] as an easy-to-compute and simplified version of a Brownian motion perturbation, see Remark 3.9 and the equivalence when $\varepsilon$ tends to 0. The model error $\nu_n(x) \sim N(\mu, \sigma^2)$ on $\mathcal{T}_x\mathcal{X}$ then acts as a multivariate normal distribution of mean $x$ and with a standard deviation of $\sigma = 20 \text{ km}/R$ where $R$ is the radius of the Earth – we here choose $R = 6367.4 \text{ km}$. The observer is initialized with $\hat{x}_{0-} = (0.4859, 0.024, 0.8503)$. For comparison, we also define the free-run trajectory as the output of (49) with the wrong initial condition $\hat{x}_{0-}$ and without any model noise; this represents the output that would have produced our naive mathematical model without taking the available measurements into account.

In the first case in Figure 3 we consider a full observation, i.e $h = \text{Id}$. The observation noise is defined similarly to the model noise with a 20 km standard deviation. For illustrating the observer convergence on a long enough time window, we overestimate the covariances during the estimation as if the standard deviation of the noises was 60 km instead of 20 km, i.e $\alpha = \delta = 60/R$. The distance between the initial estimation $\hat{x}_{0-}$ and $\tilde{x}(0)$ is approximately 2069 km, but we underestimate it by choosing a weight on this estimation as if it were 1000 km, i.e $\mathcal{Y}_0(x) = (R/1000)^2 \operatorname{dist}_{\mathcal{X}}(x, \hat{x}_{0-})^2$. In the second case in Figure 4, we keep the same parameters, but we consider observations only on the latitude position. Working with spherical coordinates $x = (\theta, \phi)$, this corresponds to the linear observation operator $h(x) = \phi$.

<footnote>2Available in the Manopt toolbox [7]</footnote>
<figure><img src="image_5.png" /><figcaption>FIGURE 1. Dynamics of the model and mesh used in the present test case</figcaption></figure>

Numerical results: The implementation of the Mortensen observer is done with a geodesic icosahedron mesh of the sphere, or icosphere, with seven subdivisions – approximately 163k vertices and 327k triangles, see Figure 1). This is a regular triangle mesh, where the distance between two vertices is $\hbar = 0.0086$, or approximately 54.7 km. The vanishing viscosity parameter is set to $\varepsilon = 10^{-3}$ and the discretization of the heat equation is done with sub-iterations, setting $\tilde{\tau} = 10^{-1}\tau$. In the full observation case, the convergence of the estimator $\hat{x}_{n-1}$ is illustrated in Figure 3. The isolines^3 of the value functions are shrinking uniformly, as expected. In the latitude observation case, the non-observability of the situation is illustrated in Figure 4; the estimator catches up with the observation, but the longitude position of the real trajectory is not retrieved. Moreover, we clearly see the impact of the manifold setting and the dynamics in bending the isovalues of the cost-to-come.

<figure><img src="image_6.png" /><figcaption>(A) Relative error of $p_1^{\tilde{\tau},\hbar}$ in $L^2(\mathcal{X})$ norm, where $\hbar = 0.016$ and the reference solution is computed with $\tilde{\tau} = 10^{-4}\tau$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(B) Relative error of $\mathcal{Y}_{-}^{\tau,\hbar}$ in $L^\infty(\mathcal{X})$ norm, where $\tau = 0.01$ and the reference solution is computed with $\hbar = 0.0086$.</figcaption></figure>

<caption>FIGURE 2. Convergence of the inf-convolution step computed from the Hopf-Lax strategy</caption>

We also want to investigate the accuracy of our discretization. Let us now set $\mathcal{Y}_0^-(x) = \text{dist}_{\mathcal{X}}^2(x, \hat{x}_{0-})$. We denote $p_1^{\tilde{\tau},\hbar}$ the solution of the heat equation at time $t = \tau$ with a time step $\tilde{\tau}$ and a spatial step $\hbar$. We are interested in the numerical convergence of $p_1^{\tilde{\tau},\hbar}$ with respect to $\tilde{\tau}$. We perform the comparison with a converged solution obtained with $\tilde{\tau} = 10^{-4}\tau$ and $\hbar = 0.016$ – corresponding to the icosphere with 6 subdivisions. In Figure 2b-(a) we plot the relative error in $L^2(\mathcal{X})$ norm and find a classical order of convergence of 1. We are also interested in the numerical convergence of $\mathcal{Y}_{-}^{\tau,\hbar}$, output of one iteration of the Mortensen observer, with respect to $\hbar$. In Figure 2b-(b) we plot the relative error in $L^\infty(\mathcal{X})$ norm, where the reference solution is obtained

<footnote>^3Visualized using the software Vizir [18]</footnote>
424 for $\hbar = 0.0086 - i.e.$ an icosphere with 7 subdivisions. For each $\hbar$, the vanishing viscosity parameter is set to

425 $\varepsilon = \hbar$, and we find a better convergence rate of approximately 1 instead of the expected $\frac{1}{2}$.

<figure><img src="image_8.png" /><figcaption>FIGURE 3. Evolution of the estimator trajectory $(\hat{x}_{n-})_{n \ge 1}$, the free run trajectory and the observations $(y_n^\delta)_{n \ge 1}$ for the full observation case. Some isolines of the function $d_{\gamma_n}(x, \hat{x}_{n-})$ are plotted.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 4. Evolution of the estimator trajectory $(\hat{x}_{n-})_{n \ge 1}$, the freerun trajectory and the target trajectory $(\tilde{x}_n)_{n \ge 1}$ for the partial observation case. Some isolines of the function $d_{\gamma_n}(x, \hat{x}_{n-})$ are plotted.</figcaption></figure>
APPENDIX A. FUNCTIONAL MINIMIZATION

We here summarize the results of variational calculus in manifolds presented in the works [37] in the context of Tonelli's theorem. Indeed, we consider a functional of the form

$$ \mathcal{J}_{t,x}(\gamma) = \mathcal{W}_0(\gamma(0)) + \int_0^t \left[ \ell(s, \gamma(s), \dot{\gamma}(s)) \right] \mathrm{d}s, \qquad (51) $$

where $\mathcal{W}_0: \mathcal{X} \to \mathbb{R}^+$ is Lipschitz and $\ell: \mathbb{R}^+ \times \mathcal{TX} \to \mathbb{R}^+$ will be specified below. Here, $(t, x) \in \mathbb{R}^+ \times \mathcal{X}$ are parameters, and we expect to minimize $\mathcal{J}_{t,x}$ in the space of absolute continuous curve defined in $[0, t]$ with value in the metric space $\mathcal{X}$, denoted $\mathrm{AC}([0, t], \mathcal{X})$. We recall – see [16] – that for all $\gamma \in \mathrm{AC}([0, t], \mathcal{X})$, we have

$$ \forall (\tilde{s}, \tilde{t}) \in [0, t], \quad \mathrm{dist}_{\mathcal{X}}(\gamma(\tilde{s}), \gamma(\tilde{t})) \le \int_{\tilde{s}}^{\tilde{t}} \|\dot{\gamma}(s)\|_{\gamma(s)} \mathrm{d}s. $$

Let us start with the case where $\ell$ is independent of $s$, which will be needed in the next appendix section.

