I have a local solution of a dynamical system $\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x})$: \begin{equation} \mathbf{x}(t) = \mathbf{g}(t;\mathbf{A}), \end{equation} where $\bf{f},\bf{g}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ and $\mathbf{A}\in\mathbb{R}^n$ are parameters.
I am currently trying to numerically solve the inverse problem of this, that is, solving for $\mathbf{A}$ given the time $t_0$ and the corresponding state $\mathbf{x}(t_0)$.
To do this , I derived the Jacobian, $J$, of the function $\mathbf{g}$ and applied the Newton's method: \begin{equation} \mathbf{A}^{n+1} = \mathbf{A}^{n}-J^{-1}(\mathbf{g}(t_0;\mathbf{A}^n)-\mathbf{x}(t_0)). \end{equation}
I have a problem solving this because some of the elements of $\mathbf{A}$ are phases, so they should be defined between $0$ and $2\pi$. Obviously, the formula above does not have such constraints.
I looked up on the Internet for this kind of Newton's method with constraints, but all I found were the ones for one variable or multivariate scaler functions (like $\mathrm{f}(x)=0$ and $\mathrm{f}(\mathbf{x})$, not $\mathrm{\mathbf{f}}(\mathbf{x})=\mathbf{0}$).
How do I formulate this vector function version of the problem with constraints and solve it numerically?