Maximum Principle for Near-Optimality of Mean-Field FBSDEs

Li, Ruijing; Hu, Chaozhu

doi:https://doi.org/10.1155/2020/8572959

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Control Problems of Nonlinear Systems with Applications 2020

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 8572959 | https://doi.org/10.1155/2020/8572959

Maximum Principle for Near-Optimality of Mean-Field FBSDEs

Ruijing Li¹and Chaozhu Hu²

Guest Editor: Rongwei Guo

Received13 Apr 2020

Revised16 May 2020

Accepted18 May 2020

Published08 Jun 2020

Abstract

The present paper concerns with a near-optimal control problem for systems governed by mean-field forward-backward stochastic differential equations (FBSDEs) with mixed initial-terminal conditions. Utilizing Ekeland’s variational principle as well as the reduction method, the necessary and sufficient near-optimality conditions are established in the form of Pontryagin’s type. The results are obtained under restriction on the convexity of the control domain. As an application, a linear-quadratic stochastic control problem is solved explicitly.

1. Introduction

Near-optimal control problems have attracted more attentions in recent years due to its distinct advantages, such as existence under minimal assumptions, availability in most practical cases, and convenience for implementation both analytically and numerically. The study of this theory can be traced back to Ekeland [1] and later greatly developed by Zhou [2–4] for deterministic and stochastic cases. Since then, many works have been devoted to the near-optimality of various stochastic control systems. Without being exhaustive, let us refer to [5–13] and the references therein.

In 2015, Zhang et al. [14] investigated the near-optimality necessary conditions for classical linear FBSDEs, where the control domain was with nonconvexity. Via convergence technique as well as reduction method, they established the near-optimal maximum principle. Soon afterwards, under the same assumptions, Zhang [15] presented the near-optimal sufficient conditions for such classical linear FBSDEs. Especially, in 2018, by defining viscosity solution with perturbation factor to dispense the illusory differentiability condition of value function, Zhang and Zhou [16] established the necessary near-optimality conditions for stochastic recursive systems by virtue of dynamic programming principle. Another noteworthy thing is that, for recent years, some authors started research studies on near-optimal control problems for delay systems. For example, Zhang [17] first studied near-optimal control problems for linear stochastic delay systems. By anticipated backward stochastic differential equations method as well as maximum principle, necessary condition and sufficient verification theorem were provided. Then, also under restriction on convexity control domain, Wang and Wu [18] investigated near-optimal control problem for nonlinear stochastic delay systems. By Ekeland’s variational principle and corresponding moment estimations, they presented the sufficient as well as necessary near-optimality conditions. For more details, refer to [19, 20] and the references therein.

However, to the best of our knowledge, few papers can be found in the literature on the near-optimality of mean-field backward stochastic differential equations (BSDEs). This new kind of mean-field BSDEs was first introduced by Buckdahn et al. [21], which were derived as a limit of some highly dimensional system of FBSDEs, corresponding to a large number of particles. It has been shown in Buckdahn et al. [22] that, such a mean-field BSDE described the viscosity solution of the associated nonlocal partial differential equations. Henceforth, many authors take into account of this system of McKean-Vlasov type (Lasry and Lions [23]) adapted for different frameworks, for example, Xu and Wu [24] presented a maximum principle for optimal control problems governed by backward stochastic partial differential equations of mean-field type, and for other related works, refer to [25–28].

As we can see that all the above literature studies are about mean-field problems involving expectations as mean-field terms. In fact, there is another line dealing with mean-field problems, which involve large-population as mean-field terms to describe the impact of the population’s collective behaviors on all agents (Huang et al. [29]) such as the work of Huang [30] and Xu and Shi [31] as well as the work of Xu and Zhang [32] all concerned with general mean-field linear-quadratic-Gaussian (LQG) games of stochastic large-population systems; through the consistency condition, they derived the decentralized strategies and further verified the asymptotic near-optimality property (namely, -Nash equilibrium) of decentralized strategies for the LQ games. On the contrary, a relevant work of Hafayed and Abbas [8] dealing with near-optimal control problems has established necessary and sufficient conditions for mean-field singular stochastic systems in the case of controlled diffusion coefficient. Particularly, in the concluding section, it is pointed out that the establishment of necessary and sufficient near-optimal conditions for mean-field FBSDEs also remains an open problem. Motivated by this fact with the addition of above described mean-field theory application background in economics and finance, this paper is to discuss near-optimal control problems for mean-field FBSDEs, where the controlled state systems are with mixed initial-terminal conditions.

