Optimal control for holonomic and Pfaffian nonholonomic systems

Holonomic and Pfaffian nonholonomic systems are dynamical systems whose generalized coordinates and velocities are subject to smooth constraints. These systems are usually described by second order differential equations of motion and algebraic equations of constraint. We present an optimal control formulation for these systems that utilizes the multiplier rule to append both the equations of motion and equations of constraint directly to the performance index. Variational calculus techniques are used to obtain the necessary conditions, and we find that, like the original state dynamical system, the costate system also represents a differential-algebraic constrained dynamical system. To numerically solve the evolution of the state and costate coupled set of differential-algebraic equations, we propose a special form of an augmented Lagrangian penalty method. Several examples are included to evaluate the merits of the new methods introduced.