Regularity results for the minimum time function with H\"ormander vector fields

TL;DR

This paper investigates the regularity of the minimum time function solving the eikonal equation with Hörmander vector fields, characterizing singular trajectories and establishing conditions for Lipschitz continuity and semiconcavity.

Contribution

It provides a characterization of singular trajectories and links their absence to Lipschitz and semiconcavity properties of the solution, with applications to specific vector fields.

Findings

Singular trajectories are characterized as points where the solution loses Lipschitz continuity.

Lipschitz continuity, semiconcavity, and absence of singular trajectories are equivalent.

The absence of singular trajectories holds when the characteristic set is a symplectic manifold.

Abstract

In a bounded domain of $\mathbb{R}^n$ with smooth boundary, we study the regularity of the viscosity solution, $T$, of the Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\ldots ,X_N\}$, subject to H\"ormander's bracket generating condition. Due to the presence of characteristic boundary points, singular trajectories may occur in this case. We characterize such trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. We then prove that the local Lipschitz continuity of $T$, the local semiconcavity of $T$, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied when the characteristic set of $\{X_1,\ldots ,X_N\}$ is a symplectic manifold. We apply our results to Heisenberg's and Martinet's vector fields.

Equations

$$\left\{\begin{array}[]{l}\sum_{j=1}^{N}(X_{j}T)^{2}(x)=1\quad\text{ in }\quad\Omega\vspace{.2cm}\\ T=0\quad\text{ on }\quad\Gamma,\end{array}\right.$$

$$\{\begin{array}{ll}{y}^{\prime }(t)={\sum }_{j=1}^N{u}_j(t)X_j(y(t))& (t\ge 0)\\ y(0)=x,& \end{array}$$

$$X(x)=\sum _{i=1}^n{f}_i(x){\partial }_{x_i}\text{mbox}andY(x)=\sum _{i=1}^n{g}_i(x){\partial }_{x_i}(x\in \Omega ),$$

$$[X,Y](x)=\sum_{i=1}^{n}h_{i}(x)\partial_{x_{i}}\quad\mbox{where}\quad h_{i}=\sum_{j=1}^{n}\Big{(}f_{j}\partial_{x_{j}}g_{i}-g_{j}\partial_{x_{j}}f_{i}\Big{)}.$$

$$\text{mbox}Lie(\{X_i{\}}_{i=1}^N)=\bigcup _{k=1}^{\infty }\text{mbox}{Lie}^k(\{X_i{\}}_{i=1}^N),$$

$$\text{mbox}{Lie}^1(\{X_i{\}}_{i=1}^N)=\mathrm{span}\{X_i{\}}_{i=1}^N$$

$$\mbox{Lie}^{k+1}(\{X_{i}\}_{i=1}^{N})\\ =\operatorname{span}\Big{(}\mbox{Lie}^{k}(\{X_{i}\}_{i=1}^{N})\cup\Big{\{}[X,X_{j}]~{}:~{}X\in\mbox{Lie}^{k}(\{X_{i}\}_{i=1}^{N}),\,j=1,\dots,N\Big{\}}\Big{)}.$$

$$\begin{cases}\mbox{Lie}^{k}(\{X_{i}\}_{i=1}^{N})[x]=\big{\{}X(x)~{}:~{}X\in\mbox{Lie}^{k}(\{X_{i}\}_{i=1}^{N})\big{\}}&\forall k\geq 1\vspace{.2cm}\\ \mbox{Lie}(\{X_{i}\}_{i=1}^{N})[x]=\big{\{}X(x)~{}:~{}X\in\mbox{Lie}(\{X_{i}\}_{i=1}^{N})\big{\}}.\end{cases}$$

$$\text{mbox}Lie(\{X_i{\}}_{i=1}^N)[x]={\mathbb{R}}^n\forall x\in \Omega .$$

$$\mathrm{span}\{X_i{\}}_{i=1}^N[x]+\mathrm{span}{\left\{\sum _{j=1}^N[X_j,X_i]\right\}}_{i=1}^N[x]={\mathbb{R}}^n\forall x\in \Omega .$$

$$\begin{array}[]{ll}(H1)&\mbox{\em$\Omega\subset{\mathbb{R}}^{n}$ is a bounded open set with boundary $\Gamma$ of class $C^{\infty}$},\\ (H2)&\mbox{\em$\{X_{1},\ldots,X_{N}\}$ is a system of H\"{o}rmander vector fields on $\overline{\Omega}$.}\end{array}$$

$$h(x,p)=\sum _{j=1}^N{X}_j(x,p{)}^2,(x,p)\in T^{\ast }{\Omega }^{\prime },$$

$$X_j(x,p):=⟨X_j(x),p⟩\forall (x,p)\in T^{\ast }{\Omega }^{\prime },\forall j=1,\dots ,N$$

$$\operatorname{Char}(X_{1},\ldots X_{N})=\big{\{}(x,p)\in\Omega^{\prime}\times({\mathbb{R}}^{n}\setminus\{0\})~{}:~{}h(x,p)=0\big{\}}.$$

$$\left\{\begin{array}[]{l}\sum_{j=1}^{N}(X_{j}T)^{2}(x)=1\quad\text{ in }\quad\Omega\vspace{.2cm}\\ T=0\quad\text{ on }\quad\Gamma\end{array}\right.$$

$$u=(u_1,\dots ,u_N):[0,+\infty [\to {\mathbb{R}}^N,$$

$$\{\begin{array}{ll}{y}^{\prime }(t)={\sum }_{j=1}^N{u}_j(t)X_j(y(t))& (t\ge 0)\\ y(0)=x.& \end{array}$$

$$\tau_{\Gamma}(x,u)=\inf\big{\{}t\geq 0~{}:~{}y^{x,u}(t)\in\Gamma\big{\}}.$$

$$T(x)=\underset{u(\cdot )}{\inf }{\tau }_{\Gamma }(x,u)(x\in \overline{\Omega }),$$

$$k(x)=\min\big{\{}k\geq 1~{}:~{}\mbox{Lie}^{k}j(\{X_{i}\}_{i=1}^{N})[x]={\mathbb{R}}^{n}\big{\}}\qquad(x\in\Omega).$$

$$r_{\Omega }=\max \{k(x):x\in \overline{\Omega }\}.$$

$$⟨A(x)DT(x),DT(x)⟩=1\text{ in }\Omega ,$$

$$N_{\Gamma }(z):=\{\lambda \nu (z):\lambda \ge 0\}.$$

$$p_k^{\prime }(t)=-\sum _{j=1}^N{u}_j(t)⟨{\partial }_{x_k}{X}_j(y(t)),p(t)⟩,⟨X_k(y(t)),p(t)⟩=0,$$

