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Matthew M. Peet

Publications and source records attributed to Matthew M. Peet.

At least 19 recordsLinked to original sources
arXiv2026

A State-Space Representation of Coupled Linear Multivariate PDEs and Stability Analysis using SDP

Recently, it has been shown that stability analysis and control of coupled Partial Differential Equations (PDEs) in a single spatial variable can be more conveniently performed using the Partial Integral Equation (PIE) representation. This PIE offers an equivalent, state-space representation of the PDE on the Hilbert space $L_{2}$, and is parameterized by an algebra of Partial Integral (PI) operators, allowing e.g. stability to be analyzed by solving a linear operator inequality on PI operator variables. In this paper, we show how this PIE framework for univariate PDEs can be extended inductively to multivariate PDEs on a hyper-rectangle. Specifically, assuming the boundary conditions defining the domain of the PDE to be decoupled along distinct spatial directions, we propose a readily verifiable condition for existence of a bijection between the PDE domain and $L_{2}$. We derive an explicit expression for the map defining this bijection, and use this map to construct an equivalent PIE representation for a broad class of linear multivariate PDEs. Next, we embed the parameters defining this PIE representation in a class of multivariate PI operators, and prove that this class forms a $*$-algebra---allowing PI operator inequalities for stability analysis, estimation, and control of univariate PDEs to be similarly formulated for multivariate PDEs. Finally, we show how such an operator inequality for stability analysis of multivariate PDEs can be solved with semidefinite programming, using a positive matrix parameterization of positive semidefinite multivariate PI operators. This framework for representation and stability analysis of multivariate PDEs is incorporated in the PIETOOLS software, and applied to analyze stability of 2D heat, wave, and plate equations, obtaining accurate bounds on the rate of decay.

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arXiv2026

Verifying Well-Posedness of Linear PDEs using Convex Optimization

Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer--Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE---a proper subspace of $L_{2}$---which can make them difficult to verify in practice. In this paper, we show how the Lumer--Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer--Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing an upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs.

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arXiv2026

Picard Iteration for Parameter Estimation in Nonlinear Ordinary Differential Equations using Low-Quality Data

We consider the problem of using experimental time-series data for parameter estimation in nonlinear ordinary differential equations, focusing on the case where the data is noisy, sparse, irregularly sampled, includes multiple experiments, and does not directly measure the system state or its time derivative. To account for such low-quality data, we propose a new framework for gradient-based parameter estimation which uses the Picard operator to reformulate the problem as constrained optimization with infinite-dimensional variables and constraints. We then formulate the Karush-Kuhn-Tucker (KKT) conditions necessary for optimality and define a convergent sequence of approximations to these KKT conditions obtained by replacing the solution map by the n-th order Picard iterate. Then, for any element of this sequence, and by exploiting the contractive properties of the Picard operator, we propose a gradient-contractive algorithm which (under regularity and convexity assumptions) is guaranteed to converge to a solution of these approximated KKT conditions. Finally, the algorithms are then tested on a battery of models and a variety of datasets in order to demonstrate robustness and improvement over alternative approaches.

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arXiv2026

Hierarchical Stability and Lyapunov Conditions for Linear PDEs

Unlike ordinary differential equations (ODEs), linear partial differential equations (PDEs) admit multiple non-equivalent notions of stability. This variety makes interpretation of Lyapunov stability results challenging. \blue{To simplify this interpretation, we propose a unified framework for hierarchical classification of notions of stability and Lyapunov conditions. To do this, for classically well-posed PDEs with admissible boundary conditions}, we define a fundamental state on $L_2$ corresponding to the minimal information needed to uniquely forward propagate the solution. Stability notions and Lyapunov functions are then defined in terms of this fundamental state. This gives rise to a hierarchy of stability notions, the weakest being fundamental state to PDE state stability. Other stability notions and Lyapunov conditions may then be interpreted relative to this weakest notion. Hierarchies are established for: Lyapunov, exponential and finite-energy stability. Sufficient Lyapunov conditions are defined in terms of operator inequalities. Illustrative examples and computational tools are provided.

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arXiv2026

Static Output Feedback Stabilization of Linear Systems with Multiple Delays

This work proposes a new procedure for the stabilization of time-delay systems using Static Output Feedback (SOF) control. A previous convex optimization approach to SOF for Ordinary Differential Equations (ODEs) is extended to time-delay systems through the use of a proposed state-space representation. This approach is based on solving two convex optimization problems, which are extensions of Linear Matrix Inequalities (LMIs) to infinite-dimensional systems. The first problem is stabilization under state feedback control; the second problem takes advantage of the Projection Lemma, which is extended here from matrices to Partial Integral (PI) operators. Finally, the results are compared with other SOF solutions for systems with delay found in the literature, showing a significant reduction in conservatism.

