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A boundary value problem for a non-linear difference equation

Lefebvre Mario

Archives of Control Sciences

2023

Vol. 33, No. 4

829--837

A boundary value problem for a non-linear difference equation of order three is considered. We show that this equation can be interpreted as the equation satisfied by the value function in a stochastic optimal control problem. We thus obtain an expression for the solution of the non-linear difference equation that can be used to find an explicit solution to this equation. An example is presented.

Minimizing the time spent in an interval by a Wiener process with uniform jumps

Lefebvre Mario

Control and Cybernetics

2019

Vol. 48, No. 3

407--415

Let Xu(t) be a controlled Wiener process with jumps that are uniformly distributed over the interval [−c, c]. The aim is to minimize the time spent by Xu(t) in the interval [a, b]. The integro- differential equation, satisfied by the value function, is transformed into an ordinary differential equation and is solved explicitly for a particular case. The approximate solution obtained is precise when c is small.

The LQG homing problem for a Wiener process with random infinitesimal parameters

Lefebvre Mario, Moutassim Abderrazak

Archives of Control Sciences

2019

Vol. 29, No. 3

413--422

The problem of optimally controlling a Wiener process until it leaves an interval (a, b) for the first time is considered in the case when the infinitesimal parameters of the process are random. When a = -∞, the exact optimal control is derived by solving the appropriate system of differential equations, whereas a very precise approximate solution in the form of a polynomial is obtained in the two-barrier case.