Consider the dynamical system $\mathbf{x}' = f(\mathbf{x,u},t)$ where $\mathbf{x} \in \mathbb{R}^2, \mathbf{u} \in \mathbb{R}^3$ and $f$ has no explicit time dependence. As conventional, $\mathbf{x}$ denotes the state vector and $\mathbf{u}$ the control vector. We need to find the optimal control $\mathbf{u}^*$ that minimizes the time of transit, $t_f$, given prescribed initial and final conditions $\mathbf{x}(0)$ and $\mathbf{x}(t_f)$. Also, we're given the additional constraints that $\mathbf{u} \geq 0$ and lies on the hyperplane $\begin{bmatrix}1 & 1& 1 \end{bmatrix}\mathbf{u}=c ,c> 0$.
How is the Hamiltonian $H(\mathbf{x,u,p},t)$ affected by these constraints on the control?
Without these constraints, it's easy to see that $H= x_1'p_1+x_2'p_2 + 1$, and we can use the canonical equations $\mathbf{p}' = \partial{H}/\partial{\mathbf{x}}$ and $\partial{H} / \partial{\mathbf{u}} =0$ to derive the optimal control. But I'm not so sure with the constraints on the control, esp. the (second) equality constraint. Can you outline the application of Pontryagin's maximum principle here?
You solve the problem (see remark 1) $$ \left\{ \begin{array}{l} \dot x = f(x,u(t)), \;\;\; 0<t<t_f, \;\;\; u(t) \in P \\ x(0) = x_0, \; x(t_f)=x_1, \\ \int\limits_{0}^{t_f} dt \to \min, \\ \end{array} \right. $$ where $P = \mathbb{R}^3_{+} \cap H$ and $H$ is a hyperplane $[1,1,1]^T u = c$. Lets find our Hamiltonian by definition (see remark 2): $$ H(x,p,u) = p \cdot f(x,u). $$ Let $(x^{\ast}(t),u^{\ast}(t), 0 \leqslant t \leqslant t^{\ast}_{f})$ be a solution of our problem. Then Pontryagin's maximum principle states that there exists a function $p^{\ast}(t)$ such that $$ \left\{ \begin{array}{l} \dot p^{\ast} = - \frac{\partial H}{\partial x}(x^{\ast}(t),p^{\ast},u^{\ast}(t)), \;\;\; 0 < t < t_f^{\ast} \\ H(x^{\ast}(t),p^{\ast}(t),u^{\ast}(t)) = \sup_{u \in P} H(x^{\ast}(t),p^{\ast}(t),u) \; \text{in points $t$ of continuity of $u^{\ast}(t)$}. \end{array} \right. $$ Since you solve the point2point problem, there are no transversality conditions in this case (see remark 3). If we have $P = \mathbb{R}^3$ then we can find optimal control from $\frac{\partial H}{\partial u} = 0$, but in general case we have to deal with supremum.
Remark 1. You wrote that $f(x,u,t)$ has no explicit time dependence. I denoted $f(x,u) \equiv f(x,u,t)$.
Remark 2. If you're solving the problem of minimizing functional $$ \int\limits_{0}^{t_f} f_0(x(t),u(t)) \, dt $$ then you should add one supplementary term to Hamiltonian and introduce a variable $p_0$: $$ H(x,p_0,p,u) = - p_0 f_0(x,u) + p \cdot f(x,u) $$ but when $f_0 \equiv 1$ we can deal with modified Hamiltonian $H'(x,p,u) = p \cdot f(x,u)$.
Remark 3. If you're solving a set2set problem from manifold $X$ to manifold $Y$ then Pontryagin's maximum principle also states that $p^{\ast}(0) \bot T_{x^{\ast}(0)}X$ and $p^{\ast}(t_f^{\ast}) \bot T_{x^{\ast}(t_f^{\ast})}Y$ but in your case this means $p^{\ast}(0), p^{\ast}(t_f^{\ast}) \in \mathbb{R}^2$ so there are no additional helpful information.