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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.08126 |
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| _version_ | 1866914545683922944 |
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| author | Ennaceur, Marwa |
| author_facet | Ennaceur, Marwa |
| contents | This paper studies Rota-Baxter operators on the matrix $C^*$-algebra $M_n(\mathbb{C})$, motivated by the discrete Toeplitz algebra (whose role is purely heuristic; see Remark~\ref{rem:toeplitz_scope}). We provide a structural classification of such operators compatible with the $C^*$-norm, analyze their induced Lie brackets, and apply them to deform system matrices in discrete-time delayed systems under sliding mode control. Lyapunov-based Bilinear Matrix Inequality conditions, together with a tractable linear reformulation via $Q=X^{-1}$, guarantee asymptotic stability on the sliding manifold and $\mathcal{L}_2$-gain stability. The effective gain from uncertainty $δ$ to state $x$ is $γ/\sqrtμ$ with $μ=λ_{\min}(-\mathcal{M})>0$ determined \emph{a posteriori}; minimizing $γ$ alone does not minimize this bound, which holds under zero extended initial conditions ($V_0=0$). We work under the standing assumption $m=n$ (square actuation); a supplementary non-degenerate example with $m=1$, $n=2$ illustrates LMI feasibility with $Π\neq0$. All algebraic results are proved directly in $M_n(\mathbb{C})$; no infinite-dimensional reduction is used. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Study of Rota-Baxter Operators in Matrix $C^*$-Algebras Motivated by Toeplitz Structures, and Applications to Sliding Mode Control Ennaceur, Marwa Rings and Algebras Dynamical Systems Functional Analysis Operator Algebras 16W99, 17B99, 46L05, 93D05, 93C55 This paper studies Rota-Baxter operators on the matrix $C^*$-algebra $M_n(\mathbb{C})$, motivated by the discrete Toeplitz algebra (whose role is purely heuristic; see Remark~\ref{rem:toeplitz_scope}). We provide a structural classification of such operators compatible with the $C^*$-norm, analyze their induced Lie brackets, and apply them to deform system matrices in discrete-time delayed systems under sliding mode control. Lyapunov-based Bilinear Matrix Inequality conditions, together with a tractable linear reformulation via $Q=X^{-1}$, guarantee asymptotic stability on the sliding manifold and $\mathcal{L}_2$-gain stability. The effective gain from uncertainty $δ$ to state $x$ is $γ/\sqrtμ$ with $μ=λ_{\min}(-\mathcal{M})>0$ determined \emph{a posteriori}; minimizing $γ$ alone does not minimize this bound, which holds under zero extended initial conditions ($V_0=0$). We work under the standing assumption $m=n$ (square actuation); a supplementary non-degenerate example with $m=1$, $n=2$ illustrates LMI feasibility with $Π\neq0$. All algebraic results are proved directly in $M_n(\mathbb{C})$; no infinite-dimensional reduction is used. |
| title | Study of Rota-Baxter Operators in Matrix $C^*$-Algebras Motivated by Toeplitz Structures, and Applications to Sliding Mode Control |
| topic | Rings and Algebras Dynamical Systems Functional Analysis Operator Algebras 16W99, 17B99, 46L05, 93D05, 93C55 |
| url | https://arxiv.org/abs/2605.08126 |