Pointwise Jacobson type necessary conditions for optimal control problems governed by impulsive differential systems

Huifu Xia; Yunfei Peng; Huifu Xia; Yunfei Peng

doi:10.3934/era.2024094

Electronic Research Archive

2024, Volume 32, Issue 3: 2075-2098. doi: 10.3934/era.2024094

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Research article

Pointwise Jacobson type necessary conditions for optimal control problems governed by impulsive differential systems

Huifu Xia ^,,
Yunfei Peng

School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

Received: 31 December 2023 Revised: 29 February 2024 Accepted: 01 March 2024 Published: 07 March 2024

This work focuses on an exploration of the pointwise Jacobson-type necessary conditions for optimal control problems governed by differential systems with impulse at fixed times; the pointwise Jacobson-type necessary optimality conditions refer to a type of pointwise second-order necessary optimality conditions for optimal singular control in the classical sense. By introducing an impulsive linear matrix Riccati differential equation, we derive the integral representation of the functional second-order variational. Based on this, the integral form of the second-order necessary conditions and the pointwise Jacobson-type necessary conditions are obtained. Incidentally, we have established the Legendre-Clebsch condition and the pointwise Legendre-Clebsch condition. Finally, an example is provided to illustrate the effectiveness of the main result.
- pointwise Jacobson-type necessary conditions,
- Legendre-Clebsch condition,
- optimal singular control,
- impulsive differential systems,
- optimal control
Citation: Huifu Xia, Yunfei Peng. Pointwise Jacobson type necessary conditions for optimal control problems governed by impulsive differential systems[J]. Electronic Research Archive, 2024, 32(3): 2075-2098. doi: 10.3934/era.2024094

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Abstract

This work focuses on an exploration of the pointwise Jacobson-type necessary conditions for optimal control problems governed by differential systems with impulse at fixed times; the pointwise Jacobson-type necessary optimality conditions refer to a type of pointwise second-order necessary optimality conditions for optimal singular control in the classical sense. By introducing an impulsive linear matrix Riccati differential equation, we derive the integral representation of the functional second-order variational. Based on this, the integral form of the second-order necessary conditions and the pointwise Jacobson-type necessary conditions are obtained. Incidentally, we have established the Legendre-Clebsch condition and the pointwise Legendre-Clebsch condition. Finally, an example is provided to illustrate the effectiveness of the main result.

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