Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint
This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameteri...
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MDPI AG
2025-04-01
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| Series: | Mathematics |
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| author | Yanrong Lu Jize Li Yonghui Zhou |
| author_facet | Yanrong Lu Jize Li Yonghui Zhou |
| author_sort | Yanrong Lu |
| collection | DOAJ |
| description | This paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided. |
| format | Article |
| id | doaj-art-b06dcf98f4c749f5a3ec3d7b5e158b75 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-b06dcf98f4c749f5a3ec3d7b5e158b752025-08-20T02:28:32ZengMDPI AGMathematics2227-73902025-04-01138132710.3390/math13081327Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality ConstraintYanrong Lu0Jize Li1Yonghui Zhou2School of Mathematics, Guizhou Normal University, Guiyang 550025, ChinaSchool of Mathematics, Guizhou Normal University, Guiyang 550025, ChinaSchool of Mathematics, Guizhou Normal University, Guiyang 550025, ChinaThis paper investigates a backward stochastic linear quadratic control problem with an expected-type equality constraint on the initial state. By using the Lagrange multiplier method, the problem with a uniformly convex cost functional is first transformed into an equivalent unconstrained parameterized backward stochastic linear quadratic control problem. Then, under the surjectivity of the linear constraint, the equivalence between the original problem and the dual problem is proven by Lagrange duality theory. Subsequently, with the help of the maximum principle, an explicit solution of the optimal control for the unconstrained problem is obtained. This solution is feedback-based and determined by an adjoint stochastic differential equation, a Riccati-type ordinary differential equation, a backward stochastic differential equation, and an equality, thereby yielding the optimal control for the original problem. Finally, an optimal control for an investment portfolio problem with an expected-type equality constraint on the initial state is explicitly provided.https://www.mdpi.com/2227-7390/13/8/1327backward stochastic optimal controlexpectation equality constraintmaximum principleRiccati equationLagrange duality theory |
| spellingShingle | Yanrong Lu Jize Li Yonghui Zhou Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint Mathematics backward stochastic optimal control expectation equality constraint maximum principle Riccati equation Lagrange duality theory |
| title | Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint |
| title_full | Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint |
| title_fullStr | Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint |
| title_full_unstemmed | Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint |
| title_short | Backward Stochastic Linear Quadratic Optimal Control with Expectational Equality Constraint |
| title_sort | backward stochastic linear quadratic optimal control with expectational equality constraint |
| topic | backward stochastic optimal control expectation equality constraint maximum principle Riccati equation Lagrange duality theory |
| url | https://www.mdpi.com/2227-7390/13/8/1327 |
| work_keys_str_mv | AT yanronglu backwardstochasticlinearquadraticoptimalcontrolwithexpectationalequalityconstraint AT jizeli backwardstochasticlinearquadraticoptimalcontrolwithexpectationalequalityconstraint AT yonghuizhou backwardstochasticlinearquadraticoptimalcontrolwithexpectationalequalityconstraint |