State-Dependent Switching Systems

• State-dependent switching systems are hybrid models where the active mode is determined by the system’s current or historical state, enabling dynamic and adaptive control.
• They encompass diverse formulations like continuous-time optimal control, discrete logical dynamics, and regime-switching diffusions to manage stability and performance.
• Applications span neural computation, animal behavior, compliant mechanism design, and distributed control, illustrating broad practical impacts across fields.

State-dependent switching systems are dynamical and hybrid systems in which the active subsystem, switching signal, or transition law depends on the current state of the system rather than on an exogenous schedule. In the literature summarized here, this dependence is realized through switching surfaces $g(x)=0$, finite-valued logical states $\theta_t$, state-dependent regime-switching rates $q_{ij}(x)$, delayed-state envelopes, or latent internal states inferred from sensory history. The resulting models span optimal control, stochastic hybrid systems, distributed synthesis, neural computation, animal behavior, and compliant mechanism design, and they include continuous-time hybrid systems, discrete-time switched linear systems, and countable-state regime-switching diffusions (Zhou et al., 2022, Zhang et al., 24 Mar 2026, Shao, 2017, Chen et al., 31 Jul 2025).

1. Canonical formulations and modeling scope

A standard continuous-time formulation treats the system as a hybrid ODE whose vector field changes when the continuous state crosses an interface. In the two-mode setting studied in optimal control, the dynamics are

$$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$

with continuous state at the switching time, $x(\tau^+) = x(\tau^-)$, and a differentiable switching surface $g(x)=0$. Once the switching time $\tau$ is fixed, the problem becomes a fixed-final-time two-phase optimal control problem (Zhou et al., 2022).

A second formulation treats switching as the output of an internal logical dynamical system. In the deterministic case,

$\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$

where $\theta_t$ is a logical state, $\gamma_t$ is a logical control input, and the active continuous subsystem is selected by the logical state through an index map $\theta_t$0. This differs from free switching because the switching sequence is state-dependent and dynamically constrained by the logical subsystem itself (Zhang et al., 24 Mar 2026).

A third formulation is the state-dependent regime-switching diffusion. Here the continuous state $\theta_t$1 evolves as a diffusion, while the discrete regime $\theta_t$2 or $\theta_t$3 changes with rates determined by the current continuous state: $\theta_t$4 This formulation underlies invariant-measure theory, Euler–Maruyama approximation, diffusion approximation, and large deviations for state-dependent switching diffusions (Shao, 2017, Sun et al., 11 Mar 2025, Hu et al., 2023).

The cited literature also studies discontinuous state dependence through finite partitions. In piecewise constant type switching, the rates are constant on regions such as

$\theta_t$5

or, in one dimension, on intervals $\theta_t$6, and $\theta_t$7. This representation is used both as a tractable model class in its own right and as an approximation scheme for more general state-dependent switching (Shao et al., 2022).

Across these formulations, the state variable that governs switching need not be only the instantaneous continuous state. The switching law may depend on a delayed-state envelope $\theta_t$8, on the current location relative to a feedback region $\theta_t$9, or on latent internal states with their own persistence and transition dynamics (Haimovich et al., 2012, Chen et al., 2018, Chen et al., 31 Jul 2025). This suggests that “state-dependent switching” is a structural description of how mode changes are organized, not a single model class.

2. Optimal control, stabilization, and delay compensation

In interface-triggered hybrid optimal control, a central issue is the behavior of the co-state at the state-triggered switching instant. For the two-mode problem with known number of switches and known switching modes, the Hamiltonian

$q_{ij}(x)$0

is continuous across the switching instant, and the co-state satisfies the usual adjoint equations on each side of the interface. The main result is a closed-form jump law: $q_{ij}(x)$1 where $q_{ij}(x)$2 is the normal to the switching surface at the optimal switching state. For time-varying interfaces $q_{ij}(x)$3, the denominator is modified by the interface time dependence. The same work introduces the numerical method GEL, which reduces the hybrid optimal control problem to two ordinary boundary-value problems solved with MATLAB’s bvp4c, while the scalar switching time $q_{ij}(x)$4 is updated by Newton’s method applied to a residual built from the jump law. In the reported examples, GEL was more efficient than ICLOCS2 brute-force search and the HMPMAS algorithm from Shaikh and Caines (Zhou et al., 2022).

