Dynamic Programming Principle and Hamilton--Jacobi--Bellman Equations for Fractional-Order Systems

Bellman optimality principle for the stochastic dynamic system on time scales is derived, which includes the continuous time and discrete time as special cases. At the same time, the Hamilton–Jacobi–Bellman (HJB) equation on time scales is obtained. Finally, an example is employed to illustrate our main results.

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Hamilton–Jacobi–Bellman equations and dynamic programming for power-maximizing relaxation of radiation

International Journal of Heat and Mass Transfer ◽

10.1016/j.ijheatmasstransfer.2006.11.018 ◽

2007 ◽

Vol 50 (13-14) ◽

pp. 2714-2732 ◽

Cited By ~ 19

Author(s):

Stanislaw Sieniutycz

Keyword(s):

Dynamic Programming ◽

Bellman Equations ◽

Hamilton Jacobi Bellman

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A BSDE Approach to Stochastic Differential Games with Regime Switching

Mathematical Problems in Engineering ◽

10.1155/2021/9930142 ◽

2021 ◽

Vol 2021 ◽

pp. 1-17

Author(s):

J. Y. Li ◽

M. N. Tang

Keyword(s):

Dynamic Programming ◽

Regime Switching ◽

Time Horizon ◽

Backward Stochastic Differential Equation ◽

Stochastic Differential Games ◽

Dynamic Programming Principle ◽

Value Functions ◽

Finite Time Horizon ◽

Hamilton Jacobi Bellman ◽

Zero Sum

In this paper, we study a two-player zero-sum stochastic differential game with regime switching in the framework of forward-backward stochastic differential equations on a finite time horizon. By means of backward stochastic differential equation methods, in particular that of the notion from stochastic backward semigroups, we prove a dynamic programming principle for both the upper and the lower value functions of the game. Based on the dynamic programming principle, the upper and the lower value functions are shown to be the unique viscosity solutions of the associated upper and lower Hamilton–Jacobi–Bellman–Isaacs equations.

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Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020070 ◽

2020 ◽

Author(s):

Juan Li ◽

Wenqiang Li ◽

Qingmeng Wei

Keyword(s):

Dynamic Programming ◽

Value Function ◽

Backward Stochastic Differential Equation ◽

Stopping Times ◽

Dynamic Programming Principle ◽

Probabilistic Interpretation ◽

Isaacs Equations ◽

Hamilton Jacobi Bellman ◽

Cost Functionals ◽

Strong Dynamic

By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function initially defined only for deterministic times $t$ and states $x$ to stopping times $\tau$ and random variables $\eta\in L^2(\Omega,\mathcal {F}_\tau,P; \mathbb{R})$. The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case.

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Optimization of the launcher ascent trajectory leading to the global optimum without any initialization: the breakthrough of the Hamilton–Jacobi–Bellman approach

Progress in Flight Dynamics, Guidance, Navigation, and Control – Volume 10 ◽

10.1051/eucass/201810159 ◽

2018 ◽

Author(s):

E. Bourgeois ◽

O. Bokanowski ◽

H. Zidani ◽

A. Désilles

Keyword(s):

Dynamic Programming ◽

High Dimension ◽

Global Optimum ◽

Dynamic Programming Principle ◽

Earth Orbit ◽

Technology Program ◽

Geostationary Earth Orbit ◽

Hamilton Jacobi Bellman ◽

Initialization Procedure ◽

Trajectory Problem

The resolution of the launcher ascent trajectory problem by the so-called Hamilton–Jacobi–Bellman (HJB) approach, relying on the Dynamic Programming Principle, has been investigated. The method gives a global optimum and does not need any initialization procedure. Despite these advantages, this approach is seldom used because of the dicculties of computing the solution of the HJB equation for high dimension problems. The present study shows that an eccient resolution is found. An illustration of the method is proposed on a heavy class launcher, for a typical GEO (Geostationary Earth Orbit) mission. This study has been performed in the frame of the Centre National d’Etudes Spatiales (CNES) Launchers Research & Technology Program.

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