The main contribution of this paper lies in the initial introduction of three first-order adjoint equations to eliminate the corresponding variational processes during dual analysis; another is rooted in the usage of reduction method to guarantee the well-posedness of the first-order adjoint equations with mixed initial-terminal conditions. Via classical convex variational technique and Ekeland’s variational principle, a necessary condition of Pontryagin’s type is derived. Then, under some additional assumptions, we prove that the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. It is remarkable that our results extend those of [5] essentially to the framework of mean-field theory.

The rest of this paper is organized as follows. In Section 2, we state some preliminaries and basic definitions. In Sections 3 and 4, we establish the main theorems and provide its detailed proof. In Section 5, an example of a linear-quadratic control problem is worked out to illustrate the theoretical applications. Finally, some concluding remarks are given in Section 6.

2. Preliminaries

Let be a filtered probability space satisfying the usual condition, on which a one-dimensional standard Brownian motion is defined, be the natural filtration generated by and augmented by all -null sets, i.e.,where is the set of all -null subsets. We now introduce some spaces of random variables and stochastic processes.

Clearly, is a Banach space. Any process in is defined by with the norm

We study the near-optimal control problem of the following controlled mean-field FBSDEs having mixed initial-terminal conditions:where ; ; ; and is a given convex closed set of . The cost functional to be minimized over the space of admissible controls takes the formwith ; .

Definition 1 (see [4]). Both a family of admissible pairs parameterized by and any element in the family are called near-optimal ifholds for sufficiently small , where is a function of satisfying as . The estimate is called an error bound. If for some independent of the constant , then is called near-optimal with order . Particularly, when , is called -optimal. The near-optimal control problem under consideration in this paper is as follows.

Problem . Find such thatSome notations and assumptions are presented before giving the well-posedness of system (3). We denote the norm by of an Euclidean space. The functions , , , and are -progressively measurable in , continuously differentiable in , and the derivatives of , , , and with respect to are bounded. Moreover, for some constant , , , and are continuously differentiable in , and the derivatives of , , and with respect to are bounded. Moreover, for some constant , There is a constant and such thatFurthermore,where and .

Remark 1. Under assumptions via Theorem 2 in [33], the mean-field stochastic system (3) admits a unique adapted solution .
In fact, due to the mixed initial-terminal conditions in the state equation, even if we have the well-posedness of the state equation via the Lyapunov operator introduced in [34], the well-posedness of the first-order adjoint equation seems to be not guaranteed. To overcome this difficulty, we introduce a reduction method inspired by the study of optimality variational principle for controlled FBSDEs with mixed initial-terminal conditions [35]. First, we pose the following problem.

Problem . Find such thatwhere is subject to the forward control system:with the mixed initial-terminal state constraints:It is remarkable that, for Problem , the mean-field system (3) has a unique solution under , which implies that is unique and completely determinate. While, for Problem , is arbitrary and viewed as a control variable. It just needs to satisfy the near-optimal state constraints at time . So, Problem is embedded into Problem . Hence, if the triple is the near-optimal control of , then is near-optimal for Problem . In the following section, we will adopt the classical convex variational technique to solve Problem .

3. Necessary Condition of Near-Optimality

This section is devoted to the study of the main theorem. For simplicity, we denote

For any and the corresponding state processes , we define the first-order adjoint equation as

Remark 2. Under assumptions , the adjoint equation (15) admits a unique adapted solution . The well-posedness of the corresponding adjoint system will be provided in the derivation process of Theorem 1.
Define a metric on bySince is closed, it can be shown that is a complete metric space. Next, we will present some continuity of the state processes and adjoint processes with respect to the metric .

Lemma 1. For any and , there is a constant such that, for any along with the corresponding trajectories and , it follows that

Proof. Applying the classical methods as Lemma 4 in [5] for dealing with mean-field FBSDEs, together with Burkholder–Davis–Gundy inequality and Gronwall’s inequality, we can logically obtain the estimates.