**Definition A.1 (Tonelli Lagrangian).** A function $\ell: \mathcal{TX} \to \mathbb{R}$ is called a Tonelli Lagrangian if:

(i) $\ell \in C^2(\mathcal{TX})$;

(ii) $\ell$ is strictly convex in each fiber, in the $C^2$ sense, i.e., the second partial vertical derivative $\partial^2\ell/\partial v^2(x, v)$ is positive definite, as a quadratic form, for all $(x, v) \in \mathcal{TX}$;

(iii) $\ell$ is superlinear in each fiber, namely

$$ \lim_{\|v\|_x \to +\infty} \frac{\ell(x, v)}{\|v\|_x} = +\infty. $$

Note that the condition (iii) is equivalent to the following condition: for each $a \in \mathbb{R}$ there exists $b(a) \in \mathbb{R}$ such that

$$ \ell(x, v) \ge a \|v\|_x - b(a), \quad \forall (x, v) \in \mathcal{TX} $$

which is satisfied with our least squares penalization. Moreover, when the manifold is compact, condition (iii) is independent of the choice of the Riemannian metric.

The main examples of Tonelli Lagrangian are typically the Riemannian Lagrangian case where $\ell(x, v) = \frac{1}{2}\|v\|_x^2$, and the Mañé Lagrangian $\ell(x, v) = \frac{1}{2}\|v - f(x)\|_x^2 + d(x)$. Typically, the Riemannian Lagrangian case is important for us when considering the prediction step of the time-discrete formulation. The Mañé Lagrangian appears in the continuous-time formulation when $f$ and $h$ are autonomous, albeit in our case our Lagrangian will still be time-dependent $\ell(s, x, v) = \frac{1}{2}\|v - f(x)\|_x^2 + d(s, x)$ as we have a time-dependency coming from the observations $s \mapsto y(s)$.

When $\ell$ also depends on $s$, we ask for the previous definition to be satisfied uniformly with respect to time [6] and following the discussion of [37] we may need to consider an additional assumption:

(iv) The Euler-Lagrange flow associated with $\ell$ is complete, namely has solutions defined in $\mathbb{R}$.

**Theorem A.2.** Let $\ell$ be a Tonelli Lagrangian. The functional $\mathcal{J}_{t,x}$ admits one minimizer over the space of absolutely continuous trajectories with value in $\mathcal{X}$ and ending point $\gamma(t) = x$.

*Proof.* The complete proof can be found in [37, Theorem 4.1.1], see also [16] and [6], we here simply recall the main steps for the existence of a minimizer in our case where the cost includes a quadratic penalization on the form $\|\dot{\gamma}(s) - f(\gamma(s))\|_{\gamma(s)}^2$. Let us considering a minimizing sequence $(\gamma_n)_{n \in \mathbb{N}}$ in the space $\mathrm{AC}([0, t], \mathcal{X})$. Then there exists a constant $C^{\mathrm{st}}$ such that

$$ \int_0^t \|\dot{\gamma}_n(s)\|_x^2 \mathrm{d}s \le C^{\mathrm{st}}. $$
Therefore, we have that for all $n \in \mathbb{N}$

$$ \forall (\tilde{s}, \tilde{t}) \in [0, t], \quad \mathrm{dist}_{\mathcal{X}}(\gamma(\tilde{s}), \gamma(\tilde{t})) \le \int_{\tilde{s}}^{\tilde{t}} \|\dot{\gamma}(s)\|_{\gamma(s)} \, ds \le C^{\mathrm{st}} |\tilde{s} - \tilde{t}|^{\frac{1}{2}}. $$

From the Arzelà-Ascoli theorem, we can extract a subsequence that we still denote $(\gamma_n)_{n \in \mathbb{N}}$ that converges uniformly in $AC([0, t], \mathcal{X})$ to $\bar{\gamma}$

By continuity of $\ell$, we have that

$$ \lim_{n \to +\infty} \int_0^t \ell(s, \gamma_n(s)) \, ds = \int_0^t \ell(s, \bar{\gamma}(s)) \, ds $$

and we can show that

$$ \lim_{n \to +\infty} \int_0^t \|\dot{\gamma}(s)\|_{\bar{\gamma}(s)}^2 \, ds \le \liminf_{n \to +\infty} \int_0^t \|\dot{\gamma}_n(s)\|_{\gamma_n(s)}^2 \, ds. $$

Therefore $s \mapsto \gamma(s)$ is an absolutely continuous path with value in $\mathcal{X}$ and a minimum for the functional $\mathcal{J}_{t,x}$. $\square$

Let us finish with this important last result proven in [16], see also [37].

**Theorem A.3.** Let $\ell$ be a time-independent Tonelli Lagrangian, then for all $t$, the minimizing function is $C^1([0, t], \mathcal{X})$.

When the Tonelli Lagrangian depends on time, [37] mentions the need of having the assumption (iv) to be satisfied to have $C^1([0, t], \mathcal{X})$ curves.

## APPENDIX B. HOPF-LAX FORMULA

In this appendix, we consider a function $\mathcal{U}: \mathbb{R}^+ \times \mathcal{X} \to \mathbb{R}$ defined as the following inf-convolution

$$ \mathcal{U}(t, x) = \inf_{z \in \mathcal{X}} \left\{ \mathcal{U}_0(z) + \frac{1}{2t} \mathrm{dist}_{\mathcal{X}}(x, z)^2 \right\}, \qquad (52) $$

As $\mathcal{X}$ is compact, the infimum is attained. Such inf-convolution has been widely studied in the literature for various manifolds, see for instance [2, 16]. We here, recall some of the most classical results used in this article, most of them available in [16]. Note however that the detailed proofs given here correspond to simplifications offered by our setting or to proofs only sketched in the literature.

**Proposition B.1.** The function $\mathcal{U}$ defined by (52) is Lipschitz.

We deduce from Rademacher's theorem that $\mathcal{U}$ is differentiable almost everywhere.

We have the following characterization.

**Proposition B.2.** The function $\mathcal{U}$ satisfies, for all $x \in \mathcal{X}$ and $0 \le s < t$

$$ \mathcal{U}(t, x) = \inf_{z \in \mathcal{X}} \left\{ \mathcal{U}(s, z) + \frac{1}{2(t-s)} \mathrm{dist}_{\mathcal{X}}(z, x)^2 \right\}. \qquad (53) $$
Proof. Let $(x, \tilde{x}) \in \mathcal{X}^2$, $t > 0$ and $0 \le s < t$. If $s = 0$, equation (53) simply reduces to the definition (52). If $s > 0$, we have

$$ 
\begin{aligned}
\left[ \frac{\mathrm{dist}_{\mathcal{X}}(x, \tilde{x})}{t} \right]^2 &\le \left[ \frac{\mathrm{dist}_{\mathcal{X}}(x, z) + \mathrm{dist}_{\mathcal{X}}(z, \tilde{x})}{t} \right]^2 \\
&\le \left[ \left(\frac{s}{t}\right) \frac{\mathrm{dist}_{\mathcal{X}}(x, z)}{s} + \left(\frac{t-s}{t}\right) \frac{\mathrm{dist}_{\mathcal{X}}(z, \tilde{x})}{t-s} \right]^2 \\
&\le \left(\frac{s}{t}\right) \frac{\mathrm{dist}_{\mathcal{X}}(x, z)^2}{s^2} + \left(\frac{t-s}{t}\right) \frac{\mathrm{dist}_{\mathcal{X}}(z, \tilde{x})^2}{(t-s)^2}
\end{aligned}
$$

the last inequality being obtained by convexity of the square function. Moreover, by choosing $z$ as the geodesic barycenter of $(x, \frac{s}{t})$ and $(\tilde{x}, \frac{t-s}{t})$, these inequalities are equalities. This gives

$$ 
\begin{aligned}
\mathcal{W}(t, x) &= \inf_{\tilde{x} \in \mathcal{X}} \left\{ \mathcal{W}_0(\tilde{x}) + \frac{t}{2} \left[ \frac{\mathrm{dist}_{\mathcal{X}}(x, \tilde{x})}{t} \right]^2 \right\} \\
&= \inf_{(\tilde{x}, z) \in \mathcal{X}^2} \left\{ \mathcal{W}_0(\tilde{x}) + \frac{1}{2s} \mathrm{dist}_{\mathcal{X}}(x, z)^2 + \frac{1}{2(t-s)} \mathrm{dist}_{\mathcal{X}}(z, \tilde{x})^2 \right\},
\end{aligned}
$$

476 hence justifies (53).

