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In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods. 

Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation has limitations. In network models with delay, the delayed channels are typically low-dimensional and accounting for this heterogeneity is challenging in the DDE framework. In addition, DDEs cannot be used to model difference equations. In this paper, we examine alternative representations for networked systems with delay and provide formulae for conversion between representations. First, we examine the Differential-Difference (DDF) formulation which allows us to represent the low-dimensional nature of delayed information. Next, we consider the coupled ODE-PDE framework and extend this to the recently developed Partial-Integral Equation (PIE) representation. The PIE framework has the advantage that it allows the H∞-optimal estimation and control problems to be solved efficiently using the recently developed software package PIETOOLS. In each case, we consider a very general class of networks, specifically accounting for four sources of delay - state delay, input delay, output delay, and process delay. Finally, we use a scalable network model of temperature control to show that the use of the DDF/PIE formulation allows for optimal control of a network with 40 users, 80 states, 40 delays, 40 inputs, and 40 disturbances.