Projection-Based Optimal Control for Nonlinear Systems with Direction-Dependent Inputs

This paper investigates a class of nonlinear optimal control problems in which the effectiveness of the control input depends not only on its magnitude, but also on its directional alignment with the instantaneous system evolution. Such situations arise when the control-induced action contributes differently depending on whether its direction is aligned or misaligned with the preferred evolution direction of the system. Consequently, control effectiveness cannot be characterized solely by input magnitude, and a directional interpretation becomes necessary. Unlike conventional formulations that primarily focus on the magnitude of the control input, a projection-based framework is introduced in which the inner product between the input-induced vector and the system direction plays a central role. By decomposing the input vector into parallel and orthogonal components, a projection-based optimality condition is derived. It is shown that the control input contributes to cost reduction only when its projection exceeds a threshold determined by the cost structure. This leads to a switching control law that emerges directly from the optimality condition. The proposed framework reformulates the control problem as a geometric optimization problem and establishes theoretical results characterizing the activation condition and switching behavior. Numerical results confirm that the proposed method achieves lower cumulative cost compared to fixed control strategies and demonstrate that the observed switching structure is a direct consequence of the derived optimality condition.