<figure><img src="image_10.png" /></figure>

477 Then the family of operator

$$ S_t(\phi) : x \mapsto \inf_{z \in \mathcal{X}} \left\{ \mathcal{W}_0(z) + \frac{1}{2t} \mathrm{dist}_{\mathcal{X}}(x, z)^2 \right\}, $$

478 is called a Lax-Oleinik semigroup as it satisfies the semigroup property

$$ \forall t \ge s \ge 0, \quad S_t(\phi) = S_{t-s} \circ S_s(\phi). \qquad (54) $$

479 **Theorem B.3 (Variational formulation).** The function $\mathcal{W}$ is solution of the variational problem

$$ \mathcal{W}(t, x) = \inf_{\substack{x(\cdot) \in \mathrm{AC}([0, t]; \mathcal{X}) \\ x(t)=x}} \left\{ \mathcal{W}(0, x(0)) + \int_0^t \frac{1}{2} \|\dot{x}(s)\|_x^2 ds \right\}. $$

480 Proof. From Jensen's inequality, we have that for all $x(\cdot) \in \mathrm{AC}([0, t]; \mathcal{X})$ such that $x(t) = x$,

$$ \frac{1}{2t} \left| \int_{0}^{t} \dot{x}(s) \, ds \right|^{2} \leq \frac{1}{t} \int_{0}^{t} \frac{1}{2} \| \dot{x}(s) \|_{x(s)}^{2} \, ds, $$

481 so that

$$ \mathcal{W}(t, x) \le \inf_{\substack{x(\cdot) \in \mathrm{AC}([0, t]; \mathcal{X}) \\ x(t)=x}} \left\{ \mathcal{W}(0, x(0)) + \int_0^t \frac{1}{2} \|\dot{x}(s)\|_{x(s)}^2 ds \right\}. $$

482 Conversely, let us consider

$$ \bar{z} = \underset{z \in \mathcal{X}}{\operatorname{argmin}} \left\{ \mathcal{W}_0(z) + \frac{1}{2t} \mathrm{dist}_{\mathcal{X}}(x, z)^2 \right\}, $$
and $\gamma : [0, 1] \to \mathbb{R}$ a geodesic path between $\bar{z}$ and $x$. Then $\bar{\gamma} : [0, t] \ni s \mapsto \gamma(\frac{s}{t}) \in \mathcal{X}$ is a path between $z$ and $x$ such that for all $s \in [0, t]$, $\|\dot{\gamma}(s)\|_{\gamma(s)} = \frac{1}{t}\|\dot{\gamma}(\frac{s}{t})\|_{\gamma(s/t)} = \frac{1}{t}\mathrm{dist}_{\mathcal{X}}(x, \bar{z})$. Therefore, we have

$$ \mathcal{U}(0, \bar{\gamma}(0)) + \int_{0}^{t} \frac{1}{2} \|\dot{\bar{\gamma}}(s)\|_{\bar{\gamma}(s)}^{2} \mathrm{d}s = \mathcal{U}_{0}(\bar{z}) + \frac{1}{2} \int_{0}^{t} \|\dot{\gamma}(s)\|_{\bar{\gamma}(s)}^{2} \mathrm{d}s \le \mathcal{U}_{0}(\bar{z}) + \frac{t}{2} \frac{\mathrm{dist}_{\mathcal{X}}(x, \bar{z})^{2}}{t^{2}} \le \mathcal{U}(t, x), $$

483

which justifies that

$$ \min_{\substack{\gamma \in \mathrm{AC}([t-\tau, t]; \mathcal{X}) \\ \gamma(t)=x}} \left\{ \mathcal{U}(0, x(0)) + \int_{0}^{t} \frac{1}{2} \|\dot{\gamma}(s)\|_{\gamma(s)}^{2} \mathrm{d}s \right\} = \mathcal{U}(t, x), $$

484

and concludes the proof.

□

485

We deduce that Lax-Oleinik semigroup is given by

$$ S_t(\phi) : x \mapsto \mathcal{U}_0(x) + \frac{1}{2} \int_0^t \|\dot{\gamma}(s)\|_{\gamma(s)}^2 \mathrm{d}s, $$

486

and justifies Bellman's Principle

$$ \mathcal{U}(t, x) = \min_{\substack{\gamma \in \mathrm{AC}([0, t]; \mathcal{X}) \\ x(t)=x}} \left\{ \mathcal{U}(t - \tau, \gamma(t - \tau)) + \frac{1}{2} \int_{t-\tau}^{t} \|\dot{\gamma}(s)\|_{\gamma(s)}^{2} \mathrm{d}s \right\}, \quad \forall t > \tau \ge 0. \quad (55) $$

487

Then, we can prove that $\mathcal{U}$ is a viscosity solution of a Hamilton-Jacobi-Bellman equation in the sense of Theorem 4.4 with here the Hamiltonian being only $\frac{1}{2}\|\mathrm{grad}\,\mathcal{U}(t, x)\|_x^2$.

488

489

**Theorem B.4.** $\mathcal{U}$ is a viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation

$$ \left\{ \begin{array}{l} \displaystyle \partial_t \mathcal{U}(t, x) + \frac{1}{2} \|\mathrm{grad} \, \mathcal{U}(t, x)\|_x^2 = 0, \quad (t, x) \in \mathbb{R}^+ \times \mathcal{X}, \\ \mathcal{U}(0, x) = \mathcal{U}_0(x). \end{array} \right. \qquad (56) $$

490

Let us recall that, in fact, we have uniqueness of the viscosity solution (56) from the following comparison principle.

491

492

**Theorem B.5.** Let $u$ be a bounded viscosity subsolution and $v$ a bounded viscosity supersolution of (56). Then, for any $(t, x) \in [0, +\infty) \times \mathcal{X}$, it holds that $u(t, x) \le v(t, x)$.

493

*Proof.* Let us assume by contradiction that

$$ \sup_{(t,x) \in [0, +\infty) \times \mathcal{X}} (u(t,x) - v(t,x)) = M > 0. $$

It means it exists $\lambda > 0$ sufficiently small such that it exist $(\bar{t}, \bar{x})$ and

$$ u(\bar{t}, \bar{x}) - v(\bar{t}, \bar{x}) - \lambda \bar{t} > 0. $$

Following the doubling variable strategy [14]; we introduce, for any $\alpha > 0$, the following auxiliary function

$$ \psi_{\alpha}(t, x, s, z) = u(t, x) - v(s, z) - \frac{\alpha}{2} \mathrm{dist}_{\mathcal{X}}(x, z)^2 - \frac{\alpha}{2} |t - s|^2 - \frac{\lambda}{2}(t + s). $$
$\psi_\alpha$ is obviously continuous and bounded above by $M$; its supremum $M_\alpha$ exists. Moreover, it holds

$$ \begin{aligned} \psi_\alpha(\bar{t}, \bar{x}, \bar{t}, \bar{x}) &= u(\bar{t}, \bar{x}) - v(\bar{t}, \bar{x}) - \lambda \bar{t} > 0, \\ \psi_\alpha(t, x, s, z) &\le 0 \text{ for any } (t, s) \notin B_\lambda = B\left(0, \frac{2}{\lambda} \left[ \sup_{\mathbb{R}^+ \times \mathcal{X}} u + \sup_{\mathbb{R}^+ \times \mathcal{X}} v \right] \right). \end{aligned} $$

