TY - GEN
T1 - A PIE Representation of Scalar Quadratic PDEs and Global Stability Analysis Using SDP
AU - Jagt, Declan
AU - Seiler, Peter
AU - Peet, Matthew
N1 - Publisher Copyright:
© 2023 IEEE.
PY - 2023
Y1 - 2023
N2 - 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 L2 - 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.
AB - 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 L2 - 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.
UR - http://www.scopus.com/inward/record.url?scp=85184830626&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85184830626&partnerID=8YFLogxK
U2 - 10.1109/CDC49753.2023.10384073
DO - 10.1109/CDC49753.2023.10384073
M3 - Conference contribution
AN - SCOPUS:85184830626
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 2950
EP - 2957
BT - 2023 62nd IEEE Conference on Decision and Control, CDC 2023
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 62nd IEEE Conference on Decision and Control, CDC 2023
Y2 - 13 December 2023 through 15 December 2023
ER -