Let $t_0<t_1$, let $U\subset\mathbb{R}^m$ be a control set, let $x_0\in\mathbb{R}^n$, and let $M\subset\mathbb{R}^n$ be a $C^1$ embedded submanifold. For each $y\in M$, let $T_yM\subset\mathbb{R}^n$ denote the tangent space of $M$ at $y$. Let $(\mathbb{R}^n)^*$ denote the dual [vector space](/page/Vector%20Space) of linear maps from $\mathbb{R}^n$ to $\mathbb{R}$. For each $v\in\mathbb{R}^n$, let $v^\flat\in(\mathbb{R}^n)^*$ denote the covector $w\mapsto v\cdot w$. Suppose $f:[t_0,t_1]\times\mathbb{R}^n\times U\to\mathbb{R}^n$, $L:[t_0,t_1]\times\mathbb{R}^n\times U\to\mathbb{R}$, and $\Phi:\mathbb{R}^n\to\mathbb{R}$ are $C^1$ in the state variable. For each $y\in\mathbb{R}^n$, let $d\Phi_y:\mathbb{R}^n\to\mathbb{R}$ denote the differential of $\Phi$ at $y$. Consider the fixed-initial-state optimal control problem of minimizing

\begin{align*} J[x,u]=\int_{[t_0,t_1]} L(t,x(t),u(t))\,d\mathcal{L}^1(t)+\Phi(x(t_1)) \end{align*}

over admissible pairs $(x,u)$, where $x:[t_0,t_1]\to\mathbb{R}^n$ is an absolutely continuous state arc and $u:[t_0,t_1]\to U$ is an admissible control, satisfying

\begin{align*} \dot{x}(t)=f(t,x(t),u(t)) \end{align*}

for $\mathcal{L}^1$-a.e. $t\in[t_0,t_1]$, with $x(t_0)=x_0$ and $x(t_1)\in M$. Assume $(x^*,u^*)$ is a local minimizer for which the normal Pontryagin maximum principle applies to this problem with cost multiplier equal to $1$. Let $p:[t_0,t_1]\to\mathbb{R}^n$ be the corresponding vector representative of the adjoint covector arc under the Euclidean identification, for the Hamiltonian

\begin{align*} H(t,x,u,p)=p\cdot f(t,x,u)-L(t,x,u). \end{align*}

Then

\begin{align*} p(t_1)+\nabla\Phi(x^*(t_1))\in N_{x^*(t_1)}M, \end{align*}

where, for $y\in M$,

\begin{align*} N_yM:=\{v\in\mathbb{R}^n: v\cdot w=0\text{ for every }w\in T_yM\}. \end{align*}