I have a control problem in which I want to minimize a quadratic function subject to the uncertain equality constraints
$$ (\mathcal{A} + \Delta\mathcal{A})z_k = 0, \ k = 0,\ldots,N-1, $$
where $N\in\mathbb{N}$ is the time horizon, and the uncertainty entering the constraints is bounded as $\|\Delta \mathcal{A}\|_{F} \leq \rho$, for some $\rho > 0$.
My complete optimization problem is:
\begin{align} &\min_{z_k} \ \mathcal{J}_{\mu} = \sum_{k=0}^{N-1} z_k^\intercal M_k z_k \\\\ &\quad \mathrm{s.t.} \quad (\mathcal{A} + \Delta \mathcal{A}) z_k = 0, \ \forall k = 0,\ldots, N - 1, \ \forall \|\Delta\mathcal{A}\|_{F} \leq \rho \\\\ &\qquad\qquad \mathcal{B} z_{N-1} = \mu_{f}, \quad \mathcal{C}z_{0} = \mu_0. \end{align}
To gain insight into this problem, I'd like to note the following points. The decision variables $z_k := [v_k; \mu_k; v_{k+1}; \mu_{k+1}]\in\mathbb{R}^{2(n + m)}$ encode the mean state and the (feed-forward) control input, and the uncertain linear constraints encode the linear dynamics of the (uncertain) system. The last constraints enforce the mean state to be fixed at its endpoints.
So the problem solves for the controller that steers the mean motion of the system from some initial mean $\mu_0$ to some terminal mean $\mu_f$. My question is if this problem can be made tractable or convexified for this unstructured, norm-bounded type of disturbance.
This would answer the question: Is it possible to design a controller that steers an linear system from some initial point to a final point under bounded model uncertainty.