Hybrid Dynamical Systems: State-Dependent Switching

• Hybrid dynamical systems with state-dependent switching are mathematical models that combine continuous flows with discrete transitions determined by state-dependent guard conditions.
• The topic emphasizes formal verification through differential dynamic logic, optimal control methods including hybrid Pontryagin principles, and identification techniques such as Hybrid-SINDy and convex optimization.
• Practical applications span robotics, power systems, epidemiology, and ecology, underscoring the significance of rigorous analysis for safety and efficient control in complex cyber-physical systems.

Hybrid dynamical systems with state-dependent switching are mathematical models in which the evolution of a system combines continuous flows and discrete switching, where the choice of the active dynamical regime at any time depends explicitly on the current state of the system. Such systems arise in diverse applications including control of robotic systems, power systems, epidemiology, stochastic processes in random environments, and model learning from data. The following sections detail the foundational modeling frameworks, formal verification techniques, optimal control theory, system identification algorithms, and key results on qualitative behavior and stability relevant to state-dependent switching hybrid systems.

1. Mathematical Formulation and Semantics

A hybrid dynamical system with state-dependent switching consists of a finite or countable set of modes (indexed by $p\in P$), each associated with its own vector field $f_p(x)$ or stochastic dynamics. The system evolves according to

$$\dot{x} = f_p(x),\qquad x\in D_p \subset \mathbb{R}^n,$$

with $p$ determined at each instant by a collection of state (and possibly time) dependent guard conditions $\chi_p(x)$. The evolution proceeds within the domain $D_p$ of the active mode until the state reaches a switching manifold (boundary), e.g., $g_{pq}(x) = 0$, at which point the mode transitions to a new regime.

The evolution may also include discrete resets upon hitting switching surfaces, modeled by deterministic or stochastic reset maps. The formal semantics of such systems can be embedded in the framework of hybrid programs from differential dynamic logic (dL). In dL, programs describe both discrete assignments and continuous ODE evolutions guarded by domain constraints: $$\alpha ::= x := e \mid ?\,\chi \mid \{x' = f(x)\;{data}\;\chi\} \mid \alpha;\beta \mid \alpha \cup \beta \mid \alpha^*.$$ For state-dependent switching with $m$ modes, the corresponding hybrid program is

$$\alpha_{sd} = \left(\bigsqcup_{p=1}^m \{x' = f_p(x)\;\mathrm{data}\;\chi_p(x)\}\right)^*,$$

encoding non-deterministic evolution in any admissible mode as long as its guard $\chi_p(x)$ holds, with instantaneous switching when guards are violated (Tan et al., 2021).

2. Verification and Invariance Principles

Formal verification of safety and invariance properties in state-dependent switched systems is facilitated by the use of differential dynamic logic (dL). A safety property $\Safe(x)$ is specified as $[\alpha_{sd}]\Safe(x)$, meaning all executions, regardless of mode-switching decisions, preserve the invariant $\Safe$.

A central proof principle is the decomposition lemma: a loop-invariant $I(x)$ for the hybrid program $\alpha_{sd}$ is equivalent to $I(x)$ being invariant for each mode under its guard: $$\forall x.\ I(x)\Longrightarrow \bigwedge_{p=1}^m [\{x' = f_p(x)\;\mathrm{data}\;\chi_p(x)\}]\,I(x) \iff \forall x.\ I(x) \Longrightarrow [\alpha_{sd}]\,I(x).$$ This allows safety and stability properties to be checked via induction and real-algebraic verification, reduced to finite-mode subproblems. For polynomial data $(f_p, \chi_p)$ and polynomial invariants $I$, the invariance condition is reducible to a decidable first-order real-arithmetic formula via Platzer–Tarski completeness (Tan et al., 2021).

3. Qualitative Dynamics and Pathological Behaviors

State-dependent switching hybrid systems can exhibit non-classical behaviors not found in purely continuous systems, especially under spatially-triggered resets. Three major pathological phenomena are observed (Clark, 15 Jul 2024):

• Beating: A trajectory on a switching surface experiences two or more simultaneous or near-simultaneous jumps before leaving the surface; the $k$-beating set quantifies the locus of such points.
• Blocking: Trajectories can be blocked on a lower-dimensional invariant set through an infinite sequence of jumps in zero time.
• Zeno: Infinitely many jumps can accumulate in finite time, causing the solution to be undefined beyond the Zeno point.

The presence of such pathologies is typical in linear and affine systems with spatially-triggered resets. Avoidance often requires the blocking set to be trivial ($\Sigma_\infty = \{0\}$); when this holds and the flow is linear, Zeno executions are ruled out for all initial conditions $x_0\ne 0$. If the set of possible simultaneous jumps reduces to the origin, all trajectories exhibit only finitely many jumps in any compact time interval (Clark, 15 Jul 2024).

4. Optimal Control and Pontryagin/Minimum Principles

Optimal control problems for state-dependent switching hybrid systems present variational challenges at switching times. The Hybrid Minimum Principle (HMP) and the hybrid Pontryagin Maximum Principle (PMP) generalize classical necessary conditions to accommodate state-triggered switching and discrete resets.