State-dependent switching is also used directly as a stabilizing feedback law. For a two-mode switched nonlinear system $q_{ij}(x)$5, $q_{ij}(x)$6, a switched equilibrium $q_{ij}(x)$7 is defined by a convex combination $q_{ij}(x)$8. Bolzern–Spinelli showed that a stabilizing state-feedback switching rule can be built from a quadratic Lyapunov function, but the resulting switching threshold is generally quadratic. Under a common quadratic Lyapunov function condition for the two subsystems, the threshold can instead be chosen linear: $q_{ij}(x)$9 The geometric argument is that Lyapunov-decreasing regions contain ellipsoidal inner approximations tangent to a hyperplane through the switched equilibrium, so a linear switching surface suffices for quadratic global stability in the Filippov sense (Makarenkov, 2018).

A further extension concerns input delay. For linear switched systems of the form

$$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$0

the predictor is implicit because the future mode sequence depends on the future predicted state. An exact predictor state can nevertheless be constructed through a mode-by-mode segmentation of the prediction horizon, followed by a backstepping transformation

$$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$1

The resulting analysis uses multiple Lyapunov functionals with exact matching across switching surfaces and yields uniform exponential stability of the closed loop (Katsanikakis et al., 20 Mar 2026).

3. Correct-by-design synthesis, distributed control, and sampled switching

A correct-by-design approach to state-dependent switching control is developed for linear discrete-time switching systems. Given a rectangular target set $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$2, the method constructs a capture set $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$3 and a state-dependent control law that steers every state in $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$4 into $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$5 in bounded time. The synthesis proceeds by iterated backward reachability, but instead of exact predecessor computation it searches over control patterns of length at most $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$6, tiled sub-rectangles, and a parametric extension $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$7. The largest admissible margin $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$8 is computed by linear programming because the dynamics are affine. The same framework yields a stability variant that keeps trajectories in $$\dot x = f(x,u,t)= \begin{cases} f_1(x,u,t), & g(x)\le 0,\ f_2(x,u,t), & g(x)\ge 0, \end{cases}$$9 indefinitely, and it extends to a distributed setting by over-approximating the unknown other subsystem state. In the Seluxit floor-heating system with 11 rooms and $x(\tau^+) = x(\tau^-)$0 switching modes, the distributed method synthesized a controller for $x(\tau^+) = x(\tau^-)$1, obtained $x(\tau^+) = x(\tau^-)$2, and required about 20 hours of CPU time (Coënt et al., 2016).

Intermittent state feedback leads to another state-dependent switched structure: the active subsystem is determined by whether the state lies in a feedback region $x(\tau^+) = x(\tau^-)$3 or in its complement $x(\tau^+) = x(\tau^-)$4. In $x(\tau^+) = x(\tau^-)$5, an observer-based controller reduces tracking and estimation errors; in $x(\tau^+) = x(\tau^-)$6, a predictor propagates the estimate without measurement correction. A Lyapunov-based analysis yields a minimum dwell time in the feedback-available region and a maximum dwell time in the feedback-denied region. These dwell-time inequalities are then used to design a switching trajectory $x(\tau^+) = x(\tau^-)$7 with smooth transitions based on the smootherstep function $x(\tau^+) = x(\tau^-)$8, so that a desired path lying in $x(\tau^+) = x(\tau^-)$9 can still be followed while maintaining global uniform ultimate boundedness of the composite error (Chen et al., 2018).

Sampled-data feedback under state-dependent regime switching introduces an additional layer: the control uses only discrete-time observations of both the continuous and discrete components. The controlled system

$g(x)=0$0

is stabilized almost surely by combining spectral estimates of exponential functionals for state-independent comparison chains with order-preserving coupling processes based on Skorokhod’s representation of the jumping process. This extends Shao’s earlier state-independent setting to the state-dependent case and replaces a moment-type conclusion by almost sure asymptotic stability (Shao et al., 2018).

These constructions share a common theme: tractability is recovered by replacing unrestricted hybrid evolution with structured objects such as macro-step patterns, dwell-time windows, over-approximated local reachable sets, or comparison chains.