$$p(T(x))\in N_{\Gamma }(y(T(x))).$$

$$H(x,p,u)=\sum _{j=1}^N{u}_j{X}_j(x,p),$$

$$\left\{\begin{array}[]{l}y^{\prime}(t)=D_{p}H(y(t),p(t),u(t))\vspace{.2cm}\\ p^{\prime}(t)=-D_{x}H(y(t),p(t),u(t))\vspace{.2cm}\end{array}\right.$$

$$H(y^{x,u}(t),p(t),u(t))=\max_{u\in\overline{B}_{1}(0)}\sum_{j=1}^{N}u_{j}\langle X_{j}(y^{x,u}(t)),p(t)\rangle\\ =\Big{(}\sum_{j=1}^{N}\big{\langle}X_{j}(y^{x,u}(t)),p(t)\big{\rangle}^{2}\Big{)}^{1/2}=\sqrt{h(y^{x,u}(t),p(t))}.$$

$$h(y^{x,u}(T(x)),p(T(x)))\ne 0$$

$$|f(x)-f(x_0)|\le L|x-x_0|\forall x\in U\cap \overline{\Omega }.$$

Full text

This work was started during the “Workshop on Hamilton-Jacobi equations” held at Fudan University, Shanghai, China (July 24-30, 2016). The first author was supported by RFO grant, Università di Bologna, Italy. The second author was partly supported by the University of Rome Tor Vergata (Consolidate the Foundations 2015) and the Istituto Nazionale di Alta Matematica “F. Severi” (GNAMPA 2016 Research Projects). The third author gratefully acknowledges the financial support of the “Vienna Graduate School on Computational Optimization” which is funded by Austrian Science Foundation (FWF, project no. W1260-N35).

1. Introduction

In a bounded open set $\Omega\subset{\mathbb{R}}^{n}$ with smooth boundary, $\Gamma$, we study the regularity of the viscosity solution, $T$, to the Dirichlet problem

[TABLE]

where $\{X_{1},\ldots,X_{N}\}$ is a system of smooth vector fields which satisfies Hörmander’s bracket generating condition ([14]). It is well known that such a solution may be interpreted as the sub-Riemannian distance from the boundary of $\Omega$ when $\{X_{1},\ldots,X_{N}\}$ span a distribution or, in control theory, as a certain minimum time function. While structural properties of the singular set and the regularity of the Riemannian distance function have been widely investigated (see, e.g., [2], [5], [6], [8], [9]), similar issues have not been much addressed for the solution of (2.7) in the sub-Riemannian case or, more generally, when the quadratic form associated with the eikonal equation in (2.7) fails to be positive definite (see, e.g., [3]). The analysis of such questions is the main purpose of this paper.

In our approach, we will often use the characterization of the unique solution, $T$, of (2.7) as the value function of the time optimal control problem with target $\Gamma$ and state equation

[TABLE]

where $u:[0,+\infty[\to\overline{B}_{1}(0)$. In particular, a crucial role in our investigation will be played by the so-called singular time-optimal trajectories of (1.2), a well-known object in geometric control theory (see, e.g., [10]). Singular trajectories are time-optimal trajectories that can be completed to a solution of the characteristic system associated with (1.1), which belongs to the characteristic set of $\{X_{1},\ldots,X_{N}\}$ and satisfies a suitable transversality condition (see Definition 3.1 below). Equivalently, singular time-optimal trajectories can be identified as those time-optimal trajectories that hit $\Gamma$ at a characteristic point, that is, a point at which $X_{1},\ldots,X_{N}$ are tangent to $\Gamma$ (see Theorem 3.1 below).

The fact that the existence of singular trajectories may destroy the smoothness (subanalyticity) of a solution of a first order Hamilton-Jacobi equation was already observed in [19], [1], and [21] (see also Remark 4.3 below). In this paper, we interested in a more basic smoothness threshold—namely Lipschitz continuity—which, as we explain below, opens the way to higher regularity properties for the solution of (2.7). Therefore, we begin our analysis by giving a necessary and sufficient condition for point-wise Lipschitz continuity, showing that $T$ fails to be Lipschitz continuous at a point $x\in\Omega$ if and only if $x$ is the starting point of a singular time-optimal trajectory (Theorem 3.2). By dynamic programming, this result implies that the set of all singular time-optimal trajectories consists—as a point set—of all the points of $\overline{\Omega}$ at which $T$ fails to be Lipschitz (Corollary 3.1).

We note that point-wise Lipschitz continuity is a very weak property. Nevertheless, Theorem 3.2 can be used to derive a classical Lipschitz regularity result whenever one can exclude the presence of singular time-optimal trajectories. This is the viewpoint leading to Theorem 4.1, which ensures the equivalence of the following three properties:

• (a)

system (1.2) admits no singular time-optimal trajectory;

• (b)

$T$ locally is semiconcave in $\Omega$;

• (c)

$T$ is locally Lipschitz in $\Omega$.

We observe that, in [7], the local semiconcavity of the sub-Riemannian distance to a point $x_{0}$, $d_{SR}(x_{0},\cdot)$, is proved assuming the absence of singular time-optimal trajectories, but without giving any estimate of the dependence on $x_{0}$ of the semiconcavity constant of $d_{SR}(x_{0},\cdot)$. Thus, the local semiconcavity of the minimum time function for a general target (i.e., (a) $\Rightarrow$ (b)) is not a direct consequence of the result of [7], even when $\{X_{1},\ldots,X_{N}\}$ generates a distribution.

We conclude the introduction by quickly mentioning further results that complete our exposition, while outlining the structure of this paper. In Section 2, we recall all required notions concerning Hörmander vector fields, characteristic sets, time optimal control problems, and viscosity solutions of related eikonal equations. In Section 3, we introduce singular time-optimal trajectories and develop our analysis of the behaviour of $T$ along such trajectories. Based on these results, in Section 4, we obtain the aforementioned characterization of the local Lipschitz regularity of $T$, we give sufficient conditions on the characteristic set of $\{X_{1},\ldots,X_{N}\}$ in order to exclude the presence of singular trajectories, and we discuss the examples of Heisenberg’s and Martinet’s vector fields as applications of our results. We conclude the paper with Appendix A, where we prove a regularity result for $T$ at the boundary of $\Omega$ (Theorem 4.2), which might be known to the reader in other forms than the one which fits the analysis of this paper.

2. Hörmander vector fields and minimum time function

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded open set and assume that the boundary of $\Omega$, $\Gamma$, is a smooth manifold of dimension $n-1$.

