Extremal controls for chained systems

Abstract

To study a time-optimal control problem with bounded controls and an optimal control problem with a quadratic functional for the so-called 2-chained control system

$$\dot x_1 = u_1 , \dot x_2 = u_{2,} \dot x_3 = \dot x_2 u_{1,} ... ,\dot x_n = \dot x_{n - 1} u_{1,} $$

we start with the first-order optimality condition, the Pontryagin maximum principle, determine normal, singular, and abnormal extremals, provide a complete description of the switching structure for bang-bang extremals of the time-optimal problem, describe their branchings, and derive Jacobi-type second-order optimality conditions for the problem with the quadratic functional. Finally we discuss the relations between 2-chained and power form systems.