Optimal control of natural second order systems
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This modest note presents the necessary conditions related to the optimal control of natural second order systems. The development includes systems subject to holonomic constraints. For natural systems, the second order form of the governing differential equations are augmented to the performance index, and as a consequence, the resulting adjoint system defining the necessary conditions of optimality is also second order in form. For natural systems subject to holonomic constraints, the second order differential equations of motion and the algebraic equations of constraint are augmented to the performance index. Following the usual methods, we find that, like the original dynamical system, the resulting adjoint system is also holonomically constrained. We propose an augmented-Lagrangian method to numerically solve the coupled set of differential-algebraic equations within the solution of the two-point boundary value problem.
