# Maximum principle for a control with mixed constraints

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?

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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.