Periodic adsorption processes have been gaining increasing commercial acceptance as energy efficient alternatives to other separation processes. Such processes are distributed in nature, with properties exhibiting spatial as well as temporal variations. Therefore, they can be mathematically represented by systems of partial differential and algebraic equations (PDAEs).The periodic nature of periodic adsorption processes arises from an externally imposed cyclic variation of the flowrates and intensive properties (e.g. pressure, temperature or composition) of the process feed streams, and of the flowrates of the product streams. This variation leads the system to a "cyclic steady state" (CSS) at which the conditions in each bed at the start and end of each cycle are identical. The traditional approach to cyclic steady state determination has been to carry out a dynamic simulation of the system, starting from a given initial condition, over a large number of cycles. In this work, a novel method is proposed to directly determine a cyclic steady state of periodic adsorption processes by replacing the initial condition specification by a periodicity condition demanding that the system states at the end of each cycle be identical to those at its start. Additional constraints are introduced to characterise the interactions between multiple beds in the process. Detailed dynamic models taking account of the spatial variation of properties within the adsorption bed(s) are used. The resulting systems of partial differential and algebraic equations, and the corresponding boundary conditions, are reduced to sets of algebraic constraints by discretisation with respect to both spatial and temporal dimensions. The performance of periodic adsorption processes is intrinsically affected by various design and operating parameters. The appropriate selection of the values of these parameters may significantly enhances the entire profit of the processes. However, the number of interacting decisions and constraints is such that obtaining an optimal solution by carrying out several dynamic simulations is laborious, if not altogether impossible. A new approach to the optimisation of periodic adsorption processes using mathematical programming is presented. It is demonstrated that the optimal operating and/or design decisions can be determined by solving a single optimisation problem with constraints representing a single bed over a single cycle of operation, irrespective of the number of adsorption beds in the process. Examples of periodic adsorption processes involving different number of adsorption beds and operating cycles are presented to illustrate the approach.