We denote by $VF(\Omega)$ the space of all $C^{\infty}$ vector fields on $\Omega$. Given any two such vector fields,

[TABLE]

with $f_{i},g_{i}\in C^{\infty}(\Omega)\,(i=1,\dots,n)$, we denote by $[X,Y]$ the Lie bracket

[TABLE]

Let $N\geq 2$ be an integer and let $X_{1},\ldots,X_{N}\in VF(\Omega)$. The Lie algebra generated by $\{X_{i}\}_{i=1}^{N}$, Lie$(\{X_{i}\}_{i=1}^{N})$, is the smallest subspace of $VF(\Omega)$, containing $\{X_{i}\}_{i=1}^{N}$, which is invariant under the action of Lie brackets. So, we have that

[TABLE]

where $\mbox{Lie}^{k}(\{X_{i}\}_{i=1}^{N})$ is defined recursively by taking

[TABLE]

and, for $k\geq 1$,

[TABLE]

For all $x\in X$ we also set

[TABLE]

We say that $\{X_{i}\}_{i=1}^{N}\subset VF(\Omega)$ is a system of Hörmander vector fields on $\Omega$ if the following bracket generating condition holds:

[TABLE]

Moreover, we say that $\{X_{i}\}_{i=1}^{N}$ is a system of Hörmander vector fields on $\overline{\Omega}$ if the bracket generating condition holds true on some open set $\Omega^{\prime}\supset\overline{\Omega}$. Finally, we say that $\{X_{i}\}_{i=1}^{N}$ is strongly bracket generating on $\Omega$ if for every $v=(v_{1},\ldots,v_{N})\in{\mathbb{R}}^{n}\setminus\{0\}$

[TABLE]

The following hypotheses $(H)$ will be assumed throughout:

[TABLE]

Let us point out that, here, $X_{1},\ldots,X_{N}$ need not be linearly independent, nor we suppose $N<n$.

We define the Hamiltonian associated with $\{X_{1},\ldots,X_{N}\}$ by

[TABLE]

where we have set, with a slight abuse of notation,

[TABLE]

(note that $T^{*}\Omega^{\prime}$ can be identified with $\Omega^{\prime}\times{\mathbb{R}}^{n}$).

The characteristic set of $\{X_{1},\ldots,X_{N}\}$ is given by

[TABLE]

Finally, a point $x\in\Gamma$ is called characteristic if the linear space generated by $X_{1}(x),\ldots,X_{N}(x)$ is contained in the tangent space to $\Gamma$ at $x$. We denote by $E\subset\Gamma$ the set of all characteristic points. The following result, essential for this paper, is due to Derridj [11].

Theorem 2.1**.**

Under assumption $(H)$, $E$ is a closed subset of $\Gamma$ of $(n-1)$-dimensional Hausdorff measure zero.

Under assumption $(H)$, consider the Dirichlet problem

[TABLE]

and observe that, since the Hamiltonian $h(x,p)$ is not strictly convex in $p$, characteristic points may appear. It is well known that (2.7) admits a unique continuous viscosity solution $T:\overline{\Omega}\to{\mathbb{R}}$. Indeed, taking $\Gamma$ as the target set, the minimum time function associated with $\{X_{1},\ldots,X_{N}\}$ is a solution of the above equation. Such a function is defined as follows. Given $x\in\overline{\Omega}$ and a measurable control

[TABLE]

taking values in the unit ball of ${\mathbb{R}}^{N}$, let us denote by $y^{x,u}(\cdot)$ the unique solution of the Cauchy problem

[TABLE]

Define the transfer time to $\Gamma$ as

[TABLE]

Clearly, $\tau_{\Gamma}(x,u)\in[0,+\infty]$. The Minimum Time Problem with target $\Gamma$ is the following:

• (MTP)

To minimize $\tau_{\Gamma}(x,u)$ over all controls $u:[0,+\infty[\to\overline{B}_{1}(0)$.

Then, the minimum time function, defined as

[TABLE]

turns out to be the unique viscosity solution of the Dirichlet problem (2.7) (see, e.g., [4]). It is well-known that Hörmander’s bracket generating condition implies that (2.8) is small time locally controllable. Hence, $T$ is finite and continuous (see, e.g., [4]).

Remark 2.1**.**

We recall that a $u(\cdot)$ is called an optimal control at a point $x\in\Omega$ if $T(x)=\tau_{\Gamma}(x,u)$. The corresponding solution of (2.8), $y^{x,u}$, is called the time-optimal trajectory at $x$ associated with $u$.

A more precise description of the continuity properties of $T$ can be given by looking at the maximal length of commutators needed to generate ${\mathbb{R}}^{n}$, that is, the function $k(\cdot)$ defined as follows:

[TABLE]

We also define

[TABLE]

Then, thanks to the Baker-Campbell-Hausdorff formula (see, e.g., [18]), $T$ turns out to be Hölder continuous of exponent $1/r_{\Omega}$ on $\overline{\Omega}$ .

Next, observe that the equation in (2.7) can be recast in the form

[TABLE]

where $A(\cdot)$ is a suitable positive semidefinite $n\times n$ matrix with smooth entries and $DT=(\partial_{x_{1}}T,\dots,\partial_{x_{n}}T)$. If $A(\cdot)$ is nondegenerate, then a classical viscosity argument ensures that any continuous viscosity solution of (2.11) is locally Lipschitz continuous in $\Omega$. Furthermore, under very weak regularity conditions222It suffices to assume that $x\mapsto\langle A(x)p,p\rangle$ is semiconvex (i.e. it can be locally written as the sum of a convex with a smooth function) uniformly w.r.t. $p$ in the unit sphere of ${\mathbb{R}}^{n}$. on the entries of $A(\cdot)$, one can show that any continuous viscosity solution of (2.11) is locally semiconcave in $\Omega$ (see [2]). We recall that a function $T:\Omega\to{\mathbb{R}}$ is locally semiconcave in $\Omega$ if for every $V\Subset\Omega$ there exists a constant $C$ such that $D^{2}T\leq CI$ in $\mathcal{D}^{\prime}(V)$ (in the sense of quadratic forms). We point out that, for the aforementioned regularity results, the strict convexity of the map $p\mapsto\langle A(x)p,p\rangle$ is a crucial assumption.

On the contrary, when $A(\cdot)$ in (2.11) is associated with a system of Hörmander’s vector fields with $N<n$, known results mostly concern the continuity of solutions. For instance, it is known that any continuous viscosity solution of (2.11) is Hölder continuous of exponent $1/r_{\Omega}$ given by (2.10) (see, e.g., [12]). On the other hand, stronger regularity results are known under more restrictive assumption. For instance, as we recalled in the introduction, the following semiconcavity result holds for the sub-Riemannian setting ([7]): if $X_{1}(x),\ldots,X_{N}(x)$ are linearly independent for every $x\in{\mathbb{R}}^{n}$ and strongly bracket generating on ${\mathbb{R}}^{n}$, then for every $x_{0}\in{\mathbb{R}}^{n}$ the minimum time function associated with the control system (2.8) and target $\{x_{0}\}$ is locally semiconcave—hence, locally Lipschitz—in ${\mathbb{R}}^{n}\setminus\{x_{0}\}$ (see also Remark 4.3 below).