Hence the supremum $M_\alpha$ is attained on the compact set $B_\lambda \times \mathcal{X}$. In particular, there exists $(t_\alpha, x_\alpha, s_\alpha, z_\alpha)$ such that

$$ \psi_\alpha(t_\alpha, x_\alpha, s_\alpha, z_\alpha) = M_\alpha. $$

However, we do not have any guarantee that $t_\alpha > 0$ and $s_\alpha > 0$, which are essential conditions in order to invoke the definition of viscosity solutions. To address this difficulty, we rely on the strategy proposed in [14, Proposition 3.7]. In fact, we consider its manifold counterpart justified in [25, Lemma 3.7]

**Step 1.** Let us define an increasing sequence of strictly positive scalars $(\alpha_k)_k$ such that $\alpha_k \to +\infty$ as $k \to +\infty$. Denote $M_{\alpha_k}$ the supremum of $\psi_{\alpha_k}$ attained in $(t_{\alpha_k}, x_{\alpha_k}, s_{\alpha_k}, z_{\alpha_k})$. Define

$$ \begin{aligned} \Phi(t, s, x, z) &= u(t, x) - v(s, z) - \frac{\lambda}{2}(t + s), \\ \Psi(t, s, x, z) &= \frac{1}{2}\mathrm{dist}_{\mathcal{X}}(x, z)^2 + \frac{1}{2}|t - s|^2, \end{aligned} $$

such that

$$ \psi_{\alpha_k}(t, s, x, z) = \Phi(t, s, x, z) - \alpha_k \Psi(t, s, x, z). $$

We have that

$$ \begin{aligned} M_{\alpha_k} &\ge \Phi(t_{\alpha_{k+1}}, x_{\alpha_{k+1}}, s_{\alpha_{k+1}}, z_{\alpha_{k+1}}) - \alpha_k \Psi(t_{\alpha_{k+1}}, x_{\alpha_{k+1}}, s_{\alpha_{k+1}}, z_{\alpha_{k+1}}), \\ &\ge M_{\alpha_{k+1}} + (\alpha_{k+1} - \alpha_k) \Psi(t_{\alpha_{k+1}}, x_{\alpha_{k+1}}, s_{\alpha_{k+1}}, z_{\alpha_{k+1}}). \end{aligned} $$

Since $\Psi > 0$, $M_{\alpha_k}$ decreases as $\alpha_k$ increases. Moreover, $M_{\alpha_k} > 0$ because $\psi_{\alpha_k}(\bar{t}, \bar{x}, \bar{t}, \bar{x}) > 0$. Therefore, $\lim_{k \to \infty} M_{\alpha_k}$ exists and is finite. Similarly, the following holds

$$ \begin{aligned} M_{\alpha_k/2} &\ge \Phi(t_{\alpha_k}, x_{\alpha_k}, s_{\alpha_k}, z_{\alpha_k}) - \frac{\alpha_k}{2} \Psi(t_{\alpha_k}, x_{\alpha_k}, s_{\alpha_k}, z_{\alpha_k}), \\ &\ge M_{\alpha_k} + \frac{\alpha_k}{2} \Psi(t_{\alpha_k}, x_{\alpha_k}, s_{\alpha_k}, z_{\alpha_k}). \end{aligned} $$

Therefore, $\alpha_k \Psi(t_{\alpha_k}, x_{\alpha_k}, s_{\alpha_k}, z_{\alpha_k}) \le 2(M_{\alpha_k/2} - M_{\alpha_k})$. At the limit everything goes to zero such that

$$ \lim_{k \to \infty} \alpha_k \mathrm{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k})^2 = 0 \quad \text{and} \quad \lim_{k \to \infty} \alpha_k |t_{\alpha_k} - s_{\alpha_k}|^2 = 0. $$

Since $\alpha_k$ increases to infinity the following also holds

$$ \lim_{k \to \infty} \mathrm{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k}) = 0 \quad \text{and} \quad \lim_{k \to \infty} |t_{\alpha_k} - s_{\alpha_k}| = 0. $$

Let us now consider $(\hat{t}, \hat{x})$ an accumulation point of $(t_{\alpha_k}, x_{\alpha_k})$, then $(\hat{t}, \hat{x})$ is also an accumulation point of $(s_{\alpha_k}, z_{\alpha_k})$. Since $M_{\alpha_k} > 0$, at the limit we have

$$ u(\hat{t}, \hat{x}) - v(\hat{t}, \hat{x}) > 0, $$

and therefore $\hat{t} > 0$ because of the initial condition of (56). Therefore, for $k$ large enough we have $t_{\alpha_k} > 0$ and $s_{\alpha_k} > 0$. In what follows, we assume that $k$ satisfies this condition.
**Step 2.** Let us consider the following test function

$$ \Phi_u(t, x) = \frac{\alpha_k}{2} \mathrm{dist}_\chi(x, z_{\alpha_k})^2 + \frac{\alpha_k}{2} |t - s_{\alpha_k}|^2 + \frac{\lambda}{2} (t + s_{\alpha_k}). $$

Therefore, $(u - \Phi_u)$ has a maximum at $(t_{\alpha_k}, x_{\alpha_k})$. In particular, recalling that $u$ is a viscosity subsolution

$$ \alpha_k(t_{\alpha_k} - s_{\alpha_k}) + \frac{\lambda}{2} + \frac{\alpha_k^2}{2} \|\log_{x_{\alpha_k}} z_{\alpha_k}\|_{x_{\alpha_k}}^2 \le 0. \quad (57) $$

**Step 3.** Let us consider the following test function

$$ \Phi_v(s, z) = -\frac{\alpha_k}{2} \mathrm{dist}_\chi(x_{\alpha_k}, z)^2 - \frac{\alpha_k}{2} |t_{\alpha_k} - s|^2 - \frac{\lambda}{2} (t_{\alpha_k} + s). $$

Therefore, $(v - \Phi_v)$ has a minimum at $(s_{\alpha_k}, z_{\alpha_k})$. In particular, recalling that $v$ is a viscosity supersolution

$$ \alpha_k(t_{\alpha_k} - s_{\alpha_k}) - \frac{\lambda}{2} + \frac{\alpha_k^2}{2} \|\log_{z_{\alpha_k}} x_{\alpha_k}\|_{z_{\alpha_k}}^2 \ge 0. \quad (58) $$

**Step 4.** Let us recall that by definition of Riemannian distance

$$ \|\log_{x_{\alpha_k}} z_{\alpha_k}\|_{x_{\alpha_k}}^2 = \mathrm{dist}_\chi(x_{\alpha_k}, z_{\alpha_k})^2 = \|\log_{z_{\alpha_k}} x_{\alpha_k}\|_{z_{\alpha_k}}^2. $$

Therefore, we combine inequalities (57) and (58) to obtain the contradiction $\lambda \le 0$. □

<figure><img src="image_11.png" /></figure>

## APPENDIX C. COMPARISON PRINCIPLE FOR CONTINUOUS-TIME MORTENSEN

Obtaining the same type of comparison principle, albeit for the more general HJB equation (59) is more intricate. Indeed, the proof of proposition B.5 was simplified by the Hamiltonian being only $\frac{1}{2}\|\mathrm{grad} \, \mathcal{V}(t, x)\|_x^2$.

