Stochastic optimal control to a nonlinear differential game
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RESEARCH
Open Access
Stochastic optimal control to a nonlinear differential game Othusitse Basimanebotlhe1,2* and Xiaoping Xue1 *
Correspondence: [email protected] 1 Department of Mathematics, Harbin Institute of Science and Technology, Nangang District, Harbin, 150001, P.R. China 2 Department of Mathematics, University of Botswana, 4775 Notwane Rd., Gaborone, Botswana
Abstract The paper studies the optimal control of a nonlinear stochastic differential game of two persons subjected to noisy measurements. The logarithmic transformation to the value function is used in trying to find the solution of the problem. The conversion of a quasilinear partial differential equation to an ordinary linear differential equation is considered. Lastly, the iterative optimal control path estimates for the minimization maximization differential game are attained. Keywords: nonlinear stochastic differential equation; Brownian motion; stochastic optimal control; Ito’s lemma
1 Introduction Control theory is a field of mathematics and engineering used in a wide range of fields and their applications, such as architecture, communications, queueing theory, robotics and in economics as evidenced in [–] just to mention a few. Control theory is a subject of much interest in today’s real world. As stated in [], optimizing a sequence of actions to attain some future goal is the general topic of control theory. Therefore, the objective of optimal control theory is to attain an optimal regulation of the system evolution []. Without the indulgence of the noise, the continuous time control problems can be solved in two ways: using Pontryagin Minimum Principle (PMP), which is a pair of ordinary differential equations, or the Hamilton-Jacobi-Bellman (HJB), which is a partial differential equation as in []. The addition of differential equations as constraints in the optimization problem leads to the property that in optimal control theory the minimum is no longer represented by one point x∗ in the state space but by a path or trajectory x∗ = (x∗i )i=,...,N , which is known as the optimal trajectory. In the presence of noise, the PMP formalism has no obvious generalization as mentioned in []. However, the inclusion of the noise in the HJB framework is mathematically quite straightforward, while the numerical solution of either deterministic or stochastic HJB equation is difficult due to the curse of dimensionality. A control problem is said to be stochastic when it is subjected to some disturbances or noise terms and time dependent, that is being uncertain of its future state. Under control theory, there lies a topic of interest to this paper which is game theory. Game theory deals with strategic interactions among several decision makers, known as players. These players have objectives that may be contradicting or non-contradicting. In situations whereby players’ aims are not in contradiction, they are said to be in coop© 2014 Basimanebotlhe and Xue; licensee Springer. This is an Open Access article distributed under the terms of the Creative Common
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