4. Stochastic analysis, approximation, and asymptotic regimes

For state-dependent regime-switching diffusions, long-time behavior requires simultaneous control of the continuous diffusion and the state-dependent jump mechanism. One approach constructs auxiliary state-independent Markov chains that dominate or are dominated by the original switching process. This makes it possible to prove existence and uniqueness of invariant measures and convergence in Wasserstein distance, and also to establish $g(x)=0$1-strong convergence of Euler–Maruyama approximations with the classical order $g(x)=0$2: $g(x)=0$3 A refined use of Skorokhod’s jump-process representation is central in controlling the probability that the true and discretized regime processes disagree (Shao, 2017).

For stochastic damping Hamiltonian systems with countably infinite state-dependent switching, a martingale approach yields global weak well-posedness under bounded measurable switching rates $g(x)=0$4, rather than continuity assumptions. The same framework establishes the strong Feller property via a killing technique and resolvent identities, exponential ergodicity through Foster–Lyapunov conditions, and large deviations principles for empirical occupation measures. Several examples, including regime-switching van der Pol and overdamped Langevin systems, are treated in detail (Xi et al., 2020).

The functional stochastic Hamiltonian setting with infinite delay and singular coefficients is more delicate because the continuous and discrete components are intertwined and correlated, and the state space of the switching process is countably infinite. In that case, the analysis proceeds in stages: first a fixed-regime martingale solution via Girsanov’s transformation, then a special decoupled switching model, and finally the full state-dependent system through a Radon–Nikodym density. Under the stated assumptions, the resulting process is well posed and has the Feller property (Xi et al., 19 Sep 2025).

Approximation theory for general state-dependent switching can itself be organized around piecewise constant switching laws. If $g(x)=0$5 approximates $g(x)=0$6 uniformly in the $g(x)=0$7-operator norm, then under the stated Lipschitz and nondegeneracy assumptions,

$g(x)=0$8

where $g(x)=0$9. The same piecewise constant framework yields explicit Lyapunov criteria for asymptotic stability near the origin and for ergodicity or transience at infinity, expressed in terms of the invariant measures of the local regime matrices that govern the relevant region of state space (Shao et al., 2022).

Multiscale limits expose another aspect of state dependence. For slow-fast SDEs with state-dependent switching and generator

$\tau$0

the slow component converges weakly to a diffusion with corrected drift and diffusion coefficients obtained from the invariant distribution of the frozen fast chain and the Poisson equation $\tau$1. The weak convergence rate is order $\tau$2, and the paper gives a one-dimensional example showing that this rate is optimal (Sun et al., 11 Mar 2025). In the Cox–Ingersoll–Ross setting with state-dependent fast switching, the matched scaling $\tau$3 leads to a path-space large deviation principle with a good action-integral rate function derived from Hamilton–Jacobi equations and a Donsker–Varadhan variational formula (Hu et al., 2023).

A distinct but related asymptotic problem concerns rare switching between metastable states under state-dependent noise rather than explicit hybrid mode laws. In semiconductor superlattices, the generalized geometric Minimum Action Method computes the maximum likelihood transition curve for

$\tau$4

and near the saddle-node threshold voltage $\tau$5 the mean lifetime satisfies

$\tau$6

This is not a state-dependent switching system in the regime-switching sense, but it addresses switching between metastable states in the presence of multiplicative noise (Heymann et al., 2010).

5. Neural, biological, and mechanical realizations

In neural computation, a finite state machine can be constructed from two coupled soft winner-take-all recurrent maps and a sparse set of transition neurons. Each map contains $\tau$7 excitatory neurons and one inhibitory neuron, and persistent memory states arise because symmetric cross-coupling $\tau$8 creates a stable nonzero attractor: $\tau$9 Transition neurons implement symbol-triggered, state-dependent edges of the automaton. The construction is systematic, scales as $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$0 in the number of states and transitions, and was reported to tolerate readout noise, synaptic weight noise, and large perturbations of $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$1 (0809.4296).

In C. elegans chemotaxis, the hierarchical state-dependent Pirouettes and Weathervaning model (staPAW) treats the latent behavioral state $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$2 as an input-driven Markov process. The model identifies two persistent states: a steer-enriched state $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$3 and a turn-enriched state $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$4, with mean dwell times of about $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$5 s in the $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$6-state and $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$7 s in the $\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$8-state. Sensory history modulates both within-state sensorimotor strategy and the transition law between states, and optogenetic stimulation of AWC$\begin{cases} \theta_{t+1}=f(\theta_t,\gamma_t),\[2mm] x_{t+1}=A_{\iota(\theta_t)}x_t+B_{\iota(\theta_t)}u_t, \end{cases}$9 increased the probability of transitioning into the $\theta_t$0-state. The data-constrained reinforcement learning model further showed that sensory-driven state switching improves gradient climbing relative to sensory-independent switching, fixed-state agents, and stateless agents (Chen et al., 31 Jul 2025).

