Stochastic optimal control and algorithm of the trajectory of horizontal wells

Abstract

This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke-Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm.

Stability and boundedness criteria for nonlinear differential systems relative to initial time difference and applications

“…Nonlinear differential dynamical systems have been widely used in many fields, such as physics, electronics, engineering [8,9], biotechnology [10], etc. In many problems of practical importance, one is not only interested in the qualitative information provided by Lyapunov stability results, but also interested in how to regulate the system behavior in the real world.…”

An Efficient Gradient Projection Method for Stochastic Optimal Control Problems

“…Theoretical investigations for the above model can be found in [4,13,17,24,3,7,15,26,32,36]. For practical applications of (1.1)-(1.2), one can refer to [7,23,26,36,38] for engineering applications, to [21,22,30,40,42] for applications in option pricing and portfolio optimization, to [1] for analysis of climate changes, and to [16] for biological and medical problems, to name a few.…”

“…For practical applications of (1.1)-(1.2), one can refer to [7,23,26,36,38] for engineering applications, to [21,22,30,40,42] for applications in option pricing and portfolio optimization, to [1] for analysis of climate changes, and to [16] for biological and medical problems, to name a few. In general the above model does not admit explicitly closed form solutions and thus efficient numerical algorithms have been widely studied in recent years.…”

“…In general the above model does not admit explicitly closed form solutions and thus efficient numerical algorithms have been widely studied in recent years. Roughly speaking, we can characterize numerical algorithms into four categories: (i) transferring the control problem into finite dimensional stochastic programming, see e.g., [9,15,18,19,26,36,39,41]; (ii) dynamic programming principle (DPP) based approach [6,25]. In this framework, one usually needs to solve the corresponding Hamilton-Bellman-Jacobin (HJB) equations, and this is one of the most widely used numerical methods [2,4,5,10,20]; (iii) martingale based methods [21,22,37]; and (iv) stochastic maximum principle (SMP) based methods, see e.g., [17] and references therein.…”

“…Gradient projection for stochastic optimal control 3 [7,26] for engineering applications, in [9] for financial applications and in [36] for an application in stochastic hybrid systems. Moreover, stochastic control (i.e., u ∈ U F ) can also be included in our approach and this will be discussed in Section 5 via numerical examples.…”