A Novel State-Centric Necessary Condition for Time-Optimal Control of Controllable Linear Systems Based on Augmented Switching Laws

Most existing necessary conditions for optimal control based on adjoining methods require both state and costate information, yet the unobservability of costates for a given feasible trajectory impedes the determination of optimality in practice. This article establishes a novel theoretical framework for time-optimal control of controllable linear systems with a single input, proposing the augmented switching law (ASL) that represents the input control and the feasibility in a compact form. Given a feasible trajectory, the perturbed trajectory under the constraints of ASL is guaranteed to be feasible, resulting in a novel state-centric necessary condition without dependence on costate information. A first-order necessary condition is proposed that the Jacobian matrix of the ASL is not of full row rank, which also results in a potential approach to optimizing a given feasible trajectory with the preservation of arc structures. The proposed necessary condition is applied to high-order chain-of-integrator systems with full box constraints, contributing to some theoretical results challenging to reason by costate-based conditions.