Section3Step 2 sub-Finsler Pontryagin Maximum Principle

In this section, the Pontryagin Maximum Principle will be rephrased in a convenient form for the purposes of Theorem 1.1. The precise statement to be proved is the following:

Remark3.2.

Up to changing the optimal control $u$ on a set of measure zero, the subdifferential condition ii may be taken to hold for all $t\in[0,T]\text{.}$

Namely, if condition ii holds on a subset $I\subset[0,T]$ of full measure, for any $t\in [0,T]\setminus I\text{,}$ pick any converging sequence $I\ni t_k\to t$ such that the limit $\lim_{k\to\infty}u(t_k)$ exists, and redefine $u(t)=\lim_{k\to\infty}u(t_k)\text{.}$ By the continuity of subdifferentials given by Lemma 2.18, it follows that $a(t)$ is a subdifferential of the squared norm at the point $u(t)\text{.}$

Remark3.3.

In the sub-Riemannian case, the squared norm $\frac{1}{2}\norm{\cdot}^2$ is differentiable at every point, and the unique subdifferential is the inner product $a(t) = \left\lt u(t),\cdot\right\gt\text{.}$ The derivative condition i then gives the usual linear ODE of controls in the implicit form

\begin{equation*} \left\lt \dot{u}(t),Y\right\gt = \frac{d}{dt}\left\lt u(t),Y\right\gt = B(u(t),Y)\quad\forall Y\in V_1\text{.} \end{equation*}

Subsection3.1General statement of the PMP

For the rest of Section 3, let $G$ be a fixed sub-Finsler Carnot group of step 2 with an arbitrary norm $\norm{\cdot}\colon V_1\to\RR\text{,}$ and let $u\colon[0,T]\to V_1$ be the control of a geodesic $\gamma\colon [0,T]\to G\text{.}$

Consider first the finite time $T\lt \infty$ case. By Definition 2.10 of the sub-Finsler distance, the control $u$ minimizes the length functional $\int_{0}^{T}\norm{u(t)}\,dt$ among all controls defining curves with the same endpoints as $\gamma\text{.}$ Since a geodesic has by definition constant speed, it follows that $u$ is also a minimizer of the energy functional $\frac{1}{2}\int_{0}^{T}\norm{u(t)}^2\,dt\text{.}$

Define the left-trivialized Hamiltonian

$$h\colon V_1\times\RR\times\mathfrak{g}^*\to\RR,\quad h(u,\xi,\lambda) = \lambda(u) + \frac{1}{2}\xi\norm{u}^2\text{.}\label{eq-hamiltonian}\tag{3.1}$$

By the Pontryagin Maximum Principle as presented in Theorem 12.10 of [2], the control $u\colon[0,T]\to V_1$ can minimize the energy $\frac{1}{2}\int_{0}^{T}\norm{u(t)}^2\,dt$ only if there is an everywhere non-zero absolutely continuous dual curve $t\mapsto (\xi,\lambda(t))\in \RR\times T^*_{\gamma(t)}G$ such that

Here $h_{v,\xi}$ and $\vec{h}_{v,\xi}\text{,}$ for $v\in V_1\text{,}$ are the left-invariant Hamiltonian and the associated Hamiltonian vector field respectively.

More explicitly, $h_{v,\xi}\colon T^*G\to\RR$ is the function defined from the left-trivialized Hamiltonian (3.1) in the natural way by

\begin{equation*} h_{v,\xi}(\lambda)=h(v,\xi,L_{g}^*\lambda),\quad \forall \lambda\in T^*_gG\text{,} \end{equation*}

and $\vec{h}_{v,\xi}$ is the Hamiltonian vector field associated with the left-invariant Hamiltonian $h_{v,\xi}$ by duality through the canonical symplectic form on the cotangent bundle, see Section 4 of [1] for more details within the context of the PMP in the sub-Riemannian setting.

Observe that if $(\xi,\lambda(t))$ is a dual curve satisfying the conditions (3.2)–(3.4) of the PMP, then also any scalar multiple $(C\xi,C\lambda(t))$ for any $C\gt 0$ satisfies the conditions (3.2)–(3.4) of the PMP. This observation allows the infinite time case $T=\infty$ to be handled as a limit of the finite time case. Namely, if $u\colon [0,\infty)\to V_1$ is the control of a geodesic, then all its finite restrictions $\restr{u}{[0,k]}\colon [0,k]\to V_1$ for $k\in\NN$ are also controls of geodesics, so by the above they have corresponding dual curves $t\mapsto (\xi_k,\lambda_k(t))\text{.}$ By taking suitable rescalings of the $(\xi_k,\lambda_k)\text{,}$ there exists a non-zero limit $(\xi_\infty,\lambda_\infty)\text{,}$ which then satisfies the conditions (3.2)–(3.4) of the PMP on the entire interval $[0,\infty)\text{.}$