3. Singular time-optimal trajectories

The notion of singular trajectory that we give below plays a key role in our analysis. For any boundary point $z\in\Gamma$ we denote by $\nu(z)$ the outward unit normal to $\Gamma$ at $z$ and we set

[TABLE]

Definition 3.1**.**

Let $x\in\Omega$ and let $y(\cdot)=y^{x,u}(\cdot)$ be a time-optimal trajectory, with $u:[0,T(x)]\to\overline{B}_{1}(0)$. We say that $y(\cdot)$ is singular if there exists an absolutely continuous arc $p:[0,T(x)]\to{\mathbb{R}}^{n}\setminus\{0\}$ such that, for a.e. $t\in[0,T(x)]$,

[TABLE]

for every $k=1,\ldots,N$, and

[TABLE]

Remark 3.2**.**

(i) Our use of the adjective ”singular” in Definition 3.1 is classical but it is also motivated by Proposition 3.1 below.

(ii) We recall that the strong bracket generating condition (2.4) ensures the absence of singular time-optimal trajectories (see, e.g., [7]).

(iii) We point out that, introducing the Control Theory Hamiltonian

[TABLE]

the optimal triple $(y,u,p)$ arising from Definition 3.1 satisfies, for a.e. $t\in[0,T(x)]$, the Hamiltonian system

[TABLE]

In other words, a time-optimal trajectory is singular if it can be lifted in the phase space in such a way that the lifted trajectory:

• •

satisfies the Hamiltonian system (3.4) (with Hamiltonian given by (3.3)) together with the transversality condition (3.2), and

• •

lies in the characteristic set $\operatorname{Char}(X_{1},\ldots,X_{N})$.

The following characterization of singular time-optimal trajectories will be used throughout the paper.

Theorem 3.1**.**

Assume $(H)$. Let $x\in\Omega$ and let $y^{x,u}$ be a time-optimal trajectory. Then, $y^{x,u}$ is singular if and only if $y^{x,u}(T(x))\in E$.

Proof.

Let $x\in\Omega$ and let $u(\cdot)$ a time-optimal control. Then, by the Pontryagin maximum principle there exists an absolutely continuous function $p:[0,T(x)]\to{\mathbb{R}}^{n}\setminus\{0\}$ (adjoint state) such that the pair $(y^{x,u},p)$ satisfies the Hamiltonian system (3.4) and the transversality condition (3.2). Furthermore, for a.e. $t\in[0,T(x)]$,

[TABLE]

Also, the function $[0,T(x)]\ni t\mapsto h(y^{x,u}(t),p(t))$ is constant.

Now, suppose $y^{x,u}(T(x))\notin E$. Then

[TABLE]

So, by the constancy of the Hamiltonian, $y^{x,u}$ cannot be singular.

Vice versa, assume $y^{x,u}(T(x))\in E$. Then, again by Pontryagin’s principle, we deduce that $(y^{x,u}(t),p(t))\in\operatorname{Char}(X_{1},\ldots,X_{N})$, for every $t\in[0,T(x)]$, that is, $y^{x,u}$ is a singular time-optimal trajectory. ∎

The following is a point-wise notion of Lipschitz continuity (see also Federer [13]).

Definition 3.2**.**

We say that a function $f:\overline{\Omega}\to{\mathbb{R}}$ is Lipschitz at a point $x_{0}\in\overline{\Omega}$ if there exists a neighbourhood $U$ of $x_{0}$ and a constant $L\geq 0$ such that

[TABLE]

Equivalently, $f$ is Lipschitz at $x_{0}$ if and only if

[TABLE]

Observe that a function may well-be Lipschitz continuous at $x_{0}$ without being Lipschitz on any neighbourhood of $x_{0}$. However, the interest of such kind of continuity is made clear by our next result.

Theorem 3.2**.**

Assume $(H)$ and let $x_{0}\in\Omega$. Then $T$ fails to be Lipschitz at $x_{0}$ if and only if there exists a singular time-optimal $y^{x_{0},u}$ .

For the proof Theorem 3.2 we borrow the following notion from nonsmooth analysis.

Definition 3.3** (Proximal normals).**

Let $S\subset{\mathbb{R}}^{n}$ be a closed set. A vector $v\in{\mathbb{R}}^{n}$ is called a proximal normal to $S$ at $x$ if there exist $\delta>0$ and $C>0$ (possibly depending on $x,v$) such that

[TABLE]

The set of all proximal normals to $S$ at $x$ will be denoted by $N^{P}_{S}(x)$.

We recall that the the hypograph of a function $f:\overline{\Omega}\to{\mathbb{R}}$ is the set

[TABLE]

Definition 3.4** (Horizontal proximal supergradients).**

The set of horizontal proximal subgradients of $f$ at a point $x\in\Omega$ is given by

[TABLE]

We note that, in order for $f$ to be Lipschitz at a point $x_{0}\in\Omega$, it is necessary (but not sufficient) that $\partial^{P,\infty}f(x_{0})=\{0\}$.

Proof of Theorem 3.2.

Assuming that $y^{x_{0},u}$ is a singular time-optimal trajectory, let us show that $\partial^{P,\infty}T(x_{0})$ contains a nonzero vector, which implies that $T$ cannot be Lipschitz at $x_{0}$. Here we adapt, to systems of Hörmander’s vector fields, an argument used in [5, Theorem 4.1] to study time optimal control problems for differential inclusions. We observe that the present context is different from the one in [5] in that the nondegeneracy assumption of [5] is not satisfied by our control system. We will use the dual arc $p(\cdot)$ for $y^{x_{0},u}$ provided by Pontryagin’s principle. Observe that, in view of (3.2), $\exists\,\delta>0$ such that

[TABLE]

or

[TABLE]

that is, $(p(T(x_{0})),0)\in N_{{\rm hypo}(T)}(y^{x_{0},u}(T(x_{0})),0)$.

We now proceed to show that

[TABLE]

or, equivalently,

[TABLE]

We have to prove that there exists a constant $C_{1}>0$ such that, for every $x\in\overline{\Omega}$ with $|T(x)-T(x_{0})|<1$, we have that

[TABLE]

Set $T_{0}=T(x_{0})$ and $T_{1}=T(x)$. We shall analyse the two cases $T_{1}\leq T_{0}$ and $T_{1}>T_{0}$ separately.