Here, in a more general configuration, we additionally require the Lipschitz continuity of $f, h$, and $y^\delta$.

$$ \partial_t \mathcal{V}(t, x) + \langle f(t, x), \mathrm{grad} \, \mathcal{V}(t, x) \rangle_x + \frac{1}{2} \|\mathrm{grad} \, \mathcal{V}(x, s)\|_x^2 - \frac{1}{2} \mathrm{dist}_y(y^\delta(t_n), h(t_n, x))^2 = 0. \quad (59) $$

**Proposition C.1.** Let $u$ be a bounded viscosity subsolution and $v$ a bounded viscosity supersolution of (59) with the same initial condition. Then it holds:

$$ u(t, x) \le v(t, x) \quad \text{for any } (t, x) \in [0, +\infty) \times \mathcal{X}. $$

*Proof.* Following the proof of Theorem B.5, let us assume by contradiction that

$$ \sup_{(t,x) \in [0, +\infty) \times \mathcal{X}} (u(t,x) - v(t,x)) = M > 0. $$

It means it exists $\lambda > 0$ sufficiently small such that it exist $(\bar{t}, \bar{x})$ and

$$ u(\bar{t}, \bar{x}) - v(\bar{t}, \bar{x}) - \lambda \bar{t} > 0. $$

For any $\alpha > 0$, we introduce the following auxiliary function

$$ \psi_{\alpha}(t, x, s, z) = u(t, x) - v(s, z) - \frac{\alpha}{2} \mathrm{dist}_{\chi}(x, z)^2 - \frac{\alpha}{2} |t - s|^2 - \frac{\lambda}{2} (t + s). $$
505 The supremum $M_\alpha$ of $\psi_\alpha$ is attained on a compact set. Therefore, there exists $(t_\alpha, x_\alpha, s_\alpha, z_\alpha)$ such that

$$ \psi_{\alpha}(t_{\alpha}, x_{\alpha}, s_{\alpha}, z_{\alpha}) = M_{\alpha}. $$

As previously, we consider an increasing sequence of strictly positive scalars $(\alpha_k)_k$ tending to infinity, and denote by $M_{\alpha_k}$ the supremum of $\Phi_{\alpha_k}$ attained at $(t_{\alpha_k}, x_{\alpha_k}, s_{\alpha_k}, z_{\alpha_k})$. We take $k$ large enough to ensure $t_{\alpha_k} > 0$ and $s_{\alpha_k} > 0$.

**Step 1.** Let us consider the following test function

$$ \Phi_u(t, x) = \frac{\alpha_k}{2} \mathrm{dist}_\chi(x, z_{\alpha_k})^2 + \frac{\alpha_k}{2} |t - s_{\alpha_k}|^2 + \frac{\lambda}{2} (t + s_{\alpha_k}). $$

Therefore, $(u - \Phi_u)$ has a maximum at $(t_{\alpha_k}, x_{\alpha_k})$. In particular, recalling that $u$ is a viscosity subsolution

$$ \alpha_k(t_{\alpha_k} - s_{\alpha_k}) + \frac{\lambda}{2} - \alpha_k \langle f(t_{\alpha_k}, x_{\alpha_k}), \log_{x_{\alpha_k}} z_{\alpha_k} \rangle_{x_{\alpha_k}} + \frac{\alpha_k^2}{2} \|\log_{x_{\alpha_k}} z_{\alpha_k}\|_{x_{\alpha_k}}^2 \\ - \frac{1}{2} \mathrm{dist}_y(y^\delta(t_{\alpha_k}), h(t_{\alpha_k}, x_{\alpha_k}))^2 \le 0. \quad (60) $$

**Step 2.** Let us consider the following test function

$$ \Phi_v(s, z) = -\frac{\alpha_k}{2} \mathrm{dist}_\chi(x_{\alpha_k}, z)^2 - \frac{\alpha_k}{2} |t_{\alpha_k} - s|^2 - \frac{\lambda}{2} (t_{\alpha_k} + s). $$

Therefore, $(v - \Phi_v)$ has a minimum at $(s_{\alpha_k}, z_{\alpha_k})$. In particular, recalling that $v$ is a viscosity supersolution

$$ \alpha_k(t_{\alpha_k} - s_{\alpha_k}) - \frac{\lambda}{2} - \alpha_k \langle f(s_{\alpha_k}, z_{\alpha_k}), -\log_{z_{\alpha_k}} x_{\alpha_k} \rangle_{z_{\alpha_k}} + \frac{\alpha_k^2}{2} \|\log_{z_{\alpha_k}} x_{\alpha_k}\|_{z_{\alpha_k}}^2 \\ - \frac{1}{2} \mathrm{dist}_y(y^\delta(s_{\alpha_k}), h(s_{\alpha_k}, z_{\alpha_k}))^2 \ge 0. \quad (61) $$

**Step 3.** Let us recall that by definition of Riemannian distance

$$ \|\log_{x_{\alpha_k}} z_{\alpha_k}\|_{x_{\alpha_k}}^2 = \mathrm{dist}_\chi(x_{\alpha_k}, z_{\alpha_k})^2 = \|\log_{z_{\alpha_k}} x_{\alpha_k}\|_{z_{\alpha_k}}^2. $$

Let us now deal with the $f$ dependent-term. Note that $\Pi_{x_{\alpha_k} z_{\alpha_k}} \log_{x_{\alpha_k}} z_{\alpha_k} = -\log_{z_{\alpha_k}} x_{\alpha_k}$. In fact, if $\gamma(t)$ is the geodesic between $x_{\alpha_k}$ and $z_{\alpha_k}$, the geodesic equation $\nabla_{\dot{\gamma}(t)}\gamma(t) = 0$ means that $\dot{\gamma}(t)$ is a parallel vector field along $\gamma(t)$. Otherwise, $\Pi_{x_{\alpha_k} z_{\alpha_k}} \dot{\gamma}(0) = \dot{\gamma}(1)$. The left term of this equality is $\Pi_{x_{\alpha_k} z_{\alpha_k}} \log_{x_{\alpha_k}} z_{\alpha_k}$ while the second is $-\tilde{\gamma}(1)$ where $\tilde{\gamma}(t) = \gamma(1-t)$, i.e. $-\log_{z_{\alpha_k}} x_{\alpha_k}$. Therefore, by isometry of the parallel transport, the following holds

$$ \langle f(t_{\alpha_k}, x_{\alpha_k}), \log_{x_{\alpha_k}} z_{\alpha_k} \rangle_{x_{\alpha_k}} - \langle f(s_{\alpha_k}, z_{\alpha_k}), -\log_{z_{\alpha_k}} x_{\alpha_k} \rangle_{z_{\alpha_k}} \\ = \langle f(t_{\alpha_k}, x_{\alpha_k}) - \Pi_{z_{\alpha_k} x_{\alpha_k}} f(s_{\alpha_k}, z_{\alpha_k}), \log_{x_{\alpha_k}} z_{\alpha_k} \rangle_{x_{\alpha_k}}. $$

The vector field $f$ being Lipschitz continuous, Cauchy-Schwarz inequality allows us to bound this quantity above by $C_1^{\mathrm{st}}|t_{\alpha_k} - s_{\alpha_k}|\mathrm{dist}_\chi(x_{\alpha_k}, z_{\alpha_k}) + C_2^{\mathrm{st}}\mathrm{dist}_\chi(x_{\alpha_k}, z_{\alpha_k})^2$, where $C_1^{\mathrm{st}}$ and $C_2^{\mathrm{st}}$ are constants independent of $\alpha_k$. For the observation terms, we have

$$ \mathrm{dist}_y(y^\delta(t_{\alpha_k}), h(t_{\alpha_k}, x_{\alpha_k}))^2 - \mathrm{dist}_y(y^\delta(s_{\alpha_k}), h(s_{\alpha_k}, z_{\alpha_k}))^2 \\ \le \mathrm{dist}_y(y^\delta(t_{\alpha_k}), y^\delta(s_{\alpha_k}))^2 - \mathrm{dist}_y(h(t_{\alpha_k}, x_{\alpha_k}), h(s_{\alpha_k}, z_{\alpha_k}))^2. $$
This is bounded above by $\le C_3^{\text{st}}|t_{\alpha_k} - s_{\alpha_k}| + C_4^{\text{st}}\text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k})$, where $C_3^{\text{st}}$ and $C_4^{\text{st}}$ are constants independent of $\alpha_k$, by Lipschitz continuity of $h$ and $y^\delta$. We combine inequalities (60) and (61) and these estimates to obtain

$$ \lambda \le \alpha_k C_1^{\text{st}} |t_{\alpha_k} - s_{\alpha_k}| \text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k}) + \alpha_k C_2^{\text{st}} \text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k})^2 + C_3^{\text{st}} |t_{\alpha_k} - s_{\alpha_k}| + C_4^{\text{st}} \text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k}). $$

