arXiv subjects

Declan S. Jagt

Publications and source records attributed to Declan S. Jagt.

8 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

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

$H_{\infty}$-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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arXiv2022

$L_2$-Gain Analysis of Coupled Linear 2D PDEs using Linear PI Inequalities

In this paper, we present a new method for estimating the $L_2$-gain of systems governed by 2nd order linear Partial Differential Equations (PDEs) in two spatial variables, using semidefinite programming. It has previously been shown that, for any such PDE, an equivalent Partial Integral Equation (PIE) can be derived. These PIEs are expressed in terms of Partial Integral (PI) operators mapping states in $L_2[Ω]$, and are free of the boundary and continuity constraints appearing in PDEs. In this paper, we extend the 2D PIE representation to include input and output signals in $\mathbb{R}^n$, deriving a bijective map between solutions of the PDE and the PIE, along with the necessary formulae to convert between the two representations. Next, using the algebraic properties of PI operators, we prove that an upper bound on the $L_2$-gain of PIEs can be verified by testing feasibility of a Linear PI Inequality (LPI), defined by a positivity constraint on a PI operator mapping $\mathbb{R}^n\times L_2[Ω]$. Finally, we use positive matrices to parameterize a cone of positive PI operators on $\mathbb{R}^n\times L_2[Ω]$, allowing feasibility of the $L_2$-gain LPI to be tested using semidefinite programming. We implement this test in the MATLAB toolbox PIETOOLS, and demonstrate that this approach allows an upper bound on the $L_2$-gain of PDEs to be estimated with little conservatism.

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arXiv2022

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

We introduce a Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in two spatial variables. PIEs are an algebraic state-space representation of infinite-dimensional systems and have been used to model 1D PDEs and time-delay systems without continuity constraints or boundary conditions -- making these PIE representations amenable to stability analysis using convex optimization. To extend the PIE framework to 2D PDEs, we first construct an algebra of Partial Integral (PI) operators on the function space L_2[x,y], providing formulae for composition, adjoint, and inversion. We then extend this algebra to R^n x L_2[x] x L_2[y] x L_2[x,y] and demonstrate that, for any suitable coupled, linear PDE in 2 spatial variables, there exists an associated PIE whose solutions bijectively map to solutions of the original PDE -- providing conversion formulae between these representations. Next, we use positive matrices to parameterize the convex cone of 2D PI operators -- allowing us to optimize PI operators and solve Linear PI Inequality (LPI) feasibility problems. Finally, we use the 2D LPI framework to provide conditions for stability of 2D linear PDEs. We test these conditions on 2D heat and wave equations and demonstrate that the stability condition has little to no conservatism.

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