Let $t_0<t_1$, let $U\subseteq\mathbb{R}^m$, let $x_0\in\mathbb{R}^n$, and let $f:[t_0,t_1]\times\mathbb{R}^n\times U\to\mathbb{R}^n$, $L:[t_0,t_1]\times\mathbb{R}^n\times U\to\mathbb{R}$, and $\Phi:\mathbb{R}^n\to\mathbb{R}$ be such that $f$ and $L$ are continuously differentiable with respect to the state variable $x$, and $\Phi\in C^1(\mathbb{R}^n)$.

Consider the fixed-time 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 $(x,u)$ satisfying $x:[t_0,t_1]\to\mathbb{R}^n$ absolutely continuous, $u:[t_0,t_1]\to U$ measurable, the state equation

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

for $\mathcal{L}^1$-a.e. $t\in[t_0,t_1]$, and the fixed initial condition $x(t_0)=x_0$. Assume that there is no terminal state constraint and no active state constraint restricting variations of $x(t_1)$, so the terminal variation space is $V_{t_1}=\mathbb{R}^n$.

Let $(x^*,u^*)$ be an admissible normal extremal for this problem, normalized with cost multiplier $p_0=-1$, in the following sense: there exists an adjoint arc $p:[t_0,t_1]\to\mathbb{R}^n$ that is absolutely continuous and satisfies the fixed-time normal maximum-principle adjoint equation and endpoint transversality condition below. Define the Hamiltonian $H:[t_0,t_1]\times\mathbb{R}^n\times U\times\mathbb{R}^n\to\mathbb{R}$ by

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

Assume the adjoint arc satisfies

\begin{align*} \dot{p}(t)=-\frac{\partial H}{\partial x}(t,x^*(t),u^*(t),p(t)) \end{align*}

for $\mathcal{L}^1$-a.e. $t\in[t_0,t_1]$, and that the endpoint transversality covector $p(t_1)+\nabla\Phi(x^*(t_1))$, identified with a vector in $\mathbb{R}^n$ by the Euclidean [inner product](/page/Inner%20Product), annihilates every terminal variation $\eta\in V_{t_1}$:

\begin{align*} \bigl(p(t_1)+\nabla\Phi(x^*(t_1))\bigr)\cdot\eta=0. \end{align*}

Then the terminal transversality condition is

\begin{align*} p(t_1)=-\nabla\Phi(x^*(t_1)). \end{align*}