We recall from previously that

$$ \lim_{k \to \infty} \alpha_k \text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k})^2 = 0 \quad \text{and} \quad \lim_{k \to \infty} \alpha_k |t_{\alpha_k} - s_{\alpha_k}|^2 = 0. $$

and

$$ \lim_{k \to \infty} \text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k}) = 0 \quad \text{and} \quad \lim_{k \to \infty} |t_{\alpha_k} - s_{\alpha_k}| = 0. $$

Therefore we obtain

$$ \lim_{k \to \infty} \alpha_k |t_{\alpha_k} - s_{\alpha_k}| \text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k}) = \lim_{k \to \infty} \left(\alpha_k \text{dist}_{\mathcal{X}}(x_{\alpha_k}, z_{\alpha_k})^2\right)^{\frac{1}{2}} \left(\alpha_k |t_{\alpha_k} - s_{\alpha_k}|^2\right)^{\frac{1}{2}} = 0. $$

Taking the limit in the previous estimate, the contradiction $\lambda \le 0$ holds. $\square$

<figure><img src="image_12.png" /></figure>

## APPENDIX D. CONVERGENCE RATE VANISHING VISCOSITY FOR DISCRETE-TIME MORTENSEN

We consider a vanishing viscosity version of (12) with

$$ \left\{ \begin{array}{l} \displaystyle \partial_t v^\varepsilon(s, x) + \frac{1}{2} \|\text{grad } v^\varepsilon(s, x)\|_x^2 = \frac{\varepsilon}{2} \Delta v^\varepsilon(s, x), \quad (s, x) \in (0, \tau) \times \mathcal{X}, \\[1.5ex] \displaystyle v^\varepsilon(0, x) = v_0(x), \end{array} \right. \qquad (62) $$

where $\varepsilon$ is a positive scalar, $v_0$ is Lipschitz function, and we formulate in the next proposition a convergence result in the closed manifold case. The proof is heavily inspired from [13]. Although we have a simpler case of Hamiltonian, the manifold settings brings small adjustments to the proof.

**Proposition D.1.** Let $v^\varepsilon$ be the unique smooth solution of (62) and $v$ the unique viscosity solution of (12) over the time window $[0, T]$. Then there exists a constant $C^{\text{st}} > 0$, independent of $\varepsilon$, such that

$$ \sup_{(t,x) \in [0,T] \times \mathcal{X}} |v^\varepsilon(t,x) - v(t,x)| \le C^{\text{st}} \varepsilon^{\frac{1}{2}}. $$

Before proving this proposition, we state the following estimate.

**Lemma D.2.** For any $(t, s, x, z) \in [0, T]^2 \times \mathcal{X}^2$, there exists a constant $C_1^{\text{st}} > 0$ independent of $\varepsilon$ such that

$$ |v^{\varepsilon}(t, x) - v^{\varepsilon}(s, x)| \le C_1^{\text{st}} \left[ \sqrt{\varepsilon} |t - s|^{\frac{1}{2}} + |t - s| \right]. \qquad (63) $$

*Proof.* We follow the proof presented in [13, Lemma 5.2]. First, we recall from that $v^\varepsilon$ satisfies the following estimates: There exist $C_a^{\text{st}}$ and $C_b^{\text{st}}$ such that

$$ \|v^{\varepsilon}\|_{L^{\infty}} \le C_a^{\text{st}} \quad \text{and} \quad \|\text{grad } v^{\varepsilon}\|_{L^{\infty}} \le C_b^{\text{st}}, $$

which in our case can be obtained from heat kernel estimates and the Hopf-Cole transform. Then, we introduce $\rho \in C^{\infty}(\mathcal{X})$ a standard mollifier supported in a ball of radius inferior to the injectivity radius and satisfying
$$ \Delta v_{\alpha} = \text{div}(\rho_{\alpha} * \text{grad } v) = \sum_{k=1}^{d} \left[ \partial_{x^k} (\rho_{\alpha} * (\text{grad } v)^k) + \rho_{\alpha} * (\text{grad } v)^k \partial_{x^k} (\log \sqrt{\det g}) \right], $$

$$ \begin{aligned} \|\partial_t v_\alpha\|_{L^\infty(X)} &\le \frac{C_b^{\text{st}}}{2} + \varepsilon \|\Delta v_\alpha\|_{L^\infty(X)} \\ &\le \left(\frac{C_b^{\text{st}}}{2} + \varepsilon C_X^{\text{st}} C_\rho^{\text{st}} C_b^{\text{st}}\right) + \frac{C_\rho^{\text{st}} \varepsilon}{\alpha} \|Dv_\alpha\|_{L^\infty(X)}, \end{aligned} $$

where $C_\rho^{\text{st}}$ depends only on $\rho$ and $C_X^{\text{st}}$ only depends on $g$. We obtain, as in [13, Lemma 5.2], that there exists $C_3^{\text{st}}$ and then $C^{\text{st}}$ such that

$$ \begin{aligned} |v(x,t) - v(x,s)| &\le |v(x,t) - v_\alpha(x,t)| + |v(x,s) - v_\alpha(x,s)| + \left(C_3^{\text{st}} + \frac{C_\rho^{\text{st}}\varepsilon}{\alpha}\right) |t-s| \\ &\le C^{\text{st}} \left(\alpha + |t-s| + \frac{\varepsilon}{\alpha}|t-s|\right), \end{aligned} $$

**Proof.** Let us define

$$ \lambda = \sup_{(t,x) \in [0,T] \times \mathcal{X}} |v^{\varepsilon}(t,x) - v(t,x)| > 0, $$

which is finite since $[0, T] \times \mathcal{X}$ is compact. For any $\alpha > 0$, we introduce the following auxiliary function, for any $\alpha > 0$:

$$ \psi_{\alpha}(t, x, s, z) = v(t, x) - v^{\varepsilon}(s, z) - \frac{\lambda}{4T}(t + s) + \left(5M + \frac{\lambda}{2}\right) \beta_{\alpha}(\text{dist}_{\mathcal{X}}(x, z), t - s), $$

where $M = \sup_{x \in \mathcal{X}} |v_0(x)|$, and $\beta_\alpha(\xi, t) = \beta\left(\frac{\xi}{\alpha}, \frac{t}{\alpha}\right)$, with $\beta$ defined such that

$$ \left\{ \begin{array}{ll} \beta(\xi, t) = 1 - \frac{|\xi|^2}{2L_{\mathcal{X}}^2} - \frac{|t|^2}{2T^2}, & \text{if } \frac{|\xi|^2}{2L_{\mathcal{X}}^2} + \frac{|t|^2}{2T^2} < \frac{1}{2}, \\ \\ \beta(\xi, t) < \frac{1}{2}, & \text{if } \frac{|\xi|^2}{2L_{\mathcal{X}}^2} + \frac{|t|^2}{2T^2} > \frac{1}{2}. \end{array} \right. $$

$$ \psi_{\alpha}(t_{\alpha}, x_{\alpha}, s_{\alpha}, z_{\alpha}) = M_{\alpha}. $$

