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1. A PIE Representation of Scalar Quadratic PDEs and Global Stability Analysis Using SDP

   Jagt, Declan; Seiler, Peter; Peet, Matthew (December 2023, Proceedings of the IEEE Conference on Decision Control)

   It has recently been shown that the evolution of a linear Partial Differential Equation (PDE) can be more conveniently represented in terms of the evolution of a higher spatial derivative of the state. This higher spatial derivative (termed the `fundamental state') lies in $$L_2$$ - requiring no auxiliary boundary conditions or continuity constraints. Such a representation (termed a Partial Integral Equation or PIE) is then defined in terms of an algebra of bounded integral operators (termed Partial Integral (PI) operators) and is constructed by identifying a unitary map from the fundamental state to the state of the original PDE. Unfortunately, when the PDE is nonlinear, the dynamics of the associated fundamental state are no longer parameterized in terms of PI operators. However, in this paper we show that such dynamics can be compactly represented using a new tensor algebra of partial integral operators acting on the tensor product of the fundamental state. We further show that this tensor product of the fundamental state forms a natural distributed equivalent of the monomial basis used in representation of polynomials on a finite-dimensional space. This new representation is then used to provide a simple SDP-based Lyapunov test of stability of quadratic PDEs. The test is applied to three illustrative examples of quadratic PDEs. 

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2. A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

   Shivakumar, S.; Das, A.; Weiland, S.; Peet, M. (January 2020, Proceedings of the IEEE Conference on Decision & Control)

   In this paper, we consider input-output 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 ODE-PDE systems using a recently developed concept of fundamental state and the associated boundary-condition-free representation. The conditions of the generalized KYP are tested using the PQRS positive matrix parameterization of operators resulting in a finite-dimensional LMI, feasibility of which implies prima facie provable passivity or L2-gain 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 finite-dimensional projection of PDEs. 

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3. Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

   https://doi.org/10.1016/j.ifacol.2023.10.538  

   Braghini, Danilo; Peet, Matthew M. (January 2023, IFAC-PapersOnLine)

   Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables where the disturbance does not enter through the boundary. For such systems, we define a notion of H2-norm using an initial state-to-output framework and show that this notion reduces to more traditional concepts under the assumption of existence of a strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE representation and for such systems show that computing a minimal upper bound on the H2-norm of delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation results are available in the literature. The numerical results demonstrate the accuracy of the computed upper bound on the H2-norm. 

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4. Analysis of a Multiscale Interface Problem Based on the Coupling of Partial and Ordinary Differential Equations to Model Tissue Perfusion

   https://doi.org/10.1137/23M1628073  

   Bociu, Lorena; Broussard, Matthew; Guidoboni, Giovanna; Strikwerda, Sarah (March 2025, Multiscale Modeling & Simulation)

   In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the surrounding blood circulation. In this paper, we propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of fluid flows through deformable porous media equations is coupled with a systemic, 0D, lumped model of the remainder of the circulation, where the fluid flow through a vascular network is described via its analog with a current flowing through an electric circuit. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the surrounding circulatory system. This PDE/ODE coupling leads to interface conditions enforcing the continuity of mass and the balance of stresses across models at different scales, and careful consideration is taken to address this interface mismatch. The resulting system involves PDEs of mixed type with interface conditions depending on nonlinear ODEs. A new result on local existence of solutions for this multiscale interface coupling is provided in this article. 

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5. H_\infty Optimal Estimation for Linear Coupled {PDE} Systems

   Das, A.; Shivakumar, S.; Weiland, S.; Peet, M. (January 2020, Proceedings of the IEEE Conference on Decision & Control)

   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 semi-separable class. We show that by redefining the state, the PDEs can be represented using operators that embed the boundary conditions and input-output relations explicitly. The optimal estimator synthesis problem is formulated as a convex optimization subject to LMIs that require no approximation or discretization. A scalable algorithm is presented to synthesize the estimator. The algorithm is illustrated by suitable examples. 

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