For a switched system with guards $g_j(x)=0$, the modewise Hamiltonian is $$H_i(x,u,\lambda,t) = \lambda^T f_i(x,u,t) + L_i(x,u,t)$$. At a switching instant $\tau$, the costate $\lambda$ can be discontinuous; the co-state jump law for state-triggered switches (no state jump) is (Zhou et al., 2022): $$\lambda(\tau^+)-\lambda(\tau^-) = -\frac{[L_+(\tau)-L_-(\tau) + \frac12 (\lambda_+ + \lambda_-)^T(f_+-f_-)] \nabla g(x(\tau))}{\nabla g(x(\tau))^T \frac{f_+ + f_-}{2}}.$$ For hybrid systems allowing state resets $x^+ = R_{ij}(x^-)$, the jump condition generalizes to include reset maps, with the new costate relating to the pushforward by the reset Jacobian and a multiplier term for the guard (Pakniyat et al., 2017).

Co-state discontinuities are essential for correct sensitivity analysis and arise from variational contributions due to the timing and manner of mode transitions. Existence and uniqueness of these jumps depend on actuation structure at the switching surface (Clark, 15 Jul 2024). Sufficiently regular settings with a known sequence of modes and transversal crossings admit analytical and efficient numerical treatment; for more general scenarios, complementarity-based formulations using MPCC techniques offer a global solution methodology (Kazi et al., 5 Mar 2025).

5. Identification from Data and Model Learning

System identification for hybrid systems with unknown switching surfaces demands estimating both the continuous dynamics in each mode and the state-dependent switching law. Two major recent frameworks are:

• Sparse regression-based (Hybrid-SINDy): Uses online/local clustering in phase-space augmented with derivative coordinates, followed by sparse regression in each cluster to identify the governing equations. Analysis of cross-cluster assignment then recovers switching surfaces, using information-theoretic model selection (AIC) for validation. Successful application to hopping mechanical systems and hybrid epidemiological models is demonstrated (Mangan et al., 2018).
• Convex optimization and bilevel frameworks: Posits a mixed-integer (or relaxed) joint estimation of mode assignment and continuous dynamics, such as SLSs and SPSs. A hierarchy of convex relaxations—including semidefinite (moment) and simplex (linear programming) approaches—enables scalable estimation and identification of the switching law via margin-based polynomial classifiers. The framework alternates SDP/LP-based mode assignment with parameter learning, with empirical results confirming accurate switching surface recovery and high rollout accuracy (Iwasaki et al., 29 Sep 2025, Rivas et al., 11 Nov 2025).

Real-time identification is also addressed via two-timescale adaptive stochastic approximation, using online deterministic annealing for partitioning and recursive regression for local model updates, with convergence and identifiability guarantees (Mavridis et al., 3 Aug 2024).

6. Stochastic Extensions, Ergodicity, and Stability

Hybrid stochastic systems with state-dependent switching extend the deterministic paradigm to random environments, models with diffusion, and Markovian regime switching. Canonical formulations involve a continuous state $x(t)$ and a discrete mode $\alpha(t)\in S$ (finite or countable), evolving under SDEs: $$dx(t) = b(x(t), \alpha(t))dt + \sigma(x(t), \alpha(t))dW(t),$$ with transitions in $\alpha$ at rates $q_{ij}(x)$ depending on the current or past state (Nguyen et al., 2017, Shao et al., 2022). Well-posedness, Feller, and strong Feller properties are established under standard Lipschitz and uniform boundedness/ellipticity assumptions.

For piecewise constant switching laws, stability and ergodicity are characterized by explicit matrix-weighted Lyapunov criteria. If the convex combination of mode–wise drift rates (weighted by the stationary law $\pi$ of the switching matrix) satisfies $\sum_i \pi_i \beta_i < 0$, stability or ergodicity follows, with sharp “cut-off” thresholds for transition to instability or transience (Shao et al., 2022).

Randomly switching hybrid systems with fast switching and small diffusion admit averaging principles: their invariant measures converge to those of the deterministic averaged ODE over the mode weights given by the stationary distributions of the Markov chain, provided the generator family is regular and irreducible (Du et al., 2019).

In evolutionary and ecological models, state-dependent switching of environments (e.g., host types or environmental regimes) modulates the stability and attracts nontrivial polymorphisms and multistability, enabling outcomes (such as stable coexistence) that cannot occur in any fixed environment (Farkas et al., 2011).

7. Applications and Illustrative Examples

Hybrid dynamical systems with state-dependent switching arise in:

• Robotic locomotion (e.g., contact mode transitions in legged systems),
• Powertrain and automotive mode management,
• Epidemic models with population-structure-driven switching of transmission rates,
• Gene regulatory networks and neuroscience,
• Stochastic models in ecology, including predator–prey dynamics and pathogen evolution in fluctuating environments.

Computationally tractable hybrid identification and analysis frameworks enable rigorous characterization of global attractors via combinatorial topology (e.g., Conley index and Morse graphs), outperforming purely statistical or Lipschitz-based outer approximation methods in high-fidelity recovery of invariant sets and their topological relations (Rivas et al., 11 Nov 2025).

Contemporary research leverages convex relaxations, bilevel optimisation, and advanced verification logics to shift hybrid modeling from black-box prediction toward formal, certifiable analysis suitable for control synthesis, safety verification, and design in complex cyber-physical and biological systems.