Problems of optimal control and differential clearance.

Summary The first part presents in detail the practical realization of a simple inverse pendulum and the feasibility study of a double inverse pendulum using the linear stochastic control theory. The originality of these pendulums is the choice of a low-end hardware, associated with a rather complex, multi-task and real time assembler programming. The power of the motors being limited, the tests on the simple pendulum show that the difficulty lies in the constraint on the control. After a general study of the feedbacks minimizing the norm of the control, on the set of stable commands, the study of the global stochastic system allows to estimate the minimal acceleration necessary for the double pendulum and to conclude that it is not feasible. The second part presents the theoretical and numerical study of a tracking game, more precisely the solution of the Isaacs equation of this differential game, thanks to the notion of viscosity solution. Moreover, it is a game modeling a pursuit in a given domain, i.e. with constraints on the edge of the domain. The value functions of this game verify the dynamic programming, are continuous, and are viscosity solutions of the same Isaacs equation. This equation with boundary conditions has a unique viscosity solution. The monotonic schemes, with differential form, and consistent with the equation, allow to approximate the solutions. The numerical codes then provide the value function of the set and thus the optimal trajectories for any initial condition.