Condition (3.2) is a binary condition $\xi=0$ or $\xi\neq 0\text{.}$ The case $\xi=0$ is the case of an abnormal control $u\text{,}$ and may be ignored in the step 2 setting, since the second order necessary criterion of the Goh condition (see e.g. Section 20 of [2]) implies that there are no strictly abnormal extremals in step 2. By rescaling $(\xi,\lambda)$ it therefore suffices to consider the normal case $\xi=-1\text{.}$

Subsection3.2The PMP in left-trivialized coordinates

Let $X_1,\ldots,X_r$ be a basis of $V_1\text{.}$ Fix a basis $X_{r+1},\ldots,X_n$ for $V_2=[V_1,V_1]$ by choosing a maximal linearly independent subset of the Lie brackets $\{[X_i,X_j]: 1\leq i\lt j\leq r\}\text{.}$ By an abuse of notation, denote also by $X_1,\ldots,X_n\text{,}$ the corresponding left-invariant frame of $TG\text{.}$ Let $\theta_1,\ldots,\theta_n$ be the dual left-invariant frame of $T^*G\text{.}$ Writing the curve $\lambda(t)$ in left-trivialized coordinates as

\begin{equation*} \lambda(t) = \sum_{i=1}^n\lambda_i(t)\theta_i(\gamma(t))\text{,} \end{equation*}

the Hamiltonian ODE (3.3) in the normal case $\xi=-1$ takes the simpler form

$$\dot{\lambda}_i(t) = \lambda(t)\Big(\Big[\sum_{j=1}^ru_j(t)X_j, X_i\Big](\gamma(t))\Big),\quad i=1,\dots,n\text{,}\label{eq-left-trivialized-hamiltonian-flow}\tag{3.5}$$

see Section 18.3 of [2] for the explicit computation.

The curve $a\colon[0,T]\to V_1^*$ will be given by restricting the linear map

$$a(t) := (L_{\gamma(t)})^*\lambda(t)\colon \mathfrak{g}\to\RR\label{eq-subdifferential-curve-defn}\tag{3.6}$$

to $V_1\text{.}$ The skew-symmetric bilinear form $B\colon V_1\times V_1\to\RR$ will be given by

$$B(X,Y) := a(t)[X,Y]\text{.}\label{eq-bilinear-form-defn}\tag{3.7}$$

The curve $a(t)$ of (3.6) has in the left-invariant frame the same coefficients as the curve $\lambda(t)\text{,}$ i.e., the coefficients of $a(t) = \sum_{i=1}^na_i(t)\theta_i(e)$ are exactly $a_i=\lambda_i\text{.}$ Left-translating the Hamiltonian ODE (3.5) to the identity shows that for almost every $t\in[0,T]\text{,}$ the components of the curve have the derivatives

$$\dot{a}_i(t) = \frac{d}{dt}\lambda_i(t) = a(t)[u(t),X_i],\quad i=1,\ldots,n\text{.}\label{eq-coordinate-ode}\tag{3.8}$$

By the step 2 assumption, $[u(t),X_i]=0$ for all the vertical components $i=r+1,\dots,n\text{,}$ so the vertical coefficients $a_{r+1},\dots,a_n$ are all constant. Therefore $a(t)[X,Y] = \sum_{i=r+1}^{n}a_i\theta_i([X,Y])$ is constant in $t\text{.}$ That is, the expression (3.7) defines a unique bilinear form $B$ independent from $t\text{.}$

Writing the system (3.8) in terms of the bilinear form $B\text{,}$ the remaining non-trivial equations are exactly

\begin{equation*} \dot{a}_i(t) = a(t)[u(t),X_i] = B(u(t),X_i),\quad i=1,\ldots,r\text{.} \end{equation*}

The derivative condition 3.1i follows by linearity, as for an arbitrary vector $Y=y_1X_1+\dots+y_rX_r\in V_1\text{,}$ the above implies that

\begin{equation*} \frac{d}{dt}a(t)Y = \frac{d}{dt}\sum_{i=1}^na_i(t)y_i = \sum_{i=1}^nB(u(t),X_i)y_i = B(u(t),Y)\text{.} \end{equation*}

The subdifferential condition 3.1ii for the linear functions $a(t)$ follows from rephrasing the maximality condition (3.4). Namely, expanding out the explicit expressions of the normal Hamiltonians $h_{u(t),-1}$ and $h_{v,-1}$ from (3.1) and reorganizing terms, the maximality condition (3.4) is equivalently stated as

This is exactly Definition 2.17 stating that the linear function $a(t)$ is a subdifferential of the squared norm $\frac{1}{2}\norm{\cdot}^2$ at the point $u(t)\in V_1\text{.}$