Systems & Control Letters, Accepted for Publication, Jan. 2009. LQ optimal control for 2-D Roesser models of finite-extent L. Ntogramatzidis and M. Cantoni Abstract This paper investigates several aspects of linear-quadratic (LQ) optimal control for Roesser models over a two-dimensional (2-D) signal-index set of finite extent. First, we consider the characterisation and computation of open-loop control laws when constraints on the system semi-states are imposed at both the south-west and north-east boundaries of the frame (i.e. signal-index set) of interest; by virtue of the quarter-plane causal structure of the Roesser model, the south-west and north-east boundary conditions are analogous to initial conditions and terminal constraints, respectively. A necessary and sufficient characterisation of optimality is obtained and explicitly computable formulae are derived to characterise the corresponding control inputs and performance index under reasonable assumptions on the problem data. In the second part of the paper, the problem of optimal LQ control via semi- state feedback is considered. A 2-D Riccati-like difference equation is introduced to characterise, in a sufficient sense, a solution to this problem. Key words: Roesser models, linear quadratic optimal control, semi-state feedback