Case 1: $T_{1}\leq T_{0}$

It suffices to verify (3.9) with $\alpha=T_{1}$. Set

[TABLE]

Notice that $T(\overline{x}_{0})=T_{0}-T_{1}\in[0,1[$ and $\overline{x}\in\overline{\Omega}$. Furthermore

[TABLE]

for a suitable constant $C_{1}>0$ depending only on $\Omega$, the $C^{1}$ norms of the vector fields $X_{1},\ldots,X_{N}$, and $\max_{\overline{\Omega}}T$. Then, we find that

[TABLE]

We now claim that, for every $t\in[0,T_{1}]$,

[TABLE]

for a suitable constant $C_{2}$ depending on $\Omega$, the $C^{2}$ norms of the vector fields $X_{1},\ldots,X_{N}$, and $\max_{\overline{\Omega}}T$. Indeed, recalling (3.1) we deduce that

[TABLE]

Hence, (3.12) follows.

Then, recalling (3.12) , (3.11), and (3.10), we find

[TABLE]

In order to bound the last term in the above inequality we observe that

[TABLE]

for a suitable constant $C_{4}$. Now, recall that

[TABLE]

Then, we have

[TABLE]

The second term can be estimated by (3.7):

[TABLE]

for some constant $C_{5}>0$. Moreover, we have that

[TABLE]

Recalling that $y^{x_{0},u}(T_{0})\in E$, i.e.

[TABLE]

we deduce that

[TABLE]

for some constant $C_{6}>0$. Thus, (3.9) follows.

Case 2: $T_{1}>T_{0}$

We have that

[TABLE]

The second term on the right-hand side of the last identity can be bounded by a constant times $|x-x_{0}|^{2}$ like in (3.12). In order to estimate the remaining term it suffices to observe that $y^{x,u}(T_{0})\in\overline{\Omega}$. So, (3.7) and (3.10) yield

[TABLE]

This shows that, if $y^{x_{0},u}$ is a singular time-optimal trajectory, then $T$ fails to be Lipschitz at $x_{0}$.

In order to prove the converse, we argue by contradiction by assuming that $T$ fails to be Lipschitz continuous at $x_{0}$ and that all optimal trajectories from $x_{0}$ are not singular. Let $y^{x_{0},u}$ be any time-optimal trajectory. So, $y^{x_{0},u}(T(x_{0}))\in\Gamma\setminus E$ by of Theorem 3.1. Then, appealing to Theorem 2.1 we can find $\delta>0$ such that $B_{\delta}(y^{x_{0},u}(T(x_{0})))\cap E=\emptyset.$ Furtheremore, taking $\delta$ small enough, we may assume that there exists a positive constant $C$ such that

[TABLE]

Since $T$ is not Lipschitz at $x_{0}$, there exists $\{x_{j}\}\subset\Omega$ such that

[TABLE]

Set

[TABLE]

Two possibilities my occur: either $T_{0}\leq T_{j}$ or $T_{0}>T_{j}$. Observe that, in both cases,

[TABLE]

where $C_{1}$ is a positive constant depending on $\Omega$, the $C^{1}$ norms of the vector fields $X_{1},\ldots,X_{N}$, and $\max_{\Omega}T$.

Case 1: $T_{0}\leq T_{j}$

We have that

[TABLE]

Since $y^{x_{0},u}(T_{0})\in\Gamma$, by (3.16) we deduce that, for $j$ large enough,

[TABLE]

Then, by (3.17) and (3.14), we find

[TABLE]

So, using (3.16) once again, we find the contradiction $j\leq CC_{1}.$

Case 2: $T_{j}<T_{0}$

Denote by $u_{j}$ a time-optimal control at $x_{j}$. After extending $u_{j}$ to $[0,T_{0}]$ by setting $u_{j}(t)\equiv 0$ for $t\in[T_{j},T_{0}]$, we choose a subsequence, still labeled $\{x_{j}\}$, such that $y^{x_{j},u_{j}}$ converges uniformly to $y^{x_{0},u_{0}}$ on all $[0,T_{0}]$, for a suitable $u_{0}(\cdot)$. Furthermore, because of $T_{j}\uparrow T_{0}$, we may assume that $y^{x_{j},u_{j}}(T_{j})\in\Gamma\cap B_{\delta}(y^{x_{0},u_{0}}(T_{0}))$ for $j$ large enough. Then, by dynamic programming, (3.15) yields

[TABLE]

Using (3.16) (with $u$ replaced by $u_{j}$) and (3.14) we find once more the contradiction

[TABLE]

This implies that $T$ cannot be Lipschitz continuous along $y^{x_{0},u_{0}}$.

In order to show that $y^{x_{0},u_{0}}$ is singular observe that otherwise $T$ would be Lipschitz continuous on a neighborhood of $y^{x_{0},u_{0}}(T(x_{0}))$. Then Lipschitz regularity would propagate backwards along $y^{x_{0},u_{0}}$, thus contradicting what we have just shown. ∎

Our next result is a straightforward consequence of Theorem 3.2 and dynamic programming.

Corollary 3.1**.**

Assume $(H)$. Let $x_{0}\in\Omega$ and let $y^{x_{0},u}$ be a singular time-optimal trajectory. Then, for any $t\in[0,T(x_{0})[$, $T$ fails to be Lipschitz continuous at $y^{x_{0},u}(t)$.

4. Regularity

In this section we study the interior and boundary regularity of the solution of equation (2.7). Here is the main theorem of this section.

Theorem 4.1** (Interior regularity).**

Under assumption $(H)$, the following properties are equivalent:

1. (1)

(MTP)* admits no singular time-optimal trajectory;* 2. (2)

$T$* is locally semiconcave in $\Omega$;* 3. (3)

$T$* is locally Lipschitz continuous in $\Omega$.*

Remark 4.3**.**

• (i)

The fact that the existence of singular optimal trajectories may destroy the regularity of a solution of a first order Hamilton-Jacobi equation was already observed by Sussmann (in an implicit form) in [19] and (explicitly) by Agrachev in [1]. The regularity these authors consider is subanalyticity of the point-to-point distance function associated with real analytic distributions. The aforementioned subanalyticity results were extended to solutions of the Dirichlet problem in [21].

• (ii)

We observe that, as shown in [10], there exists an open dense set $\mathcal{O}$ in $(VF(\Omega))^{N}$ (equipped with the $C^{\infty}$ topology) such that condition (1) in Theorem 4.1 holds whenever $(X_{1},\ldots,X_{N})\in\mathcal{O}$.

• (iii)

In [7], the local semiconcavity of the sub-riemannian distance to a point $x_{0}$, $d_{SR}(x_{0},x)$, is proved without giving any estimate of the dependance on $x_{0}$ of the semiconcavity constant of $d_{SR}(x_{0},\cdot)$. Therefore, the semiconcavity of

[TABLE]

is not a direct consequence of the result of [7].

The proof of Theorem 4.1 is based on methods from optimal control.

