Extension of the Partial Integral Equation Representation to GPDE Input-Output Systems

TL;DR

PIE representations for a large class of such PDE models, including higher order derivatives, boundary-valued inputs, and coupling with ordinary differential equations are proposed, including higher order derivatives, boundary-valued inputs and coupling with ordinary differential equations.

Abstract

It has been shown that the existence of a Partial Integral Equation (PIE) representation of a Partial Differential Equation (PDE) simplifies many numerical aspects of analysis, simulation, and optimal control. However, the PIE representation has not previously been extended to many of the complex, higher-order PDEs such as may be encountered in speculative or data-based models. In this paper, we propose PIE representations for a large class of such PDE models, including higher-order derivatives, boundary-valued inputs, and coupling with Ordinary Differential Equations. The main technical contribution which enables this extension is a generalization of Cauchy's rule for repeated integration. The process of conversion of a complex PDE model to a PIE is simplified through a PDE modeling interface in the open-source software PIETOOLS. Several numerical tests and illustrations are used to demonstrate the results.

Figures (9)

• Figure 1: Depiction of the ODE subsystem for use in defining a GPDE. All external input signals in the GPDE model pass through the ODE subsystem and are labeled as $u$ and $w$, corresponding to control input and disturbance/forcing input. All external outputs pass through the ODE subsystem and are labeled $y$ and $z$, corresponding to measured output and regulated output. All interaction with the PDE subsystem is routed through two vector-valued signals: $r$ the sole output of the PDE subsystem and $v$ the sole input to the PDE subsystem.
• Figure 2: Depiction of the PDE subsystem for use in defining a GPDE. All interaction of the PDE subsystem with the ODE subsystem is routed through the two vector-valued signals: $r(t)$ an output of the PDE subsystem (and input to the ODE subsystem) and $v(t)$ is input to the PDE subsystem (and output from the ODE subsystem).
• Figure 3: A GPDE is the interconnection of an ODE subsystem (an ODE with finite-dimensional inputs $w,u,v$ and outputs $z,y,r$) with a PDE subsystem ($N$th-order PDEs and BCs with finite-dimensional input $r$ and output $v$). The BCs and internal dynamics of the PDE subsystem are specified in terms of all well-defined spatially distributed terms as encoded in $\mathcal{ F} \hat{\mathbf{ x}}(t)$ and all well-defined limit values as encoded in $\mathcal{ B} \hat{\mathbf{ x}}(t)$.
• Figure 4: Definitions based on $n \in \mathbb{N}^{N+1}$ and the parameters of $\mathbf{ G}_{\mathrm b}:=\{B$, $B_{I}$, $B_v\}$ used in \ref{['thm:T_map']}.
• Figure 5: Definitions based on the PDE and GPDE parameters in $\mathbf{ G}_{\mathrm{p}}$$=$$\{A_{0},$$A_{1},$$A_{2},$$B_{xv},$$B_{xb},$$C_{r},$$D_{rb}\}$ and $\mathbf{ G}_{\mathrm{o}}$$=$$\{A,$$B_{xw},$$B_{xu},$$B_{xr},$$C_z,$$D_{zw},$$D_{zu},$$D_{zr},$$C_y,$$D_{yw},$$D_{yu},$$D_{yr},$$C_v,$$D_{vw},$$D_{vu}\}$, the Definitions from $\mathbf{ G}_{\mathrm b}$ as listed in \ref{['fig:Gb_definitions']} and the map $\mathbf{ P}_\times^4$ as defined in \ref{['eq:pi_comp_map']}.

Theorems & Definitions (82)

• Definition 1: Separable Function
• Definition 2: 3-PI operators, $\Pi_3$
• Definition 3: 4-PI operators
• Definition 4: *-subalgebras of $\Pi_i$ with polynomial parameters
• Definition 5: Admissible solution for a PIE system
• Definition 6: Admissible solution for an ODE subsystem
• Definition 7: Admissible solution for a PDE subsystem
• Definition 8: Admissible solution for a GPDE model
• Definition 9: PIE-compatible
• Theorem 10