Dynamic Programming Viscosity Solution Approach and Its Applications to Optimal Control Problems

Sun, Bing; Tao, Zhen-Zhen; Wang, Yangyang

doi:10.1007/978-3-030-12232-4_12

Cited by 9 publications

(5 citation statements)

References 64 publications

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“…To sum up, we have obtained the Pontryagin maximum principle (20) for the optimal control problem (5) subject to the system (4). Theorem 3.3.…”

Section: Optimal Control Investigationmentioning

confidence: 99%

“…However, in view of the complexity caused by both the investigational system and the necessary optimality condition, we, by the Pontryagin maximum principle along with an iterative algorithm, only give the profile for numerically solving the optimal control problem for the transverse vibration of a moving string with time-varying lengths in fixed final horizon case, i.e., the problem (5). For further discussion on the numerical solutions of optimal control problems via the Pontryagin maximum principle, we can refer to [20,21] and so on. We can also find the feedback control formulation in [13,19], to mention but a few.…”

Section: Optimal Control Investigationmentioning

confidence: 99%

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Optimal control of transverse vibration of a moving string with time-varying lengths

Sun¹

2022

MCRF

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<p style='text-indent:20px;'>In this article, we are concerned with optimal control for the transverse vibration of a moving string with time-varying lengths. In the fixed final time horizon case, the Pontryagin maximum principle is established for the investigational system with a moving boundary, owing to the Dubovitskii and Milyutin functional analytical approach. A remark then follows for discussing the utilization of obtained necessary optimality condition.</p>

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“…To sum up, we have obtained the Pontryagin maximum principle (20) for the optimal control problem (5) subject to the system (4). Theorem 3.3.…”

Section: Optimal Control Investigationmentioning

confidence: 99%

Section: Optimal Control Investigationmentioning

confidence: 99%

Optimal control of transverse vibration of a moving string with time-varying lengths

Sun¹

2022

MCRF

Self Cite

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“…To recapitulate briefly, there are mainly three kinds of numerical methods for discretizing optimal control problems. They are, respectively, the finite difference ( [36]), the finite element ( [5,6,12,26,28]), and the spectral methods ( [7,8]). Among them, the finite element approximation is the most commonly used one, while the finite difference method is not universally valid enough.…”

Section: Introductionmentioning

confidence: 99%

Error estimates for spectral approximation of flow optimal control problem with <inline-formula><tex-math id="M1">$ L^2 $</tex-math></inline-formula>-norm control constraint

Tao

Sun

2023

JIMO

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<p style='text-indent:20px;'>In this paper, we are concerned with the Galerkin spectral approximation of an optimal control problem governed by the Stokes equation with <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm constraint on the control variable. By means of the derived optimality conditions for both the original control system and its spectral approximation one, we establish a priori error estimates and then obtain a posteriori error estimator. A numerical example is, subsequently, executed to illustrate the effectiveness of method and the high performance of estimators. Furthermore, we conjecture that the similar conclusions should hold for optimal control of the Navier-Stokes equation. It is then confirmed by another numerical example.</p>

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“…Hence, researchers put numerical solutions into their perspective. There are several proposed approximate solution techniques in papers such as power series expansion (PSE), 2,3 Adomian decomposition, 4 dynamic solution, 5,6 Galerkin, 7 improved Patchy solution, 8 viscosity, 9 reinforcement learning, 10,11 differential transformation, 12 state-dependent Riccati equation (SDRE), 13 and so on. 14,15 Authors have used these techniques to design optimal controllers for a broad range of systems, from economical to mechanical ones, which have had satisfactory results.…”

Section: Introductionmentioning

confidence: 99%

A new modified polynomial-based optimal control design approach

Monfared

Fakharian

Menhaj

2020

Proceedings of the Institution of Mechanical Engineers, Part I:

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The main concern of this article is addressing a new modified approach to design a nonlinear optimal controller. The modification focuses on proposing a new approximate solution for the Hamilton–Jacobi–Bellman nonlinear partial differential equation. The introduced solution works based on the state-dependent power series expansion presentation of the involved functions in the Hamilton–Jacobi–Bellman partial differential equation. Applying this technique results in releasing a set of free state-dependent functions in the controller structure that can be adjusted to fulfill some special control missions in addition to the optimization objectives. They are formed based on the specific formulation of the candidate Lyapunov function. The proposed approach is exemplified for an intricate biological system, immunogenic tumor-immune cell interaction in the human body, to clarify the mechanism of designing the controller and adjusting the arrays of the free matrices. The closed-loop system by presented optimal state feedback controller meets the predefined optimization objectives without getting feedback from a hard-measurable state. It is achieved by adjusting the aforementioned released functions such that an optimal output feedback controller is obtained. To have some insights into the performance of the system and the effectiveness of the controller, the positiveness of the system’s states is proved and checked numerically by applying the differential transformation method to the system’s differential equations. Finally, to highlight the abilities of the proposed approach from different aspects, some simulations are carried out.

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Dynamic Programming Viscosity Solution Approach and Its Applications to Optimal Control Problems

Cited by 9 publications

References 64 publications

Optimal control of transverse vibration of a moving string with time-varying lengths

Optimal control of transverse vibration of a moving string with time-varying lengths

Error estimates for spectral approximation of flow optimal control problem with <inline-formula><tex-math id="M1">$ L^2 $</tex-math></inline-formula>-norm control constraint

A new modified polynomial-based optimal control design approach

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