Chaotic heteroclinic networks provide a deterministic alternative to noise-driven or Markov models of switching behavior. Here the system lingers near saddle invariant states, then exits along unstable directions into other states through heteroclinic connections. By perturbing a time-$\theta_t$1 map so that stable and unstable manifolds intersect nontrivially, the model produces branching, dwell-time variability, and apparent randomness through sensitive dependence on initial conditions rather than stochastic forcing. The framework was used to reconstruct four-state and eight-state C. elegans behavioral data from Nichols et al. and Linderman et al. (Morrison et al., 2022).

Mechanical state-dependent switching can be implemented without discrete contacts. A single-input building block combines a bistable state element, a buckling column near bifurcation, and a nonlinear spring with quadratic stiffness

$\theta_t$2

which yields a cubic force law. The current state of the bistable element acts as internal information, effectively as an extra input, allowing the same cyclic actuation to trigger different transitions depending on configuration. The reported monolithic compliant prototype, fabricated in polyamide-12 using Multi Jet Fusion, exhibited the intended four-step switching cycle [(Hu et al., 2023)?]

The last citation must be corrected: the mechanical system is described in "A Single-Input State-Switching Building Block Harnessing Internal Instabilities" (Wolde et al., 2023).

These examples address a common misconception: state-dependent switching need not be purely exogenous, memoryless, or noise-driven. In the cited biological and neural models, persistence of internal state is part of the mechanism; in the compliant mechanism, internal geometry and instability encode the switching logic itself (0809.4296, Chen et al., 31 Jul 2025, Wolde et al., 2023, Morrison et al., 2022).

6. Structural properties, invariant sets, and relations to Lyapunov methods

Some classes of switching systems admit constructive stability and stabilizability analysis because their matrix sets have a strong order structure. For positive linear discrete-time switching systems built from $\theta_t$3-sets, the hourglass alternative implies

$\theta_t$4

so asymptotic stability and stabilizability reduce to maximizing or minimizing the spectral radius over the admissible set. The class is closed under Minkowski addition, multiplication, and positive scaling, so serial and parallel interconnections preserve the “constructive resolvability” property. The same framework supports step-by-step construction of positive trajectories with the greatest rate of convergence to zero (Kozyakin, 2015).

For continuous-time switching linear systems with delayed-state-dependent perturbations, a componentwise method yields transient bounds, ultimate bounds, and invariant regions without using norms. After a similarity transformation $\theta_t$5, the method works with a Metzler matrix $\theta_t$6 and nonlinear or affine overbounds on the perturbation. In the affine case, if

$\theta_t$7

then the ultimate bound

$\theta_t$8

is globally valid. The same paper clarifies the relation to common quadratic Lyapunov functions: the class of switching systems for which the componentwise method applies is a strict subclass of the class admitting a common quadratic Lyapunov function. Thus applicability of the componentwise method implies existence of a common quadratic Lyapunov function, but not conversely (Haimovich et al., 2012).

This relation is important because many stabilization and synthesis results in state-dependent switching rely on Lyapunov structure, yet the relevant Lyapunov objects differ substantially across problem classes. In switched-equilibrium stabilization they appear as common quadratic Lyapunov functions for the subsystems; in delayed predictor feedback they appear as multiple Lyapunov functionals matched exactly on switching surfaces; in piecewise constant switching they appear as local or tail Lyapunov functions weighted by invariant distributions of regime matrices; and in distributed synthesis they are replaced by set inclusions, over-approximations, and linear programs (Makarenkov, 2018, Katsanikakis et al., 20 Mar 2026, Shao et al., 2022, Coënt et al., 2016).

A plausible implication is that the main mathematical difficulty in state-dependent switching is not simply the presence of multiple modes, but the closure of the feedback loop between state evolution and the switching law itself. The literature repeatedly restores tractability by imposing structure on that loop: differentiable interfaces, irreducible finite-state chains, piecewise constant partitions, logical algebraic representations, monotone comparison chains, or recursively closed matrix families.