Proof.

It is well-known that the local semiconcavity in $\Omega$ of $T$ yields the local Lipschitz continuity of $T$ in the same set. Thus, in order to prove the theorem, it suffices to show that the following implications hold true:

• (i)

if (MTP) admits no singular time-optimal trajectories then $T$ is locally semiconcave in $\Omega$;

• (ii)

if $T$ is locally Lipschitz continuous in $\Omega$ then (MTP) admits no singular time-optimal trajectories.

Now, (ii) follows from Corollary 3.1. We therefore proceed to prove (i). First of all, let us recall that for a smooth controlled system the minimum time function is locally semiconcave whenever the target is a noncharacteristic smooth compact manifold. This result can be proved arguing as in [8]. For the reader convenience we give a sketch of the proof. It is well-known that since the target $\Gamma$ is a smooth manifold, the Euclidean distance function is smooth on a neighborhood of $\Gamma$. Furthermore, since $\Gamma$ is a noncharacteristic hypersurface, the minimum time function can be estimated from above by the Euclidean distance function in some neighborhood of $\Gamma$ (see [8, Prop. 2.2]). Owing to these two facts, the minimum time function inherits the semiconcavity of the distance function in a neighborhood of $\Gamma$. Finally, by standard techniques which use time-optimal trajectories to “propagate” regularity, one can show that the semiconcavity of $T$ holds true even away from the target.

Now, let $x_{0}\in\Omega$ and consider all the time-optimal trajectories starting at $x_{0}$. We denote by $\{u_{\alpha}\}$, for $\alpha$ running in a suitable set of indices $\mathcal{A}(x_{0})$, the set of all the optimal controls at $x_{0}$. By Theorem 3.1, we have that $y^{x_{0},u_{\alpha}}(T(x_{0}))\notin E$, for every $\alpha\in\mathcal{A}$. We claim that there exist a smooth compact submanifold $\Gamma_{0}\subset\Gamma$, with $\Gamma_{0}\cap E=\emptyset$, and a neighborhood of $x_{0}$, $V\Subset\Omega$, such that

[TABLE]

Indeed, thanks to Theorem 2.1, for any $x_{0}\in\Omega$ there exists a smooth compact submanifold $\Gamma_{0}\subset\Gamma$ such that

[TABLE]

Arguing by contradiction, let us assume that there exist a sequence $x_{j}\in\Omega$, converging to $x_{0}$, and a sequence of optimal controls $u_{j}\in\mathcal{A}(x_{j})$ such that $y^{x_{j},u_{j}}(T(x_{j}))\notin\Gamma_{0}$. Then, possibly taking a subsequence, we deduce that there exists $\bar{u}\in\mathcal{A}(x_{0})$ such that

[TABLE]

in contradiction with (4.2). Thus, (4.1) holds true.

This shows that the restriction of $T$ to the set $V$ coincides with the restriction to $V$ of the minimum time function for the control system (2.8) with target $\Gamma_{0}$. Since $\Gamma_{0}$ is a smooth compact manifold, by the first part of this proof we deduce that $T$ is semiconcave in $V$. A compactness argument then shows that $T$ is semiconcave on every set $C\Subset\Omega$, thus completing the proof of (i). ∎

Next, we give an example showing that singular time-optimal trajectories may well occur for the problem of interest to this paper.

Example 1**.**

In ${\mathbb{R}}^{3}$ consider vector fields

[TABLE]

Then, there exists a bounded open set $\Omega\subset{\mathbb{R}}^{3}$, with $C^{\infty}$ boundary, such that the viscosity solution of the Dirichlet problem

[TABLE]

fails to be locally Lipschitz in $\Omega$.

Proof.

For fixed $a>0$, consider the subset of ${\mathbb{R}}^{3}$

[TABLE]

and the controlled system

[TABLE]

with initial condition $y(0)=(0,0,0)$. We denote by $T_{\Gamma_{a}}$ the minimum time function associated with system (4.4) and target $\Gamma_{a}$.

We claim that, for $a<1$, $T_{\Gamma_{a}}(0)=a$ and $(0,t,0)$, $t\in[0,a]$, is a singular time-optimal trajectory. Indeed:

• •

taking $p(t)=(0,0,-1)$, $y(t)$ can be lifted to a solution of (3.4) (in the characteristic set of the vector fields) satisfying the transversality condition (3.2);

• •

$y(t)$ reaches $\Gamma_{a}$ in time $a$;

• •

in order to reach the target, starting at the origin, $\dot{y}_{2}$ should be positive whilst $\dot{y}_{3}$ should be negative (this can be done provided that $1-y_{1}<0$ but the minimal time, in order to have $1-y_{1}<0$, is larger than $1$.)

Hence, for any $a\in]0,1[$ we have that $T_{\Gamma_{a}}(0)=a$.

In order to complete the proof it remains to define the set $\Omega$. In ${\mathbb{R}}^{3}$, consider the Euclidean open ball with center at $(0,a,0)$ and radius $a/2$ and denote such a ball by $B$. Then, for every $b\in]a/2,a[$,

[TABLE]

It is clear that the hypersurface $\Gamma_{a}\cap B$ can be smoothly extended outside $B$ and we can construct a $C^{\infty}$ hypersurface $\Gamma$ such that $\Gamma=\partial\Omega$ (for a suitable open bounded set $\Omega$). For $b\in]a/2,a[$ small enough we have that $(0,b,0)\in\Omega$ and $T_{\Gamma}(0,b,0)=a-b$. As noted above, this implies that the (MTP) associated with the controlled system (4.4) and target $\Gamma$ admits a singular time-optimal trajectory. Then, owing to Theorem 4.1, $T_{\Gamma}$ is not a locally Lipschitz function. ∎

We complete this section with the study of the regularity of $T$ at the boundary of $\Omega$. As we did for Lipschitz continuity, we begin by introducing a point-wise notion of Hölder regularity.

Definition 4.1**.**

We say that a function $f:\overline{\Omega}\to{\mathbb{R}}$ is Hölder continuous of exponent $\alpha\in]0,1]$ at a point $x_{0}\in\Omega$ if there exist a neighborhood $U\subset\Omega$ of $x_{0}$ and a constant $C\geq 0$ such that

[TABLE]

The following result shows that, for the Dirichlet problem, the maximal length of commutators $k(\cdot)$ gives the“optimal” Hölder exponent at characteristic boundary points.

Theorem 4.2** (Boundary regularity).**

Assume $(H)$. Then:

1. (1)

for any $x\in\Gamma\setminus E$, $T$ is $C^{\infty}$ in a neighborhood of $x$; 2. (2)

for any $x\in E$, $T$ is Hölder (not Lipschitz) continuous at $x$ of exponent $1/k(x)$, with $k(x)$ given by (2.9).

