Opuscula Mathematica

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ISSN:: 1232-9274
Category:: MATHEMATICS
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Wintner-type asymptotic behavior of linear differential systems with a proportional derivative controller

2026-01-27

Author(s): Kazuki Ishibashi. Abstract: This study investigated the asymptotic behavior of linear differential systems incorporating a proportional derivative-type (PD) differential operator. Building on the classical asymptotic convergence property of Wintner, a generalized Wintner-type asymptotic result was established for such systems. The proposed framework encompasses a wide class of time-varying coefficient matrices and extends classical asymptotic theory to equations governed by PD operators. An illustrative example is presented to demonstrate the applicability of the proposed theorem. Keywords: Wintner-type, asymptotic behavior, linear differential systems, proportional-derivative controller. Mathematics Subject Classification: 34D05, 26A24. Journal: Opuscula Mathematica. Citation: Opuscula Math. 46 , no. 1 (2026), 41-54, https://doi.org/10.7494/OpMath.202512101 .  

Parametric formal Gevrey asymptotic expansions in two complex time variable problems

2026-01-27

Author(s): Guoting Chen, Alberto Lastra, Stéphane Malek. Abstract: The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution. Keywords: singularly perturbed, formal solution, several complex variables, Cauchy problem. Mathematics Subject Classification: 35C10, 35R10, 35C15, 35C20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 46 , no. 1 (2026), 5-40, https://doi.org/10.7494/OpMath.202512271 .  