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arXiv2026

$H_2$-Optimal Estimation of Linear Delayed and PDE Systems

The $H_2$ norm is a commonly used performance metric in the design of estimators. However, $H_2$-optimal estimation of most PDEs is complicated by the lack of transfer function and state-space representations. To address this problem, we first re-characterize the $H_2$-norm in terms of a map from initial condition to output. We then leverage the Partial Integral Equation (PIE) state-space representation of systems of linear PDEs coupled with ODEs to recast this characterization of $H_2$-norm as a convex optimization problem defined in terms of Linear Partial Integral (LPI) inequalities. We then parameterize a class of PIE-based observers and solve the associated $H_2$-optimal estimation problem. The observer synthesis problem is then recast as an LPI, and the resulting observers are validated using numerical simulation.

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arXiv2026

Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems

Abel's classic transformation shows that any well-posed system with time-varying delay is equivalent to a parameter-varying system with fixed delay. The existence of such a parameter-varying constant delay representation then simplifies the problems of stability analysis and optimal control. Unfortunately, the method for construction of such transformations has been ad-hoc -- requiring an iterative time-stepping approach to constructing the transformation beginning with a seed function subject to boundary-value constraints. Moreover, a poor choice of seed function often results in a constant delay representation with large time-variations in system parameters -- obviating the benefits of such a representation. In this paper, we show how the set of all feasible seed functions can be parameterized using a basis for $L_2$. This parameterization is then used to search for seed functions for which the corresponding time-transformation results in smaller parameter variation. The parameterization of admissible seed functions is illustrated with numerical examples that contrast how well-chosen and poorly chosen seed functions affect the boundedness of a time transformation.

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arXiv2026

Time-Transformation-Based Analysis of Systems with Periodic Delay via Perturbative Expansion

It is difficult to analyze the stability of systems with time-varying delays. One approach is to construct a time-transformation that converts the system into a form with a constant delay but with a time-varying scalar appearing in the system matrices. The stability of this transformed system can then be analyzed using methods to bound the effect of the time-varying scalar. One issue is that this transformation is non-unique and requires the solution of an Abel equation. A specific time-transformation typically must be computed numerically. We address this issue by computing an explicit, although approximate, time-transformation for systems where the delay has a constant plus small periodic term. We use a perturbative expansion to construct our explicit solutions. We provide a simple numerical example to illustrate the approach. We also demonstrate the use of this time-transformation to analyze stability of the system with this class of periodic delays.

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arXiv2026

Lyapunov Functions can Exactly Quantify Rate Performance of Nonlinear Differential Equations

Pointwise-in-time stability notions for Ordinary Differential Equations (ODEs) provide quantitative metrics for system performance by establishing bounds on the rate of decay of the system state in terms of initial condition -- allowing stability to be quantified by e.g. the maximum provable decay rate. Such bounds may be obtained by finding suitable Lyapunov functions using, e.g. Sum-of-Squares (SOS) optimization. While Lyapunov tests have been proposed for numerous pointwise-in-time stability notions, including exponential, rational, and finite-time stability, it is unclear whether these characterizations are able to provide accurate bounds on system performance. In this paper, we start by proposing a generalized notion of rate performance -- with exponential, rational, and finite-time decay rates being special cases. Then, for any such notion and rate, we associate a Lyapunov condition which is shown to be necessary and sufficient for a system to achieve that rate. Finally, we show how the proposed conditions can be enforced using SOS programming in the case of exponential, rational, and finite-time stability. Numerical examples in each case demonstrate that the corresponding SOS test can achieve tight bounds on the rate performance with accurate inner bounds on the associated regions of performance.

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arXiv2024

Efficient Convex Algorithms for Universal Kernel Learning

The accuracy and complexity of machine learning algorithms based on kernel optimization are determined by the set of kernels over which they are able to optimize. An ideal set of kernels should: admit a linear parameterization (for tractability); be dense in the set of all kernels (for robustness); be universal (for accuracy). Recently, a framework was proposed for using positive matrices to parameterize a class of positive semi-separable kernels. Although this class can be shown to meet all three criteria, previous algorithms for optimization of such kernels were limited to classification and furthermore relied on computationally complex Semidefinite Programming (SDP) algorithms. In this paper, we pose the problem of learning semiseparable kernels as a minimax optimization problem and propose a SVD-QCQP primal-dual algorithm which dramatically reduces the computational complexity as compared with previous SDP-based approaches. Furthermore, we provide an efficient implementation of this algorithm for both classification and regression -- an implementation which enables us to solve problems with 100 features and up to 30,000 datums. Finally, when applied to benchmark data, the algorithm demonstrates the potential for significant improvement in accuracy over typical (but non-convex) approaches such as Neural Nets and Random Forest with similar or better computation time.

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arXiv2024

Model Predictive Bang-Bang Controller Synthesis via Approximate Value Functions

In this paper, we propose a novel method for addressing Optimal Control Problems (OCPs) with input-affine dynamics and cost functions. This approach adopts a Model Predictive Control (MPC) strategy, wherein a controller is synthesized to handle an approximated OCP within a finite time horizon. Upon reaching this horizon, the controller is re-calibrated to tackle another approximation of the OCP, with the approximation updated based on the final state and time information. To tackle each OCP instance, all non-polynomial terms are Taylor-expanded about the current time and state and the resulting Hamilton-Jacobi-Bellman (HJB) PDE is solved via Sum-of-Squares (SOS) programming, providing us with an approximate polynomial value function that can be used to synthesize a bang-bang controller.

