Arbitrary finite spectrum assignment and stabilization of bilinear systems with multiple lumped and distributed delays in state

Vasilii Zaitsev; Inna Kim; Vasilii Zaitsev; Inna Kim

doi:10.3934/math.2025317

AIMS Mathematics

2025, Volume 10, Issue 3: 6934-6951. doi: 10.3934/math.2025317

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Arbitrary finite spectrum assignment and stabilization of bilinear systems with multiple lumped and distributed delays in state

Vasilii Zaitsev ^,,
Inna Kim

Laboratory of Mathematical Control Theory, Udmurt State University, Izhevsk, 426034, Russia

Received: 08 December 2024 Revised: 23 February 2025 Accepted: 18 March 2025 Published: 26 March 2025
MSC : 93B55, 93D23, 93C43, 34H15

We consider a bilinear control system defined by a linear time-invariant system of differential equations with multiple lumped and distributed delays in the state variable. A problem of finite spectrum assignment is studied. One needs to construct control vectors such that the characteristic function of the closed-loop system is equal to a polynomial with arbitrary given coefficients. We obtain conditions on coefficients of the system under which the criterion was found for solvability of this finite spectrum assignment problem. This criterion is expressed in terms of rank conditions for matrices of the special form. Corollaries on stabilization of a bilinear system with delays are obtained. An illustrative example is presented.
- linear time-invariant differential system,
- time-delay systems,
- spectrum assignment problem,
- stabilization,
- bilinear system
Citation: Vasilii Zaitsev, Inna Kim. Arbitrary finite spectrum assignment and stabilization of bilinear systems with multiple lumped and distributed delays in state[J]. AIMS Mathematics, 2025, 10(3): 6934-6951. doi: 10.3934/math.2025317

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Abstract

We consider a bilinear control system defined by a linear time-invariant system of differential equations with multiple lumped and distributed delays in the state variable. A problem of finite spectrum assignment is studied. One needs to construct control vectors such that the characteristic function of the closed-loop system is equal to a polynomial with arbitrary given coefficients. We obtain conditions on coefficients of the system under which the criterion was found for solvability of this finite spectrum assignment problem. This criterion is expressed in terms of rank conditions for matrices of the special form. Corollaries on stabilization of a bilinear system with delays are obtained. An illustrative example is presented.

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