Lemma 2. Let hold, for any and satisfying , and there is a constant such that, for any , along with the corresponding trajectories and and the solutions and of the corresponding adjoint equation (15), it holds that

Proof. Applying the classical methods as Lemma 5 in [5] for dealing with mean-field FBSDEs, we can naturally obtain the estimates.

Theorem 1. Let hold, for any , is an -optimal control of problem . Then, for any , there exist three parameters , , and with , and holds thatwhere is the solution of the first-order adjoint equation (15) corresponding to .

Proof. Under the assumption , it is easy to check that is lower semicontinuous on , which is a complete metric space under the following metric:By Ekeland’s variational principle [1], there exists an admissible control such thatwhereIt means that is optimal for system (12) with the new cost functional . On the contrary, due to the mixed initial-terminal endpoint constraints in problem , we need to introduce the penalty functional to transform the original problem with endpoint constraints to the penalized optimal control problem with no endpoint constraints.
Let be an optimal control of problem , with the corresponding optimal state process . Without loss of generality, we assume that . For any and , we define the penalty functional:Obviously,By Ekeland’s variational principle, there exists a 3-tuple such thatTherefore, is optimal for system (13) with the new cost functional:So far, we have transformed the original problem with endpoint constraints to the penalized optimal control problem with no endpoint constraints, and the optimal 3-tuple approaches as . In the following, a convex perturbation is employed to obtain a maximum principle for . To this end, let such that ; then, for any ,Let be the state processes corresponding to , and the processes be the solution of the following variational equations:where for simplicity of notations, we still use corresponding to , . Then, we have the following estimates, whose proofs are similar to those given in [27]Noting that , from the last relation in (25), we derivewithandIt is necessary to point out that , and . Thus, there exists a subsequence still denoted by convergent, i.e.,We claim that . The detailed illustration of this point refers to [35]. Here, is called the Lagrange multiplier of the corresponding optimal 3-tuple . On the contrary,From and (29), we can deducewhere , . Similarly,Then, taking notice of (30), we can further obtainLet us introduce the following first-order BSDEs:where , . Applying Itô’s formula to fulfillsCombining (37) and (39), yieldsTo derive the first-order adjoint equation with mixed initial-terminal conditions, divide in (40) and then send , and we see thatwhich impliesMeanwhile, by taking in (40), dividing this inequality by , and then sending , , the variational inequality follows:where . On the contrary, from (38) and (42), we can present the adjoint equation with mixed initial-terminal conditions as follows:whose well-posedness can be deduced directly virus the above derivative process (Remark 2). Now, consider (43) again but with , etc., replaced by , etc. We are about to derive an estimate for the term similar to the right side of (43) with respect to , etc. To this end, we first estimate the following difference:Due to Lemma 2, for any , by using the similar arguments as developed in [7] the proof of Theorem 1, we can also prove thati.e.,Similarly, via Lemma 2, we also haveTherefore, the desired result (20) follows immediately by combining (43)–(48).
Since is arbitrary, we draw the desired conclusion and summarize it as follows.

Theorem 2. Suppose hold. For any , is an -optimal control of problem . Then, for any , there exist three nonnegative parameters , , and with and such that, for any , , and , the necessary condition (20) holds a.e. a.s., where is the solution of (15) corresponding to .
Define the Hamiltonian bythen we have the following form of necessity conditions.

Corollary 1. Under the assumptions of Theorem 2, it holds that

Proof. According to the definition of the control , the point can be replaced by any admissible control , and the subsequent arguments still go through. Therefore, the conclusion in Theorem 2 holds for any , which is an easy variant of our corollary.

Remark 3. If the coefficients of system (3) do not depend on the expected values of the states, Theorem 2 reduces to the near-maximum condition for the classical system under convex control domain.

Remark 4. For exact optimality, the integral form and the pointwise form of the maximum condition is equivalent; however, it is not the case for near-optimality. We can only deduce the near maximum condition in an integral form.

Remark 5. If , we can obtain a stochastic maximum principle for controlled mean-field FBSDEs with the control domain of convexity assumption.

