Optimal Control of Two-Dimensional Systems

Necessary and sufficient conditions for the existence and the uniqueness of the solution of the optimal control problem of discrete-time linear time invariant two-dimensional systems are determined. Given a system that satisfies these conditions, the optimal control law is obtained using an algebraic Riccati equation with coefficients in the polynomial ring $R[z]$. Since the feedback implementation of this law does not preserve the causal structure of the system, suboptimal control laws are also discussed that lead to a weakly causal two-dimensional system. Finally, some important connections between optimal control in an $\mathfrak{l}_2 $ setting and $\mathfrak{l}_\infty $ stabilization are investigated.