Time-Optimal Switching Surfaces

• Time-optimal switching surfaces are geometric loci in state or cotangent space where abrupt control switches occur, crucial for synthesizing bang–bang and bang–singular–bang solutions.
• They are defined by the zero set of switching functions derived from Pontryagin’s Maximum Principle, with explicit constructions in linear systems and chains of integrators.
• Modern computational algorithms, such as the Switch Point Algorithm and Manifold-Intercept Method, leverage these surfaces to optimize minimum-time control trajectories under constraints.

Time-optimal switching surfaces are geometric loci in state or state-costate space across which the optimal control for a minimum-time problem undergoes an abrupt change, typically switching between its allowed extrema. These surfaces are algorithmically and structurally central in the synthesis and computational analysis of bang–bang and bang–singular–bang solutions, providing an explicit or implicit partition of the admissible state-space into regions governed by discrete control behaviors. Their precise characterization depends on system class (finite-dimensional ODEs, infinite-dimensional PDEs, chains of integrators) and the control and state constraints imposed.

1. Core Definitions and Geometric Role

In time-optimal control problems, the objective is to steer the system from an initial to a terminal state in minimum time, subject to dynamics $\dot{x} = f(x,u)$ and control/state constraints. According to Pontryagin's Maximum Principle (PMP), the optimal control is determined by maximizing the Hamiltonian with respect to the control at each time, leading—under compact, polyhedral, or ball-type control sets—to "bang-bang" laws where the control alternates instantaneously between boundary values.

The locus at which the switching occurs is mathematically described by the zero set of a switching function, typically a function of state and costate (i.e., co-state variables arising in the adjoint equations of the PMP). The time-optimal switching surfaces are the sets in $(x,p)$ or $x$ where the switching function vanishes, and crossing these surfaces triggers a discontinuity in optimal control.

For linear finite-dimensional systems $\dot{x} = Ax + Bu$ with $\|u\|\leq 1$, switching occurs as the state trajectory $x(t)$ crosses affine hyperplanes $\Sigma_t = \{x \mid B^Te^{A^T(T-t)}z^* = 0\}$, with $z^*$ being the terminal costate. The number, geometry, and regularity of such surfaces are dictated by the system's controllability and structure (Qin et al., 2019, Qin et al., 2020).

2. Analytical Characterizations in Finite-Dimensional Systems

2.1 Linear Systems

In classic linear ODEs with ball-type input constraints, the switching function $\varphi(t) = B^Tp(t)$, with $p$ solving $\dot{p} = -A^Tp,\, p(T)=z^*$, determines the control: $$u(t) = \begin{cases} \frac{\varphi(t)}{\|\varphi(t)\|}, & \varphi(t)\neq 0 \ \text{undetermined in }B_1(0), & \varphi(t)=0 \end{cases}$$ Switching surfaces $\ker(B^Te^{A^T(T-t)}z^*)$ partition state-space into regions of constant control direction, and each control switch is a 180° reversal in direction (Qin et al., 2019).

In multi-input or high-rank cases, the switching function is vector-valued and surfaces become hyperplanes of codimension $m$ (for $m$-input systems). Analyticity of the costate and switching function provides upper bounds on the number of switches via analytic function zero-counts (Qin et al., 2019, Qin et al., 2020).

2.2 Affine Systems and Lie Bracket Conditions

For affine control systems on a smooth manifold $M$, with drift $f_0$ and input vector fields $f_1,\ldots,f_k$, and ball-type constraints $u\in B^k(0,1)$, switching surfaces are given by the "singular locus": $$A = \{\lambda \in T^*M : h_1(\lambda) = \cdots = h_k(\lambda) = 0\}$$ where $h_i(\lambda) = \langle\lambda, f_i(q)\rangle$. For $k=n-1$, explicit Lie-bracket characterizations (vector $a(q)$ and matrix $A(q)$ built from $\det[f_1,\ldots,[f_0,f_i]]$ and $[f_i,f_j]$) determine whether an optimal trajectory remains smooth (no switch, $a(q)\in A(q)\cdot B^{n-1}$) or undergoes an isolated switching event (transversal intersection) (Agrachev et al., 2016, Agrachev et al., 2016).

In $n=3, k=2$ systems, the codimension-2 locus in $T^*M$ projects to a 2D surface in $M$ dictating where switching occurs; bracket-derived invariants $h_{01}, h_{02}, h_{12}$ determine the occurrence and uniqueness of switches (Agrachev et al., 2016).

3. High-Order Integrator Chains and Explicit Manifold Constructions

In $n$-order chain-of-integrators $\dot{x}_1 = u$, $\dot{x}_k = x_{k-1}$ ($k=2,\ldots,n$), time-optimal switching surfaces are implicitly defined by the zeros of certain costate polynomials, with degree up to $n-1$. Switching patterns (or "laws") for admissible controls correspond to sequences of "mode switches" in state-control variables, e.g., between bang arcs at control/external variable bounds, singular (zero) arcs, and tangent/limit arcs at active state constraints (Wang et al., 2023, Wang et al., 22 Jan 2026).

For $n=3$ (triple integrator) under full box constraints, each elementary switching surface is given as an explicit polynomial hypersurface in $(a,v,p)$ space (acceleration-velocity-position). Constructing all possible switching surfaces (e.g., $\mathcal{S}_{\bar{0}}$, $\mathcal{S}_{1\bar{0}}$, $\mathcal{S}_{0\bar{0}}$, etc.) by backward integration from the terminal point partitions state-space into regions of constant control sequence ("Augmented Switching Laws"), each characterized by a distinct order of bang, singular, and tangent arcs (Wang et al., 22 Jan 2026).