4. Sufficient Condition of Near-Optimality

In this section, we will prove that the near-maximum condition of the Hamiltonian in the integral form is sufficient for near-optimality under some additional assumptions. Let , , , and be differentiable in , and there exists a constant such that

Theorem 3. Let hold, and assume that the Hamiltonian is concave for a.e. , -a.s., is concave, and and are convex. Let be the solution of the adjoint equation (15) associated with . If, for any and some ,holds, then we havewhere is a constant independent of .

Proof. Fix , for any , , and define a new metric on as follows:with . Obviously, is a complete metric on as a weighted norm. Define a functional on byA simple calculation shows thatTherefore, is continuous on with respect to . Then, by using (52) and Ekeland’s variational principle, there exists a such thatandwhereThe integral-form maximum condition (58) implies a pointwise maximum condition, that is, for a.e. and -a.s.,Then, by Lemma 2.3 of Yong and Zhou [36], we have . By using (59) and the fact that the generalized gradient of the sum of two functions is contained in the sum of the generalized gradients of the two functions, we deduceFurthermore, since is differentiable in , there exists a , such thatConsequently, by , we can prove thatBy the concavity of , we havefor any , where , .
Taking integrations, from (57) and (63), followsApplying Itô’s formula to , we obtainSince is concave, we haveSimilarly, by the convexity of and yieldingCombining (65) and (67)–(69), we can finally get the following result:Since is arbitrary, we can draw the conclusion.

Corollary 2. Under the assumptions of Theorem 3, a sufficient condition for an admissible control to be -optimal is

5. A Linear-Quadratic Problem

Consider the near-optimal control problem.

Problem . Minimize , subject towhere is a small parameter, and the control domain is limited as .
In view of the practical point, it is difficult to find a near-optimal control for directly. So, we adopt the approach, in which is firstly approximated by a simpler one (Problem ), and then an optimal control for is obtained. Finally, it is proved that is near-optimal for . This method has been applied to the hierarchical controls of stochastic manufacturing systems [37].

Problem . Minimize , subject toLet be an optimal admissible control of , and the corresponding optimal trajectory is denoted by . Set , for a given admissible triple , the corresponding first-order adjoint equation is presented asFrom Remark 5 and Theorem 2, the candidate optimal control should satisfyThen, we havewith .
We claim that the control defined by (76) is optimal for Problem , which will be illustrated in the following proposition. Now, we are about to show that the same optimal control is near-optimal for when is sufficiently small. Denote by the optimal state and optimal control under (76) and the corresponding solution of (74). Then, the Hamiltonian function for isSince is optimal, it necessarily maximizes the Hamiltonian function a.s., namely,However, the Hamiltonian function for isObviously, it is concave. Moreover, it is maximized at , which satisfiesHence,According to Theorem 3, is near-optimal for with an error order of when is sufficiently small.

Proposition 1. The control defined by (76) together with the corresponding trajectory is an optimal solution for Problem .

Proof. Suppose is the trajectory of system (73) controlled by . By the convexity of a function, we haveApplying Itô’s formula to , we obtainThen,From (76), we haveThe inequality above implies thatTherefore, is the optimal control for , and is the optimal trajectory.

6. Conclusion

This paper discussed near-optimal control problems for mean-field FBSDEs with mixed initial-terminal conditions. Firstly, we initially introduce three first-order adjoint equations to operate dual analysis with corresponding variational processes. Secondly, the reduction method is adopted to guarantee the well-posedness of the first-order adjoint equations also with mixed initial-terminal conditions. Furthermore, by introducing the penalty functional, the original problem with endpoint constraints is transformed as penalized optimal control problem with no endpoint constraints. Via convex variational technique as well as Ekeland’s variational principle, the necessary condition of Pontryagin’s type is established. Finally, to illustrate the application of our theoretical results, a linear-quadratic problem is worked out. In our future work, we will develop the research on near-optimality to solve both theoretical and practical problems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11901112), and National Natural Science Foundation of China (Grant no. 11701161), and High level introduction of talent research start-up fund of Hubei University of Technology (Grant no. BSQD14067).

