We present a framework for stability analysis of systems of coupled linear PartialDifferential Equations (PDEs). The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichelet, Neuman and mixed boundary conditions. The results in this paper apply to systems with a single spatial variable and assume existence and continuity of solutions except in such cases when existence and continuity can be inferred from existence of a Lyapunov function. Our approach is based on a new concept of state for PDE systems which allows us to express the derivative of the Lyapunov function as amore »
A Generalized LMI Formulation for InputOutput Analysis of Linear Systems of ODEs Coupled with PDEs
In this paper, we consider inputoutput properties
of linear systems consisting of PDEs on a finite domain coupled
with ODEs through the boundary conditions of the PDE. This
framework can be used to represent e.g. a lumped mass fixed
to a beam or a system with delay. This work generalizes the
sufficiency proof of the KYP Lemma for ODEs to coupled
ODEPDE systems using a recently developed concept of
fundamental state and the associated boundaryconditionfree
representation. The conditions of the generalized KYP are
tested using the PQRS positive matrix parameterization of
operators resulting in a finitedimensional LMI, feasibility of
which implies prima facie provable passivity or L2gain of
the system. No discretization or approximation is involved at
any step and we use numerical examples to demonstrate that
the bounds obtained are not conservative in any significant
sense and that computational complexity is lower than existing
methods involving finitedimensional projection of PDEs.
 Award ID(s):
 1935453
 Publication Date:
 NSFPAR ID:
 10113773
 Journal Name:
 Proceedings of the IEEE Conference on Decision & Control
 ISSN:
 25762370
 Sponsoring Org:
 National Science Foundation
More Like this


In this work, we present a Linear Matrix Inequality (LMI) based method to synthesize an optimal H1 estimator for a large class of linear coupled partial differential equations (PDEs) utilizing only finite dimensional measurements. Our approach extends the newly developed framework for representing and analyzing distributed parameter systems using operators on the space of square integrable functions that are equipped with multipliers and kernels of semiseparable class. We show that by redefining the state, the PDEs can be represented using operators that embed the boundary conditions and inputoutput relations explicitly. The optimal estimator synthesis problem is formulated as a convexmore »

Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and blocktridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistorcapacitorinductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MORmore »

For a given PDE problem, three main factors affect the accuracy of FEM solutions: basis order, mesh resolution, and mesh element quality. The first two factors are easy to control, while controlling element shape quality is a challenge, with fundamental limitations on what can be achieved. We propose to use prefinement (increasing element degree) to decouple the approximation error of the finite element method from the domain mesh quality for elliptic PDEs. Our technique produces an accurate solution even on meshes with badly shaped elements, with a slightly higher running time due to the higher cost of highorder elements. Wemore »

We study the Schr{\"o}dinger bridge problem (SBP) with nonlinear prior dynamics. In controltheoretic language, this is a problem of minimum effort steering of a given joint state probability density function (PDF) to another over a finite time horizon, subject to a controlled stochastic differential evolution of the state vector. For generic nonlinear drift, we reduce the SBP to solving a system of forward and backward Kolmogorov partial differential equations (PDEs) that are coupled through the boundary conditions, with unknowns being the ``Schr\"{o}dinger factors". We show that if the drift is a gradient vector field, or is of mixed conservativedissipative nature,more »