An essential feature in the presence of box constraints is the inclusion of "tangent markers" at the locus where a state constraint becomes active, e.g., $(p=\underline{p}, v=0, a\geq 0)$. A previously absent analytic condition for when such a tangent marker interrupts an unconstrained switching law is derived as the root of a polynomial constraint involving integration variables, ensuring completeness of the partition into admissible control sequences (Wang et al., 22 Jan 2026).

4. General Upper Bounds, Regularity, and Switch Structure

The maximum number of switches in time-optimal control is sharply bounded by system algebraic structure. For a controllable $n$-state, $m$-input linear system, the number of switches is at most $q_{A,B}-1$ on any interval of length $\leq d_A$, with $q_{A,B}$ the controllability index and $d_A$ the inverse imaginary part of a nonreal eigenvalue of $A$ (Qin et al., 2019). If all eigenvalues are real ($d_A=+\infty$), the bound applies globally.

For fully controllable, single-input systems in companion form, switching surfaces are $n-1$ explicit hyperplanes, and the control flips direction (by antipodal mapping on the unit ball) at each surface. For multi-input systems, switching occurs only on a proper subspace and, when $B$ is invertible, no switches occur (optimal control is constant) (Qin et al., 2019).

Generically, time-optimal controls in affine systems exhibit either smooth (switch-free) bang arcs or isolated, transversal switchings. The system's bracket data at a given point discriminates the regimes. Chattering (Zeno) phenomena are ruled out under rank and transversality assumptions (Agrachev et al., 2016, Agrachev et al., 2016). Singular arcs only arise where the switching function and a prescribed number of its derivatives vanish (Aghaee et al., 2020).

For high-order chains of integrators, the necessary and sufficient condition for a switching law to be time-optimal is that its pattern "uses up" all $n$ degrees of freedom (dimension matching), obeys the explicit sign-flipping rules (sign-structure theorem), and yields a state-space trajectory respecting all bounds (Wang et al., 2023).

5. Computational Algorithms and Practical Implementation

Modern algorithms for time-optimal trajectory generation, such as the Switch Point Algorithm (Aghaee et al., 2020) and the Manifold-Intercept Method (MIM) (Wang et al., 2023), systematically exploit the parametrization of switching surfaces:

• The Switch Point Algorithm reduces a general bang–bang or singular problem to an unconstrained optimization over switch times, initial costate, and terminal time. Sensitivities of the objective (final time) to each switch are computed from a single forward-backward integration sweep. The switching surfaces are reconstructed as the set of times where the switching function changes sign (Aghaee et al., 2020).
• MIM, applicable to high-order chains-of-integrators under box constraints, recursively enumerates all combinatorially admissible switching laws, classifies their switching surfaces via costate polynomials, and evaluates their feasibility and time-optimality. Polynomial systems of degree $n$ are solved for each candidate law, with the switching surfaces acting as barriers for forward or backward trajectory propagation (Wang et al., 2023).

For triple integrator systems, all switching surfaces can be expressed as closed-form polynomials in $(a,v,p)$, and the time-optimal trajectory can be constructed in $O(1)$ runtime, using the explicit partitioning of state-space by these surfaces (Wang et al., 22 Jan 2026). This enables robust, globally optimal synthesis for industrial applications with hard state and input constraints.

6. Special Classes and Terminal Sliding Mode Control

A particular class of time-optimal switching surface arises in sliding mode and terminal sliding mode (TSM) control for second order (double integrator) systems. Here, the optimal switching surface coincides with the time-optimal deceleration parabola, $x_1 = -(m/2U)x_2^2$, where $x_1$ is position and $x_2$ velocity. The non-singular TSM manifold $$S(x_1,x_2) = x_1 + \alpha(m/U) x_2^2 \operatorname{sign}(x_2)$$, for $\alpha=0.5$, yields a unique, time-optimal switching law (the “½-law”) and exactly one switch in the unperturbed case (Ruderman, 2020). For $\alpha > 0.5$, robust sliding is ensured even under bounded disturbances, and the design parameter allows for trade-off between ideal sliding and boundary-layer twisting.

The connection between TSM and classical time-optimal switching problems (Fuller's problem) underscores the geometric congruence of the switching surface with fundamental structures in minimum-time control (Ruderman, 2020).

7. Synthesis, Visualization, and Structural Summary

Time-optimal switching surfaces provide a geometric and algebraic partition of the state (and occasionally cotangent) space into operational regimes, each determined by a constant (often extremal) control law. Their explicit or implicit construction is fundamental for the analysis of optimality, robustness, regularity, computational trajectory synthesis, and feasibility under constraints.

In summary:

• Switching surfaces are determined by the zeros of analytically or algebraically constructed switching functions, as dictated by Pontryagin's Maximum Principle.
• For linear and integrator-chain systems with full state constraints, explicit polynomial representations are available, and combinatorial classification of switching sequences is tractable.
• Local and global bounds on the number of switches follow from system algebraic structure and regularity of switching function zeros.
• For affine nonlinear systems, Lie bracket data at each point governs the presence, uniqueness, or absence of local switching surfaces.
• Advanced construction and realization techniques yield time-optimal control algorithms of low computational complexity for high-order, fully constrained systems.
• Sliding mode and terminal sliding mode control for second-order systems leverage coincidence of switching surfaces with optimal trajectories for robustness and finite-time convergence.

Key advances in explicit representation, analytic bounding, and algorithmic realization of switching surfaces continue to deepen the theoretical and applied capabilities of time-optimal control synthesis (Qin et al., 2019, Ruderman, 2020, Agrachev et al., 2016, Agrachev et al., 2016, Qin et al., 2020, Aghaee et al., 2020, Wang et al., 2023, Wang et al., 22 Jan 2026).