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

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.

scholarly journals A BSDE Approach to Stochastic Differential Games with Regime Switching

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.

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

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.

scholarly journals Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator

2018 ◽  
Vol 24 (1) ◽  
pp. 355-376 ◽  
Author(s):  
Jiangyan Pu ◽  
Qi Zhang
Keyword(s):  
Dynamic Programming ◽  
Control Problem ◽  
Bellman Equation ◽  
Backward Stochastic Differential Equation ◽  
Dynamic Programming Principle ◽  
Unknown Variable ◽  
Hamilton Jacobi Bellman Equation ◽  
Hamilton Jacobi Bellman ◽  
The Stability ◽  
The Value Function

In this work we study the stochastic recursive control problem, in which the aggregator (or generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem for backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein−Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.

Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic controlled systems

2020 ◽  
Vol 26 ◽  
pp. 81
Author(s):  
Mingshang Hu ◽  
Shaolin Ji ◽  
Xiaole Xue
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
Adjoint Equations ◽  
Dynamic Programming Principle ◽  
The Maximum Principle ◽  
Controlled Systems ◽  
Local Case ◽  
Hamilton Jacobi Bellman ◽  
The Relationship ◽  
Fully Coupled

Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 56 (2018) 4309–4335] and the dynamic programming principle (DPP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 57 (2019) 3911–3938] for a fully coupled forward–backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton–Jacobi–Bellman (HJB) equation combine an algebra equation respectively. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward–backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first-order and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622–1662] and Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249–259], and obtain the relationship between the MP from Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249–259] and the DPP from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622–1662].

scholarly journals Data association and uncertainty pruning for tracks determined on short arcs

2020 ◽  
Vol 132 (1) ◽  
Author(s):  
Laura Pirovano ◽  
Gennaro Principe ◽  
Roberto Armellin
Keyword(s):  
Optimization Problems ◽  
Data Association ◽  
Earth Orbit ◽  
Orbital State ◽  
Geostationary Earth Orbit ◽  
Multiple Observations ◽  
Observation Strategies

AbstractWhen building a space catalogue, it is necessary to acquire multiple observations of the same object for the estimated state to be considered meaningful. A first concern is then to establish whether different sets of observations belong to the same object, which is the association problem. Due to illumination constraints and adopted observation strategies, small objects may be detected on short arcs, which contain little information about the curvature of the orbit. Thus, a single detection is usually of little value in determining the orbital state due to the very large associated uncertainty. In this work, we propose a method that both recognizes associated observations and sequentially reduces the solution uncertainty when two or more sets of observations are associated. The six-dimensional (6D) association problem is addressed as a cascade of 2D and 4D optimization problems. The performance of the algorithm is assessed using objects in geostationary Earth orbit, with observations spread over short arcs.

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