**Preliminary results.** First, we have

$$ M_{\alpha} \ge \sup_{(t,x) \in [0,T] \times \mathcal{X}} |v(t,x) - v^{\varepsilon}(t,x)| - \frac{\lambda}{4T}(T+T) + \left(5M + \frac{\lambda}{2}\right) \beta_{\alpha}(0, T-T). $$

<footnote n="519">519 $\int_X \rho(x) \, dx = 1$, $\rho_\alpha(x) = \alpha^{-d}\rho(x/\alpha)$, and $v_\alpha = \rho_\alpha * v$. We have $|\partial_t v_\alpha - \varepsilon\Delta v_\alpha| \le \frac{C_b^{\text{st}}}{2}$ and, with $g$ denoting the space metric,</footnote>

<footnote n="521">521 leads, for all $t \ge 0$, to</footnote>

<footnote n="522">522 Setting $\alpha = \varepsilon^{1/2}|t-s|^{1/2}$ finishes the proof. $\square$</footnote>

<footnote n="523">523 We would like to mention that an alternative proof using heat kernel estimated could be envisioned from [12, Lemma 2.5]. Let us now return to the proof of Proposition D.1.</footnote>

<footnote n="525">525 In particular, we have $0 \le \beta \le 1$ and $\beta(0, 0) = 1$. The function $\psi_\alpha$ is continuous on the compact set $[0, T] \times \mathcal{X}$; therefore its supremum $M_\alpha$ exists and is attained. There exists $(t_\alpha, x_\alpha, s_\alpha, z_\alpha)$ such that</footnote>

<footnote>527</footnote>

<footnote>528</footnote>
529 Therefore,

$$M_{\alpha} \geq 5M + \lambda. \tag{64}$$

From this estimate we also obtain

$$5M + \lambda \leq M_{\alpha} \leq \sup_{(t,x) \in [0,T] \times \mathcal{X}} |v(t,x)| + \sup_{(s,z) \in [0,T] \times \mathcal{X}} |v^{\varepsilon}(s,z)| + \left(5M + \frac{\lambda}{2}\right) \beta_{\alpha}(\mathrm{dist}_{\mathcal{X}}(x_{\alpha}, z_{\alpha}), t_{\alpha} - s_{\alpha}).$$

By the maximum principle for (62) and (12), both suprema are bounded by M. Hence,

$$\beta_{\alpha}(\mathrm{dist}_{\mathcal{X}}(x_{\alpha}, z_{\alpha}), t_{\alpha} - s_{\alpha}) \geq \frac{6M + 2\lambda}{10M + \lambda} \geq \frac{3}{5}.$$

530 This implies

$$1 - \frac{\mathrm{dist}_{\mathcal{X}}(x_{\alpha}, z_{\alpha})^2}{2L_{\mathcal{X}}^2 \alpha^2} - \frac{|t_{\alpha} - s_{\alpha}|^2}{2T^2 \alpha^2} \geq \frac{3}{5},$$

531 so that

$$\mathrm{dist}_{\mathcal{X}}(x_{\alpha}, z_{\alpha})^2 + |t_{\alpha} - s_{\alpha}|^2 \leq C^{\mathrm{st}} \alpha^2, \tag{65}$$

532 where $C^{\mathrm{st}} > 0$ is independent of $\alpha$. Moving on to the main part of the proof, we separate cases.

533 **Case 1**, $t_{\alpha} > 0$ and $s_{\alpha} > 0$. This case follows arguments similar to those used in comparison principles proofs.

534 Consider the test function

$$\Phi_1(t, x) = \frac{\lambda}{4T}(t + s_{\alpha}) - \left(5M + \frac{\lambda}{2}\right) \beta_{\alpha}(\mathrm{dist}_{\mathcal{X}}(x, z_{\alpha}), t - s_{\alpha}).$$

535 The function $v - \Phi_1$ attains its maximum at $(t_{\alpha}, x_{\alpha})$. Since $t_{\alpha} > 0$ and $v$ is a viscosity subsolution, it follows that

$$\frac{\lambda}{4T} + \left(5M + \frac{\lambda}{2}\right) \frac{t_{\alpha} - s_{\alpha}}{T^2 \alpha^2} + \frac{1}{2} \left(5M + \frac{\lambda}{2}\right) \frac{\| \mathrm{Log}_{x_{\alpha}} z_{\alpha} \|_{x_{\alpha}}^2}{L_{\mathcal{X}}^2 \alpha^2} \leq 0.$$

537 Now consider the test function

$$\Phi_2(s, z) = -\frac{\lambda}{4T}(t_{\alpha} + s) + \left(5M + \frac{\lambda}{2}\right) \beta_{\alpha}(\mathrm{dist}_{\mathcal{X}}(x_{\alpha}, z), t_{\alpha} - s).$$

538 The function $v^{\varepsilon} - \Phi_2$ attains its minimum at $(s_{\alpha}, z_{\alpha})$. Since $s_{\alpha} > 0$, it follows that

$$\partial_t v^{\varepsilon}(s_{\alpha}, z_{\alpha}) + \frac{1}{2} \| \mathrm{grad} v^{\varepsilon}(s_{\alpha}, z_{\alpha}) \|_{z_{\alpha}}^2 = \partial_t \Phi_2(s_{\alpha}, z_{\alpha}) + \frac{1}{2} \| \mathrm{grad} \Phi_2(s_{\alpha}, z_{\alpha}) \|_{z_{\alpha}}^2,$$

539 and

$$\Delta v^{\varepsilon}(s_{\alpha}, z_{\alpha}) \geq \Delta \Phi_2(s_{\alpha}, z_{\alpha}).$$

540 Invoking the equation satisfied by $v^{\varepsilon}$, we obtain

$$-\frac{\lambda}{4T} + \left(5M + \frac{\lambda}{2}\right) \frac{t_{\alpha} - s_{\alpha}}{T^2 \alpha^2} + \frac{1}{2} \left(5M + \frac{\lambda}{2}\right) \frac{\| - \mathrm{Log}_{z_{\alpha}} x_{\alpha} \|_{z_{\alpha}}^2}{L_{\mathcal{X}}^2 \alpha^2} \geq \frac{\varepsilon}{2} \Delta \Phi_2(s_{\alpha}, z_{\alpha}).$$

541 Putting these inequalities together, we find

$$\frac{\lambda}{2T} \leq \frac{\varepsilon}{\alpha^2} \frac{1}{4L_{\mathcal{X}}^2} (\Delta_x \mathrm{dist}_{\mathcal{X}}(x, z_{\alpha})^2)|_{x=x_{\alpha}}.$$

542 Taking $\alpha = \varepsilon^{1/4}$ yields the desired estimate.
Before moving on to the other cases, we need estimates on $\text{dist}_X(x_\alpha, z_\alpha)$ and $|t_\alpha - s_\alpha|$. This will be useful when invoking Lipschitz continuity of $v$ and Lemma D.2.

