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Peer-reviewedCC BY 4.0Axioms

The Well-Posedness and Attainability of an LQ Optimal Control Problem for Mean-Field Linear Stochastic Systems Controlled by Impulses

  • Vasile Drăgan1,2
  • Ivan Ganchev Ivanov3
  1. 1Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
  2. 2Academy of the Romanian Scientists, 3 Ilfov, 050044 Bucharest, Romania
  3. 3Faculty of Economics and Business Administration, Sofia University “St. Kl. Ohridski”, 125 Tzarigradsko Chaussee Blvd., bl. 3, 1113 Sofia, Bulgaria
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Abstract

In this work, the well-posedness and attainability of a finite horizon time linear quadratic (LQ) optimal control problem for a class of McKean–Vlasov stochastic linear differential equations controlled by impulses are addressed. No assumption regarding the sign of the weighting matrices involved in the performance criterion is made. First, employing an orthogonal decomposition of the state variables and of the control parameters, we obtain an equivalent version of the original problem of the optimal control. Then, the new optimal control problem is embedded into a so-called extended LQ optimal control problem. The well-posedness and the closed-loop attainability properties initially defined for the original LQ optimal control problem are then adapted to the extended optimal control problem. The main tool involved in the analysis of the well-posedness and the closed-loop attainability properties of the extended LQ optimal control problem is the solution of a system of two coupled backward jump matrix Lyapunov-type differential equations with a Riccati-type jumping operator. This solution is subject to a set of constraints due to the absence of information about the rank and the sign of the weighting matrices from the objective functions. We show that the existence of such a solution is a necessary and sufficient condition for the closed-loop attainability of the extended LQ optimal control problem. Explicit formulae of the gain matrices of the optimal feedback are provided.

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Cite

BibTeXRIS
BibTeX
@article{dragan2026wellposedness,
  title = {{The Well-Posedness and Attainability of an LQ Optimal Control Problem for Mean-Field Linear Stochastic Systems Controlled by Impulses}},
  author = {Vasile Drăgan and Ivan Ganchev Ivanov},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100729},
  url = {https://www.mdpi.com/2075-1680/15/10/729},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - The Well-Posedness and Attainability of an LQ Optimal Control Problem for Mean-Field Linear Stochastic Systems Controlled by Impulses
AU  - Vasile Drăgan
AU  - Ivan Ganchev Ivanov
PY  - 2026
DA  - 2026/10/01
JO  - Axioms
DO  - 10.3390/axioms15100729
UR  - https://www.mdpi.com/2075-1680/15/10/729
AB  - In this work, the well-posedness and attainability of a finite horizon time linear quadratic (LQ) optimal control problem for a class of McKean–Vlasov stochastic linear differential equations controlled by impulses are addressed. No assumption regarding the sign of the weighting matrices involved in the performance criterion is made. First, employing an orthogonal decomposition of the state variables and of the control parameters, we obtain an equivalent version of the original problem of the optimal control. Then, the new optimal control problem is embedded into a so-called extended LQ optimal control problem. The well-posedness and the closed-loop attainability properties initially defined for the original LQ optimal control problem are then adapted to the extended optimal control problem. The main tool involved in the analysis of the well-posedness and the closed-loop attainability properties of the extended LQ optimal control problem is the solution of a system of two coupled backward jump matrix Lyapunov-type differential equations with a Riccati-type jumping operator. This solution is subject to a set of constraints due to the absence of information about the rank and the sign of the weighting matrices from the objective functions. We show that the existence of such a solution is a necessary and sufficient condition for the closed-loop attainability of the extended LQ optimal control problem. Explicit formulae of the gain matrices of the optimal feedback are provided.
SN  - 2075-1680
ER  -