This paper considers a general class of 2D continuous-discrete linear systems of both systems theoretic and applications interest. The focus is on the development of a comprehensive control systems theory for members of this class in a unified manner based on analysis in an appropriate algebraic and operator setting. In particular, important new results are developed on stability, controllability, stabilization, and optimal control.

Abstract. Differential repetitive processes are a distinct class of continuousdiscrete 2D linear systems of both systems theoretic and applications interest. The feature which makes them distinct from other classes of such systems is the fact that information propagation in one of the two independent directions only occurs over a finite interval. Applications areas include iterative learning control and iterative solution algorithms for classes of dynamic nonlinear optimal control problems based on the maximum principle, and the modelling of numerous industrial processes such as metal rolling, and long-wall cutting etc. The new results in is paper solve a general optimal problem in the presence of non-stationary dynamics.

Abstract. Differential repetitive processes are a distinct class of continuous-discrete twodimensional linear systems of both systems theoretic and applications interest. These processes complete a series of sweeps termed passes through a set of dynamics defined over a finite duration known as the pass length, and once the end is reached the process is reset to its starting position before the next pass begins. Moreover the output or pass profile produced on each pass explicitly contributes to the dynamics of the next one. Applications areas include iterative learning control and iterative solution algorithms, for classes of dynamic nonlinear optimal control problems based on the maximum principle, and the modeling of numerous industrial processes such as metal rolling, long-wall cutting, etc. In this paper we develop substantial new results on optimal control of these processes in the presence of constraints where the cost function and constraints are motivated by practical application of iterative learning control to robotic manipulators and other electromechanical systems. The analysis is based on generalizing the well-known maximum and -maximum principles to them.Key words. two-dimensional systems, optimal control, constraints AMS subject classifications. 93C05, 93C15 DOI. 10.1137/060668298 1. Introduction. Repetitive processes are a distinct class of two-dimensional (2D) systems of both systems theoretic and applications interest. The unique characteristic of such a process is a series of sweeps, termed passes, through a set of dynamics defined over a fixed finite duration known as the pass length. On each pass an output, termed the pass profile, is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. This, in turn, leads to the unique control problem in that the output sequence of pass profiles generated can contain oscillations which increase in amplitude in the pass-to-pass direction.To introduce a formal definition, let α < +∞ denote the pass length (assumed constant). Then in a repetitive process the pass profile y k (t), 0 ≤ t ≤ α, generated on pass k acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile y k+1 (t), 0 ≤ t ≤ α, k ≥ 0.Physical examples of repetitive processes include long-wall coal cutting and metalrolling operations (see, for example, the references cited in [17]). Also in recent years applications have arisen where adopting a repetitive process setting for analysis has distinct advantages over alternatives. Examples of these so-called algorithmic applications include classes of iterative learning control (ILC) schemes (see, for example, [13]) and iterative algorithms for solving nonlinear dynamic optimal control problems based on the maximum principle [15]. In the case of iterative learning control for the linear

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This paper introduces a model for the dynamics of a sorption process from the industrial water supply and sewage treatment industries that is a continuous version of the Roesser state-space model for 2D discrete systems. Conditions for unique solvability and the representation formula are then developed together with the solution of an optimization problem using boundary control. The solution of this optimization problem by state feedback is also developed.