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arXiv2024

$H_{\infty}$-Optimal Estimator Synthesis for Coupled Linear 2D PDEs using Convex Optimization

Any suitably well-posed PDE in two spatial dimensions can be represented as a Partial Integral Equation (PIE) -- with system dynamics parameterized using Partial Integral (PI) operators. Furthermore, $L_2$-gain analysis of PDEs with a PIE representation can be posed as a linear operator inequality, which can be solved using convex optimization. In this paper, these results are used to derive a convex-optimization-based test for constructing an $H_{\infty}$-optimal estimator for 2D PDEs. In particular, we first use PIEs to represent an arbitrary well-posed 2D PDE where sensor measurements occur along some boundary of the domain. An associated Luenberger-type estimator is then parameterized using a PI operator $\mathcal{L}$ as the observer gain. Examining the error dynamics of this estimator, it is proven that an upper bound on the $H_{\infty}$-norm of these error dynamics can be minimized by solving a linear operator inequality on PI operator variables. Finally, an analytical formula is proposed for inversion of a class of 2D PI operators, which is then used to reconstruct the Luenberger gain $\mathcal{L}$. Results are implemented in the PIETOOLS software suite -- applying the methodology and simulating the resulting observer for an unstable 2D heat equation with boundary observations.

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arXiv2024

Constructive Representation of Functions in $N$-Dimensional Sobolev Space

A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed derivative, as well as their lower-order derivatives evaluated along suitable boundaries of the domain. The proposed expansion is proven to be invertible, uniquely identifying any function in the Sobolev space with its derivatives and boundary values. Since these boundary values are either finite-dimensional, or exist in the space of square-integrable functions, this offers a bijective relation between the Sobolev space and $L_{2}$. Using this bijection, it is shown how approximation of functions in Sobolev space can be performed in the less restrictive space $L_{2}$, reconstructing such an approximation of the function from an $L_{2}$-optimal projection of its boundary values and highest-order derivative. This approximation method is presented using a basis of Legendre polynomials and a basis of step functions, and results using both bases are demonstrated to exhibit better convergence behavior than a direct projection approach for two numerical examples.

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arXiv2023

Representation of PDE Systems with Delay and Stability Analysis using Convex Optimization -- Extended Version

Partial Integral Equations (PIEs) have been used to represent both systems with delay and systems of Partial Differential Equations (PDEs) in one or two spatial dimensions. In this paper, we show that these results can be combined to obtain a PIE representation of any suitably well-posed 1D PDE model with constant delay. In particular, we represent these delayed PDE systems as coupled systems of 1D and 2D PDEs, obtaining a PIE representation of both subsystems. Taking the feedback interconnection of these PIE subsystems, we then obtain a 2D PIE representation of the 1D PDE with delay. Next, based on the PIE representation, we formulate the problem of stability analysis as convex optimization of positive operators which can be solved using the PIETOOLS software suite. We apply the result to PDE examples with delay in the state and boundary conditions.

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arXiv2023

Employing Feature Selection Algorithms to Determine the Immune State of a Mouse Model of Rheumatoid Arthritis

The immune response is a dynamic process by which the body determines whether an antigen is self or nonself. The state of this dynamic process is defined by the relative balance and population of inflammatory and regulatory actors which comprise this decision making process. The goal of immunotherapy as applied to, e.g. Rheumatoid Arthritis (RA), then, is to bias the immune state in favor of the regulatory actors - thereby shutting down autoimmune pathways in the response. While there are several known approaches to immunotherapy, the effectiveness of the therapy will depend on how this intervention alters the evolution of this state. Unfortunately, this process is determined not only by the dynamics of the process, but the state of the system at the time of intervention - a state which is difficult if not impossible to determine prior to application of the therapy. To identify such states we consider a mouse model of RA (Collagen-Induced Arthritis (CIA)) immunotherapy; collect high dimensional data on T cell markers and populations of mice after treatment with a recently developed immunotherapy for CIA; and use feature selection algorithms in order to select a lower dimensional subset of this data which can be used to predict both the full set of T cell markers and populations, along with the efficacy of immunotherapy treatment.

stat.ML↗
arXiv2023

Integral Quadratic Constraints with Infinite-Dimensional Channels

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such input-output analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinite-dimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinite-dimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.

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arXiv2023

A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose 1-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.

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arXiv2023

A New Treatment of Boundary Conditions in PDE Solution with Galerkin Methods via Partial Integral Equation Framework

We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing equation. The framework is based on a Partial Integral Equation (PIE) representation of PDEs, where a PDE equation is transformed into an equivalent PIE formulation that does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized PIE-Galerkin approximation methodology for a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries. The significance of this result is that solution to almost any linear PDE can now be constructed in a form of an analytical approximation based on a series expansion using a suitable set of basis functions, such as, e.g., Chebyshev polynomials of the first kind, irrespective of the boundary conditions. In many cases involving homogeneous or simple time-dependent boundary inputs, an analytical integration in time is also possible. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

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