Construction of the attainability set of a Brockett integrator

Methods of optimal control theory are used to solve the problem of constructing attainability sets for a non-linear dynamical system known as the non-holonomic Brockett integrator. It is proved that the boundaries of attainability sets are characterized by points of optimal trajectories constructed for a control problem whose integral performance index defines the area of the figure bounded by a trajectory of motion of the controlled system. The problem is to maximize that area. This formulation is similar to that of Dido's problem in the calculus of variations, namely, to construct a figure of maximum area for a given perimeter. An algorithm is proposed for constructing optimal trajectories and their properties are investigated. The algorithm is based on results obtained by solving a special case of the optimal control problem with a closed trajectory of motion. The necessary optimality conditions of Pontryagin's Maximum Principle are verified for the optimal trajectories constructed. Analytical formulae are derived for the value function of the control problem and the necessary and sufficient conditions for optimally are verified using Subbotin's minimax inequalities for Hamilton-Jacobi equations.

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