References

I. Ekeland, “Non convex minimization problems,” Bulletin of the American Mathematical Society, vol. 1, no. 3, pp. 443–475, 1979.
View at: Publisher Site | Google Scholar
X. Y. Zhou, “Deterministic near-optimal control, part 1: necessary and sufficient conditions for near-optimality,” Journal of Optimization Theory and Applications, vol. 85, no. 2, pp. 473–488, 1995.
View at: Publisher Site | Google Scholar
X. Y. Zhou, “Deterministic near-optimal controls. Part II: dynamic programming and viscosity solution approach,” Mathematics of Operations Research, vol. 21, no. 3, pp. 655–674, 1996.
View at: Publisher Site | Google Scholar
X. Y. Zhou, “Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality,” SIAM Journal on Control and Optimization, vol. 36, no. 3, pp. 929–947, 1998.
View at: Publisher Site | Google Scholar
K. Bahlali, N. Khelfallah, and B. Mezerdi, “Necessary and sufficient conditions for near-optimality in stochastic control of FBSDEs,” Systems & Control Letters, vol. 58, no. 12, pp. 857–864, 2009.
View at: Publisher Site | Google Scholar
R. Guo, “Projective synchronization of a class of chaotic systems by dynamic feedback control method,” Nonlinear Dynamics, vol. 90, no. 1, pp. 53–64, 2017.
View at: Publisher Site | Google Scholar
M. Hafayed, A. Abba, and S. Boukaf, “On Zhou’s maximum principle for near-optimal control of mean-field forward-backward stochastic systems with jumps and its applications,” International Journal of Modelling, Identification and Control, vol. 25, no. 1, pp. 1–16, 2016.
View at: Publisher Site | Google Scholar
M. Hafayed and S. Abbas, “On near-optimal mean-field stochastic singular controls: necessary and sufficient conditions for near-optimality,” Journal of Optimization Theory and Applications, vol. 160, no. 3, pp. 778–808, 2014.
View at: Publisher Site | Google Scholar
J. Huang, X. Li, and G. Wang, “Near-optimal control problems for linear forward-backward stochastic systems,” Automatica, vol. 46, no. 2, pp. 397–404, 2010.
View at: Publisher Site | Google Scholar
E. Hui, J. Huang, X. Li, and G. Wang, “Near-optimal control for stochastic recursive problems,” Systems & Control Letters, vol. 60, no. 3, pp. 161–168, 2011.
View at: Publisher Site | Google Scholar
M. Tang, “Stochastic maximum principle of near-optimal control of fully coupled forward-backward stochastic differential equation,” Abstract and Applied Analysis, vol. 2014, Article ID 361259, 12 pages, 2014.
View at: Publisher Site | Google Scholar
S. Yang, “The necessary and sufficient conditions for stochastic differential systems with multi-time state cost functional,” Systems & Control Letters, vol. 114, pp. 11–18, 2018.
View at: Publisher Site | Google Scholar
X. Yi, R. Guo, and Y. Qi, “Stabilization of chaotic systems with both uncertainty and disturbance by the UDE-based control method,” IEEE Access, vol. 8, no. 1, pp. 62471–62477, 2020.
View at: Publisher Site | Google Scholar
L. Zhang, J. Huang, and X. Li, “Necessary condition for near optimal control of linear forward-backward stochastic differential equations,” International Journal of Control, vol. 88, no. 8, pp. 1594–1608, 2015.
View at: Publisher Site | Google Scholar
L. Zhang, “Sufficient condition for near-optimal control of general controlled linear forward-backward stochastic differential equations,” International Journal of Dynamics and Control, vol. 5, no. 2, pp. 306–313, 2017.
View at: Publisher Site | Google Scholar
L. Zhang and Q. Zhou, “Near-optimal control of stochastic recursive systems via viscosity solution,” Journal of Optimization Theory and Applications, vol. 178, no. 2, pp. 363–382, 2018.
View at: Publisher Site | Google Scholar
F. Zhang, “Maximum principle for near-optimality of stochastic delay control problem,” Advances in Difference Equations, vol. 2017, no. 1, p. 98, 2017.
View at: Publisher Site | Google Scholar
Y. Wang and Z. Wu, “Necessary and sufficient conditions for near-optimality of stochastic delay systems,” International Journal of Control, vol. 91, no. 8, pp. 1–24, 2017.