**Bounds on** $\text{dist}_X(x_\alpha, z_\alpha)$ **and** $|t_\alpha - s_\alpha|$. Differentiating $\beta_\alpha$, we obtain

$$ \frac{\partial \beta_{\alpha}}{\partial \xi} (\text{dist}_{\chi}(x_{\alpha}, z_{\alpha}), t_{\alpha} - s_{\alpha}) = - \frac{\text{dist}_{\chi}(x_{\alpha}, z_{\alpha})}{L_{\chi}^2 \alpha^2}, $$

and

$$ \frac{\partial \beta_{\alpha}}{\partial t} (\text{dist}_{\chi}(x_{\alpha}, z_{\alpha}), t_{\alpha} - s_{\alpha}) = - \frac{t_{\alpha} - s_{\alpha}}{T^2 \alpha^2}. $$

In other words, it suffices to bound these derivatives to obtain estimates on $\text{dist}_X(x_\alpha, z_\alpha)$ and $|t_\alpha - s_\alpha|$. Observe that the function

$$ x \mapsto v(t_{\alpha}, x) + \left(5M + \frac{\lambda}{2}\right) \beta_{\alpha} (\text{dist}_{\chi}(x, z_{\alpha}), t_{\alpha} - s_{\alpha}), $$

attains its maximum at $x = x_\alpha$. Consequently,

$$ \left(5M + \frac{\lambda}{2}\right) \left[ \beta_{\alpha}(\xi, t_{\alpha} - s_{\alpha}) - \beta_{\alpha}(\xi_{\alpha}, t_{\alpha} - s_{\alpha}) \right] \leq v(t_{\alpha}, x_{\alpha}) - v(t_{\alpha}, x) \leq L_{v}^{x} \text{dist}(x, x_{\alpha}), $$

where $\xi = \text{dist}_X(x, z_\alpha)$ and $\xi_\alpha = \text{dist}_X(x_\alpha, z_\alpha)$, and $L_v^x$ is the Lipschitz constant of $v$ with respect to $x$. We may choose $x$ on the geodesic joining $x_\alpha$ and $z_\alpha$ so that $\text{dist}(x, x_\alpha) = |\xi_\alpha - \xi|$. Taking the limit $x \to x_\alpha$ gives

$$ \left(5M + \frac{\lambda}{2}\right) \left| \frac{\partial \beta_{\alpha}}{\partial \xi} (\text{dist}_{\chi}(x_{\alpha}, z_{\alpha}), t_{\alpha} - s_{\alpha}) \right| \leq L_{v}^{x}. $$

It follows that

$$ \text{dist}_{\chi}(x_{\alpha}, z_{\alpha}) \leq \alpha^2 \frac{2L_{\chi}^2 L_v^x}{10M + \lambda}. \qquad (66) $$

Similarly, observe that the function

$$ t \mapsto v(t, x_{\alpha}) - \frac{\lambda}{4T}(t + s_{\alpha}) + \left(5M + \frac{\lambda}{2}\right) \beta_{\alpha}(\xi_{\alpha}, t - s_{\alpha}), $$

attains its maximum at $t = t_\alpha$. It follows that, if $t_\alpha > 0$ and for $h > 0$ small enough,

$$ \left(5M + \frac{\lambda}{2}\right) \left[ \beta_{\alpha}(\xi_{\alpha}, t_{\alpha} - h - s_{\alpha}) - \beta_{\alpha}(\xi_{\alpha}, t_{\alpha} - s_{\alpha}) \right] \leq v(t_{\alpha}, x_{\alpha}) - v(t_{\alpha} - h, x) - \frac{\lambda}{4T}h. $$

Therefore,

$$ -\left(5M + \frac{\lambda}{2}\right) \frac{\partial \beta_{\alpha}}{\partial t} (\text{dist}_{\chi}(x_{\alpha}, z_{\alpha}), t_{\alpha} - s_{\alpha}) \leq L_{v}^{t} - \frac{\lambda}{4T}. $$

where $L_v^t$ is the Lipschitz constant of $v$ with respect to $t$. Similarly, if $t_\alpha < T$, we obtain

$$ \left(5M + \frac{\lambda}{2}\right) \frac{\partial \beta_{\alpha}}{\partial t} (\text{dist}_{\chi}(x_{\alpha}, z_{\alpha}), t_{\alpha} - s_{\alpha}) \leq L_{v}^{t} + \frac{\lambda}{4T}. $$

We conclude that

$$ |t_{\alpha} - s_{\alpha}| \leq \alpha^2 \frac{2T^2(L_v^t - \lambda/(4T))}{10M + \lambda}, \quad \text{if } t_{\alpha} = T, \qquad (67) $$

and

$$ |t_{\alpha} - s_{\alpha}| \leq \alpha^2 \frac{2T^2(L_v^t + \lambda/(4T))}{10M + \lambda}, \quad \text{if } 0 < t_{\alpha} < T. \qquad (68) $$
**Case 2**, $s_\alpha = 0$ and $t_\alpha \ge 0$. From (64) we have

$$5M + \lambda \le M_\alpha \le |v(t_\alpha, x_\alpha) - v(t_\alpha, z_\alpha)| + |v(t_\alpha, z_\alpha) - v^\epsilon(0, x_\alpha)| + 5M + \frac{\lambda}{2}.$$

Since $v$ and $v^\epsilon$ coincide at $t=0$, it follows that

$$\frac{\lambda}{2} \le |v(t_\alpha, x_\alpha) - v(t_\alpha, z_\alpha)| + |v(t_\alpha, z_\alpha) - v(0, x_\alpha)|,$$

and we may invoke the Lipschitz continuity of $v$ to deduce there exists a constant $C^{\text{st}} > 0$ independent of $\alpha$ such that

$$\frac{\lambda}{2} \le C^{\text{st}} (\text{dist}_\chi(x_\alpha, z_\alpha) + t_\alpha).$$

The estimate (66) holds, and either $t_\alpha = 0$ or one of (67)–(68) applies. Therefore, we obtain

$$\lambda \le \tilde{C}^{\text{st}} \alpha^2,$$

where $\tilde{C}^{\text{st}}$ is independent of $\alpha$. Taking $\alpha = \varepsilon^{1/4}$ yields the desired result.

**Case 3**, $t_\alpha = 0$ and $s_\alpha > 0$. From $M_\alpha \ge \psi_\alpha(0, x_\alpha, 0, z_\alpha)$ we get

$$5M [\beta_\alpha (\text{dist}_\chi(x_\alpha, z_\alpha), 0) - \beta_\alpha (\text{dist}_\chi(x_\alpha, z_\alpha), -s_\alpha)] \le v^\epsilon(0, z_\alpha) - v^\epsilon(s_\alpha, z_\alpha).$$

We invoke (63) and the definition of $\beta_\alpha$ to get the estimate

$$\frac{5M s_{\alpha}^{2}}{2T^{2}} \leq C_{1}^{\text{st}}(\sqrt{\varepsilon}\sqrt{s_{\alpha}} + s_{\alpha}).$$

This is a biquadratic inequation in the variable $\sqrt{s_\alpha}$ from which we can get the estimate

$$s_{\alpha} \le C^{\text{st}} \left( \varepsilon^{\frac{1}{3}} \alpha^{\frac{4}{3}} + \alpha^2 \right), \qquad (69)$$

with $C^{\text{st}}$ independent of $\alpha$.

Now, we can produce a similar estimate as in the previous case. We have

$$5M + \lambda \le M_\alpha \le |v^\epsilon(0, x_\alpha) - v^\epsilon(0, z_\alpha)| + |v^\epsilon(0, z_\alpha) - v^\epsilon(s_\alpha, z_\alpha)| + 5M + \frac{\lambda}{2}.$$

As previously, it follows that

$$\frac{\lambda}{2} \le |v(0, x_\alpha) - v(0, z_\alpha)| + |v^\epsilon(0, z_\alpha) - v^\epsilon(s_\alpha, z_\alpha)|.$$

Using Lipschitz continuity of $v$, estimate (66) and Lemma D.2 combined with (69), it follows that

$$\frac{\lambda}{2} \le C_2^{\text{st}} \alpha^2 + C_3^{\text{st}} (\sqrt{\varepsilon}\sqrt{s_\alpha} + s_\alpha),$$

where $C_2^{\text{st}}$ and $C_3^{\text{st}}$ are independent of $\alpha$ and $\varepsilon$. Taking $\alpha = \varepsilon^{1/4}$, we conclude that there exists a constant $\tilde{C}^{\text{st}} > 0$

independent of $\varepsilon$ such that

$$\lambda \le \tilde{C}^{\text{st}} \varepsilon^{\frac{1}{2}}.$$
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