Insight into the Stability Analysis of the Reaction-Diffusion Equation Interconnected with a Finite-Dimensional System Taking Support on Legendre Orthogonal Basis

Abstract

This chapter deals with the stability analysis of the reaction-diffusion subject to dynamic boundary conditions. More particularly, the objective is to propose a linear matrix inequality criterion which ensures the stability of such infinite-dimensional system. By the use of Fourier-Legendre series, the Lyapunov functional is split into an augmented finite-dimensional state including within it the first Fourier-Legendre coefficients and the residual part. A link between this modeling and Padé approximation is briefly highlighted. Then, from Bessel and Wirtinger inequalities applied to the Fourier-Legendre remainder and using its orthogonality properties, a sufficient condition of stability expressed in terms of linear matrix inequalities is obtained. This efficient and scalable stability condition is finally performed on examples.

Appendix

Derived from the matrix inversion and determinant lemmas, a usefull lemma can also be formulated.

Lemma 7

For any $u\in \mathbb {R}^n$ with a non-zero first component, $v\in \mathbb {R}^n$ not equal to the zero vector and $L\in \mathbb {R}^{n\times n}$ a strictly lower triangular matrix with non-zero under diagonal coefficients (i.e. $L_{i+1,i}\ne 0\;\forall i\in \{1,\dots n-1\}$), one obtains

$$\begin{aligned} 1-v^T((s-\lambda )I_n+L+uv^T)^{-1}u=\underset{s\rightarrow \lambda }{O}{(s^n)}. \end{aligned}$$

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Proof

The matrix inversion lemma applied to vectors $u\in \mathbb {R}^n$, $v\in \mathbb {R}^n$ and non singular matrix $M=sI_n+L\in \mathbb {R}^{n\times n}$ gives, for any $s\in \mathbb {C}\backslash \{0\}$,

$$\begin{aligned} 1-v^T((s-\lambda )I_n+L+uv^T)^{-1}u = (1+v^T((s-\lambda )I_n+L)^{-1}u)^{-1}, \end{aligned}$$

and the matrix determinant lemma leads to

$$\begin{aligned} 1-v^T((s-\lambda )I_n+L+uv^T)^{-1}u&= \frac{\mathrm {det}((s-\lambda )I_n+L)}{\mathrm {det}((s-\lambda )I_n+L+uv^T)}. \end{aligned}$$

Then, since L is strictly lower triangular, we have

$$\begin{aligned} \mathrm {det}((s-\lambda )I_n+L) = \mathrm {det}((s-\lambda )I_n) = (s-\lambda )^n. \end{aligned}$$

and, because L has non-zero under diagonal coefficients and under the hypothesis done on vectors u, v, matrix $L+uv^T$ has full rank which means $\mathrm {det}(L+uv^T)\ne 0$. That yields the result for s tending to $\lambda$.