View at: Publisher Site | Google Scholar
G. Ren and B. Liu, “Near-optimal control for a singularly perturbed linear stochastic singular system with Markovian jumping parameters,” European Journal of Control, vol. 50, pp. 88–95, 2019.
View at: Publisher Site | Google Scholar
Y. Zhang, S. Li, and L. Liao, “Near-optimal control of nonlinear dynamical systems: a brief survey,” Annual Reviews in Control, vol. 47, pp. 71–80, 2019.
View at: Publisher Site | Google Scholar
R. Buckdahn, B. Djehiche, J. Li, and S. Peng, “Mean-field backward stochastic differential equations: a limit approach,” The Annals of Probability, vol. 37, no. 4, pp. 1524–1565, 2009.
View at: Publisher Site | Google Scholar
R. Buckdahn, J. Li, and S. Peng, “Mean-field backward stochastic differential equations and related partial differential equations,” Stochastic Processes and Their Applications, vol. 119, no. 10, pp. 3133–3154, 2009.
View at: Publisher Site | Google Scholar
J.-M. Lasry and P.-L. Lions, “Mean field games,” Japanese Journal of Mathematics, vol. 2, no. 1, pp. 229–260, 2007.
View at: Publisher Site | Google Scholar
R. M. Xu and T. T. Wu, “Mean-field backward stochastic evolution equations in Hilbert spaces and optimal control for BSPDEs,” Mathematical Problems in Engineering, vol. 2014, Article ID 718948, 18 pages, 2014.
View at: Publisher Site | Google Scholar
D. Andersson and B. Djehiche, “A maximum principle for SDEs of mean-field type,” Applied Mathematics & Optimization, vol. 63, no. 3, pp. 341–356, 2011.
View at: Publisher Site | Google Scholar
R. Buckdahn, B. Djehiche, and J. Li, “A general stochastic maximum principle for SDEs of mean-field type,” Applied Mathematics & Optimization, vol. 64, no. 2, pp. 197–216, 2011.
View at: Publisher Site | Google Scholar
J. Li, “Stochastic maximum principle in the mean-field controls,” Automatica, vol. 48, no. 2, pp. 366–373, 2012.
View at: Publisher Site | Google Scholar
Y. Shen and T. K. Siu, “The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 86, pp. 58–73, 2013.
View at: Publisher Site | Google Scholar
J. Huang, S. Wang, and Z. Wu, “Backward mean-field linear-quadratic-Gaussian (LQG) games: full and partial information,” IEEE Transactions on Automatic Control, vol. 61, no. 12, pp. 3784–3796, 2016.
View at: Publisher Site | Google Scholar
M. Huang, “Large-population LQG games involving a major player: the Nash certainty equivalence principle,” SIAM Journal on Control and Optimization, vol. 48, no. 5, pp. 3318–3353, 2010.
View at: Publisher Site | Google Scholar
R. Xu and J. Shi, “-Nash mean-field games for linear-quadratic systems with random jumps and applications,” International Journal of Control, pp. 1–11, 2019.
View at: Publisher Site | Google Scholar
R. M. Xu and F. Zhang, “-Nash mean-field games for general linear-quadratic systems with applications,” Automatica, vol. 114, Article ID 108835, 2020.
View at: Publisher Site | Google Scholar
A. Bensoussan, S. C. P. Yam, and Z. Zhang, “Well-posedness of mean-field type forward-backward stochastic differential equations,” Stochastic Processes and Their Applications, vol. 125, no. 9, pp. 3327–3354, 2015.
View at: Publisher Site | Google Scholar
J. M. Yong, “Forward-backward stochastic differential equations with mixed initial-terminal conditions,” Transactions of the American Mathematical Society, vol. 362, no. 2, pp. 1047–1096, 2010.
View at: Publisher Site | Google Scholar
J. Yong, “Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions,” SIAM Journal on Control and Optimization, vol. 48, no. 6, pp. 4119–4156, 2010.
View at: Publisher Site | Google Scholar
J. M. Yong and X. Y. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, Berlin, Germany, 1999.
X. Y. Zhou and S. P. Sethi, “A sufficient condition for near-optimal stochastic controls and its application to manufacturing systems,” Applied Mathematics & Optimization, vol. 29, no. 1, pp. 67–92, 1994.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Ruijing Li and Chaozhu Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

270

Downloads

574

Citations