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

TL;DR

The paper develops a state-centric framework for time-optimal control of controllable single-input linear systems with state constraints by introducing the augmented switching law (ASL), which compactly encodes both input structure and feasibility. A first-order necessary condition is derived: the Jacobian of the ASL-induced equality constraints with respect to keypoint times is not full row rank at the optimum, enabling costate-free optimality reasoning. The approach yields theoretical results for high-order chain-of-integrator systems, including bounds on the number of arcs and a recursive description of chattering under 2nd-order state constraints, complemented by numerical experiments that demonstrate how ASL-guided perturbations can improve or certify optimality while preserving arc structure. The work offers a practical alternative to costate-based methods, with potential to enhance trajectory optimization by maintaining feasibility guarantees and reducing oscillations and open-loop errors in high-order systems.

Abstract

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 paper 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.

Figures (9)

• Figure 1: Graphical roadmap of the proposed theoretical framework. Main contributions regarding representations and optimal conditions are highlighted in red.
• Figure 2: A feasible BBS trajectory under the constraint ${\boldsymbol{x}}\geq{\boldsymbol{0}}$. The trajectory consists of 3 arcs, i.e., ${\mathcal{S}}_1$, ${\mathcal{S}}_2$, and ${\mathcal{S}}_3$. ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ is connected at ${\boldsymbol{0}}$, denoted as $\mathcal{E}_1$. The trajectory is tangent to $\left\{x_3=0\right\}$ at $\mathcal{T}_1^{(1)}$, $\mathcal{T}_1^{(2)}$, and $\mathcal{T}_2^{(2)}$.
• Figure 3: Two examples of ASLs. For a 4th-order chain-of-integrator system with full box state constraints, let ${u_\mathrm{m}}=1$, $x_{{\mathrm{m}}1}=1$, $x_{{\mathrm{m}}2}=1.5$, $x_{{\mathrm{m}}3}=4$, $x_{{\mathrm{m}}4}=15$, ${\boldsymbol{x}}_0=-x_{{\mathrm{m}}4}{\boldsymbol{e}}_4$, and ${{\boldsymbol{x}}_\mathrm{f}}=x_{{\mathrm{m}}4}{\boldsymbol{e}}_4$. (a) A non-chattering feasible trajectory planned by wang2025time. (b) A chattering optimal trajectory planned by wang2025chattering. (b2) is the plot of $\hat{x}_3\left(t\right)=\left(x_{{\mathrm{m}}3}-x_3\left(t\right)\right)\left(t_\infty-t\right)^{-3}$ where $t_\infty\approx6.0732$ is a chattering limit time.
• Figure 4: A feasible BBS trajectory with system behaviors, tangent markers, and additional end-constraints. In (d), $\hat{\lambda}\triangleq\left(2\lambda_3-\lambda_4\right)\xi\left(t\right)$ where $\xi\left(t\right)>0$ serves as a scaling factor to enhance the visibility of ${\mathrm{sgn}}\left(2\lambda_3-\lambda_4\right)$.
• Figure 5: Optimizing a 4th-order near-optimal trajectory. (a) Plot of the locally minimal $t_{\mathrm{f}}'-t_{\mathrm{f}}$ and $t_4'-t_4$. (b) The original and the optimized trajectories.

Theorems & Definitions (55)

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• Example 2