Even though part of the above conclusions may be known to the reader, for completeness we provide a proof of Theorem 4.2 in Appendix A.

In order to state a more general regularity result, we introduce the set

[TABLE]

which consists of all points at which $T$ fails to be Lipschitz.

Proposition 4.1**.**

Assume $(H)$ and let $T$ be the viscosity solution of (2.7). Then, $\operatorname{sing}_{L}T$ is a closed set.

Proof.

Let $x_{j}\in\operatorname{sing}_{L}T$, $j\in{\mathbb{N}}$, be a sequence of points converging to $x\in\Omega$. We claim that $x\in\operatorname{sing}_{L}T$. Indeed, since $T$ fails to be Lipschitz at $x_{j}$, we have that there exist control functions $u_{j}$ such that:

1. (1)

$y^{x_{j},u_{j}}$ is a singular time-optimal trajectory (this is a consequence of Theorem 3.2); 2. (2)

possibly taking a subsequence, $y^{x_{j},u_{j}}$ converges uniformly to an optimal trajectory $y^{x,u}$.

We claim that $y^{x,u}$ is singular. Indeed, Theorem 2.1 implies that $E$ is a closed set and $y^{x_{j},u_{j}}(T(x_{j}))\in E$, by Theorem 3.1. Hence, we find that

[TABLE]

So, by using once again Theorem 3.1, we deduce that $y^{x,u}$ is a singular time-optimal trajectory.

In order to complete the proof we observe that, if $x\in\Gamma$, then

[TABLE]

and the conclusion follows by Theorem 4.2(2). ∎

Our next result fully describes the regularity of the solution to (2.7).

Theorem 4.3**.**

Assume $(H)$ and let $T$ be the viscosity solution of (2.7). Then, $T$ fails to be Lipschitz at all points of the closed set $\operatorname{sing}_{L}T$ and is semiconcave on the relatively open set $\overline{\Omega}\setminus\operatorname{sing}_{L}T$.

The proof of Theorem 4.3 is similar to the one of Theorem 4.1 and is omitted.

Remark 4.4**.**

An immediate consequence of the above result is that $\operatorname{sing}_{L}T\setminus E$ coincides with the Lipschitz singular support of $T$. Indeed, $x\in\Omega\setminus\operatorname{sing}_{L}T$ if and only if there exists a neighborhood of $x$, $V$, such that $T$ is Lipschitz continuous on $V$.

4.1. Sufficient conditions for regularity

In this section, we give conditions to guarantee the absence of singular time-optimal trajectories. Let us recall that, in canonical coordinates333More in general a symplectic form in a $C^{\infty}$ manifold is a non-degenerate, closed $C^{\infty}$ two form., the symplectic form in $T^{*}\Omega$ is the $2$-form

[TABLE]

Furthermore, for any $\rho\in T^{*}\Omega$, given a vector space $W\subset T_{\rho}(T^{*}\Omega)$ we denote by $W^{\sigma}$ the symplectic orthogonal to $W$, i.e.

[TABLE]

Finally, we say that a manifold $M\subset T^{*}\Omega$ is symplectic if the restriction of $\sigma$ to $M$ is nondegenerate, i.e.

[TABLE]

We have the following result.

Theorem 4.4**.**

Assume (H). Then (MTP) admits no singular time-optimal trajectory if any of the following conditions is satisfied:

• (i)

$\Gamma$* is noncharacteristic (i.e. $E=\emptyset$);*

• (ii)

$\operatorname{Char}(X_{1},\ldots,X_{N})$* is a symplectic manifold.*

Proof.

For noncharacteristic $\Gamma$ the conclusion follows from Theorem 3.1.

Let us assume that $\operatorname{Char}(X_{1},\dots,X_{N})$ is a symplectic manifold. We have that

[TABLE]

(here “$\perp$” stands for orthogonality w.r.t. the Euclidean scalar product). Hence, we find

[TABLE]

On the other hand, since $\operatorname{Char}(X_{1},\ldots,X_{N})$ is a symplectic manifold, using the identity

[TABLE]

we conclude that either the broken Hamiltonian flow (3.4) is transveral to $\operatorname{Char}(X_{1},\ldots,X_{N})$, or it is a stationary flow, i.e., there are no singular time-optimal trajectories. This completes our proof. ∎

Remark 4.5**.**

As a consequence of our results and [21, Theorem 3.4], assuming that $\{X_{1},\ldots,X_{N}\}$ are real analytic and $\partial\Omega$ is a real analytic submanifold, we have that, if $\operatorname{Char}(X_{1},\ldots,X_{N})$ is a symplectic manifold, then $T$ is a subanalytic function in $\Omega$ (see, e.g. [20], for the definition and basic properties of subanalytic sets and functions).

4.2. Examples

In this section, we discuss problems to which Theorem 4.4 can be directly applied as well as examples where the assumptions of Theorem 4.4 are not fulfilled, but one can still use Theorem 4.1 to study the regularity of $T$.

In the example below we consider Heisenberg vector fields and use Theorem 4.4 to show that this problem admits no singular time-optimal trajectory. It is worth noting that the same conclusion follows from the fact that such vector fields are strongly bracket generating.

Example 2** (Heisenberg vector fields).**

In ${\mathbb{R}}^{3}$ consider vector fields

[TABLE]

and let $\Omega$ be a bounded open set with $C^{\infty}$ boundary. We have that

[TABLE]

is a smooth submanifold of ${\mathbb{R}}^{6}$ of codimension $2$. Furthermore, the restriction of $\sigma$ to $\operatorname{Char}(X_{1},X_{2})$ is nondegenerate, i.e. $\operatorname{Char}(X_{1},X_{2})$ is a symplectic manifold. Then, by Theorem 4.4 (ii), we conclude that (MTP) has no singular time-optimal trajectory.

Since we consider a boundary value problem for the eikonal equation, due to the interaction of the boundary of $\Omega$ with $\operatorname{Char}(X_{1},\ldots,X_{N})$, even systems of vector fields admitting, in general, singular time-optimal trajectories may have a better behaviour when $\Omega$ enjoys specific properties. In this regard, we consider once more the system of vector fields in Example 1. We point out that such a system, introduced in [17], admits, in general, strictly abnormal geodesics—in the language of the geometric control theory. We will show that, whenever $\Omega$ is convex, we can exclude the occurrence of singular time-optimal trajectories.

Example 3**.**

In ${\mathbb{R}}^{3}$ consider vector fields

[TABLE]

and let $\Omega$ be a bounded convex open set with $C^{\infty}$ boundary. Then, (MTP) admits no singular time-optimal trajectory.

Proof.

