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.
Highlights
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
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
Summary
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. 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. The main difficulty associated with stochastic minimax dynamic games studied here is the presence of the noise in the dynamical constraints and the solution to a nonlinear second order Hamilton Jacobi Bellman equation (HJB) as mentioned in [ ]. (iii) V (x), ∇xV , L, and φ are bounded, where V is defined as the value of the game that each control tries to optimize from.
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