Consider a constrained optimal control problem given as
$$J(u_1,u_2) = \int_0^{t_f} L(x_1,x_2,u_1,u_2)\,dt$$
subject to
$$\dot{x}_1 = f_1(x_1,u_1),$$
$$\dot{x}_2 = f_2(x_2,u_2),$$
$$u_1,u_2 \in \textit{U}$$
where $U$ is constrained convex set. The above problem can be solved by maximum principle, which states that in optimal u is one which minimizes the Hamiltonian i.e.
$$u^* = \min_u H(x,u,\lambda,t)$$
So for the problem mentioned above, the maximum principle will imply $$u^* = \min_{u_1,u_2} H(x_1,x_2,u_1,u_2,\lambda_1,\lambda_2,t)$$
Now my question is if $x_1^*,x_2^*$ are the optimal solution to the above $u^*$, then if I do the following:
$$u_1^* = \min_{u_1} H(x_1,x_2,u_1,u_2,\lambda_1,\lambda_2,t)$$ and keep the other input constant. Once the system is evolved, I again perform the maximum principle for the other input on the new reconstructed Hamiltonian keeping the first input as constant now(already optimal u). Will the local minima be the same. If yes, can I show it with a proof.
Also this problem is disjoint( system dynamics are function of $x_i,u_i$ only, can I show it in general setting for coupled dynamics.