We argue by contradiction assuming the existence of a singular time-optimal trajectory. Let us write the two systems appearing in (3.4) in this specific case. We have

[TABLE]

The only possible initial point $\bar{x}\in\Omega$ of a singular time-optimal trajectory must be of the form $(0,\bar{x}_{2},\bar{x}_{3})$, $\bar{x}_{2},\bar{x}_{3}\in\mathbb{R}$. Then the corresponding solution of (4.6) is given by

[TABLE]

where $(0,\bar{x}_{2},\bar{x}_{3},0,0,\bar{p}_{3})$ are the “initial” conditions for the Hamiltonian system in (4.6). Let $x_{2}^{*}$ be such that $y(T)=(0,x_{2}^{*},\bar{x}_{3})$ and note that $(0,x_{2}^{*},\bar{x}_{3})\in\Gamma$ is a characteristic point. Since $\Omega$ is a smooth convex set, there exists a smooth function $\Phi=\Phi(x_{1},x_{2},x_{3})$ such that

[TABLE]

and

[TABLE]

for a suitable $\delta>0$. Now, the transversality condition (3.2) in Pontryagin’s maximum principle implies that $D\Phi(y(T))=\lambda(0,0,1),$ for some $\lambda\not=0$. In particular, we deduce that $\partial_{x_{3}}\Phi(y(T))\not=0.$ By appealing to the implicit function theorem and convexity assumption, we can suppose that, near $(0,x^{*}_{2},\bar{x}_{3})$, $\Omega$ is the hypograph of a smooth concave function $(x_{1},x_{2})\mapsto\varphi(x_{1},x_{2})$, which attains its maximum at $(0,x^{*}_{2})$ and such that $\varphi(0,x^{*}_{2})=\bar{x}_{3}$. This implies that $(0,x^{*}_{2},\bar{x}_{3})$ cannot be the final point of a trajectory starting from $\Omega$ because the line segment $(0,x^{*}_{2}+s,\bar{x}_{3})$, $s\in{\mathbb{R}}$, can at most belong to the boundary of $\Omega$. We have thus reached a contradiction. ∎

As shown in the next example, in some cases, even if the characteristic set is not a symplectic manifold but it can be splitted into a disjoint union of symplectic submanifolds, our approach can be applied.

Example 4**.**

Let $\Omega\subset{\mathbb{R}}^{3}$ be a bounded open set with smooth boundary and let $k$ be a positive integer. In ${\mathbb{R}}^{3}$, consider vector fields

[TABLE]

Then no singular time-optimal trajectories exists.

Proof.

The characteristic set is given by

[TABLE]

Therefore, $\operatorname{Char}(X_{1},X_{2})$ can be split into the connected submanifolds

[TABLE]

and

[TABLE]

Moreover, all these submanifolds are symplectic (the rank of the symplectic form is constant and the symplectic form is nondegenerate on these sets). So, there is no singular time-optimal trajectory. ∎

Observe that, in the above example, the characteristic set is a manifold of condimension $2$ in ${\mathbb{R}}^{3}$, but the rank of the symplectic form is not constant, i.e., $\operatorname{Char}(X_{1},X_{2})$ is not a symplectic manifold. Then, since we are not assuming that $E=\emptyset$, none of the conditions in Theorem 4.4 is satisfied.

We complete this section with an application of Theorem 4.3.

Example 5**.**

Let $\Omega\subset$ be a bounded open set with smooth boundary and consider vector fields

[TABLE]

Then, $\operatorname{sing}_{L}T$ is a set of measure zero and $T$ has a second order Taylor expansion at a.e. point of $\Omega$.

Proof.

We observe that, if we show that $\operatorname{sing}_{L}T$ is a set of measure zero, then $T$ turns out to be semiconcave in $\Omega\setminus\operatorname{sing}_{L}T$ by Theorem 4.3. So, the conclusion follows by the Alexandrov theorem on twice differentiability of a convex function. Hence, let us show that $\operatorname{sing}_{L}T$ is a set of measure zero. We observe that all singular time-optimal trajectories can be lifted to the characteristic set

[TABLE]

On the other hand, the characteristic set can be decomposed as follows

[TABLE]

with

[TABLE]

[TABLE]

and

[TABLE]

We observe that the only solutions of (3.4) in the set $V_{1}$ are stationary points (it is a symplectic manifold). Hence, a nontrivial lifting of a singular time-optimal trajectory can only belong to the set $V_{2}\cup V_{3}$. By Theorem 3.2, we deduce that

[TABLE]

and the conclusion follows. ∎

Appendix A Proof of Theorem 4.2

The fact that $T$ is of class $C^{\infty}$ in a neighborhood of every non-characteristic point can be proved arguing as follows. Let $x_{0}\in\Gamma\setminus E$. Then, by Theorem 2.1, there exists a neighborhood of $x_{0}$ in $\Omega^{\prime}$, $W$, such that $W\cap E=\emptyset$. Consider the Dirichlet problem

[TABLE]

with initial condition $v=0$ in $\Gamma\cap W$. This is a noncharacteristic Dirichlet problem with smooth data. Then, possibly replacing $W$ with a smaller neighborhood of $x_{0}$, we have that there exists a unique solution $v\in C^{\infty}(W\cap\Omega)$. Since the characteristics of the above problem are time-optimal trajectories, we conclude that $T\in C^{\infty}(W\cap\Omega)$.

Let us now prove Hölder regularity at a characteristic boundary point. Take $x_{0}\in E$ and denote by $T_{\{x_{0}\}}$ the minimum time function for the controlled system (2.8) with target $\{x_{0}\}$. Let $W$ be a neighborhood of $x_{0}$ in $\Omega^{\prime}$ such that

[TABLE]

We want to show that $T$ is Hölder continuous of exponent $1/k(x_{0})$ at $x_{0}$. It is well known ([4, Theorem 1.15, Chap. IV]) that there exists a constant $C>0$ such that

[TABLE]

By possibly taking a smaller set $W$, we may assume that

[TABLE]

and

[TABLE]

(in the viscosity sense). So, the comparison principle implies that

[TABLE]

Then, we deduce that

[TABLE]

Let us show that, if $x_{0}\in E$, then $T$ cannot be Lipschitz continuous at $x_{0}$. We argue by contradiction and assume $\exists\,C>0$ such that

[TABLE]

for some $\delta>0$. Then, by Theorem 2.1, there exists a sequence $x_{j}\in\Gamma\cap B_{\delta}(x_{0})\setminus E$ such that $x_{j}\to x_{0}$ as $j\to\infty$. Then

[TABLE]

By (A.1), $\{\langle DT(x_{j}),\nu(x_{j})\rangle\}_{j}$ is a bounded sequence. Hence, possibly taking a subsequence, we find that, as $j\to\infty$,

[TABLE]

for some $c\in{\mathbb{R}}$. Thus, we find the contradiction

[TABLE]