Hybrid System Formulation

• Hybrid system formulation is a unified framework that combines continuous dynamics and discrete events to accurately model complex, real-world systems.
• It employs methodologies like differential dynamic logic, mixed-integer programming, and occupation measures to optimize control and ensure safety.
• The approach supports efficient simulation, planning, and verification in diverse applications such as robotics, power electronics, and cyber-physical systems.

A hybrid system formulation rigorously encodes dynamical systems featuring both continuous-time evolution and discrete transitions (events, mode switches, resets) within a unified mathematical structure. Hybrid system techniques underpin high-fidelity modeling in control, robotics, power electronics, networked systems, cyber-physical verification, and multi-domain field theory, and multiple formalisms have been developed to support analysis, control synthesis, simulation, and composition. Approaches span analytic formulations (PDE, complementarity, occupation measures), algorithmic modeling frameworks (differential dynamic logic, category-theoretic composition), and numerically tractable planning/control toolchains (mixed-integer programs, semidefinite relaxations, and hybrid zonotopes). This article surveys the canonical elements, highlighted methodologies, and major application contexts as crystallized by leading research on arXiv.

1. Canonical Frameworks for Hybrid System Formulation

Hybrid system models are characterized by their state space, modal (discrete) structure, admissible controls/inputs, and discrete or continuous evolution laws. Fundamental definitions include:

• State and modes: The state $x \in \mathbb{R}^n$ evolves according to discrete modes $q \in \mathcal{I}$, and the system may be indexed over a directed graph or combinatorial structure representing allowed discrete transitions (Culbertson et al., 2019).
• Continuous evolution: In each mode, continuous dynamics are specified: e.g., $\dot x = f_i(x,u)$ on a domain $\mathcal{D}_i$, with input $u(t)$ constrained by $\mathcal{U}$ (Westenbroek et al., 2015).
• Discrete events: At mode-dependent guard sets (e.g., $x(t) \in Z_e$, $u \in U$), the system may instantaneously transition (jump) to new mode/state via a reset map $r_e$ (Culbertson et al., 2019, Heemels et al., 16 Jan 2026).
• General hybrid time: The trajectory unfolds over a hybrid time domain $E \subset \mathbb{R}_{+} \times \mathbb{N}$ to encode continuous evolution on $[t_j, t_{j+1}] \times \{j\}$ (flows) punctuated by jumps (Heemels et al., 16 Jan 2026, Leudo et al., 2024).

Hybrid systems are thus formalized as tuples $(C, D, f, g, \mathcal{U})$—flow set, jump set, flow map, jump map, and input class. Existence and completeness of solutions are underpinned by careful viability-theoretic or tangent-cone conditions, expressing that from any initialized state, the system can either flow (satisfying $C$) or jump (satisfying $D$), and that maximal solutions cannot be “blocked” without leaving the feasible phase-space (Heemels et al., 16 Jan 2026).

2. Hybrid System Modeling in Control, Optimization, and Verification

Hybrid system formulations underpin a spectrum of control, scheduling, and verification workflows:

• Optimal control via relaxed or mixed-integer programming: For continuous-trajectory hybrid systems, the optimal control problem is posed for $(x(\cdot), u(\cdot), \omega(\cdot))$, with $\omega_i(t)$ as mode indicator (pure or relaxed, e.g., convex combinations, $\omega(\cdot) \in \Sigma_r^M$). Control laws are optimized by projecting relaxed solutions onto pure mode sequences to ensure practical feasibility while enabling numerical tractability (Westenbroek et al., 2015).
• Piecewise affine and complementarity modeling: Any continuous piecewise-affine system (PWA) can be equivalently formulated as a one-step mathematical program with complementarity constraints (MPCC), facilitating efficient NLP-based optimal control that bypasses the combinatorial complexity of traditional mixed-integer approaches (Hempel et al., 2015).
• Mixed-integer quadratic programming (MIQP) and convexification: For finite-horizon control, strong MIQP models incorporate tight convex-hull descriptions, feasibility cuts, and nonlinear perspective cuts to greatly amplify branch-and-bound efficiency (Lee et al., 2024). Hybrid zonotope formulations enable memory-efficient reachability and planning for hybrid PWA systems in embedded settings (Robbins et al., 19 Feb 2026).
• Hybrid games and robust control: Two-player zero-sum games with hybrid system dynamics are formulated via cost functionals on hybrid arcs, controlled flow/jump sets, and terminal sets. The value function is characterized by hybrid Hamilton–Jacobi–Bellman–Isaacs PDEs (HJBI), enabling optimal strategy synthesis and robustness analysis (Leudo et al., 2024).

3. Hybrid System Formulation in Verification: Differential Dynamic Logic and Invariants

Formal verification of hybrid systems leverages logic-based approaches capable of expressing both discrete program flow and continuous ODE evolution:

• Differential dynamic logic (dL): Hybrid programs encode alternative compositions—assignments, tests, ODE evolutions, nondeterministic choices, loops—with program-modal formulas $\dbox{\alpha}{P}$ expressing safety properties for all possible executions (Tan et al., 2021).
• Switched systems as hybrid programs: Complex switching schemes are encoded as nondeterministic repetition of ODE evolutions with mode-dependent domains. Invariant properties of such systems are proved compositional: a set $I$ is invariant for the full hybrid program if and only if it is invariant under each constituent ODE with its guard; this yields decidable verification via quantifier elimination in semialgebraic structures for polynomial systems (Tan et al., 2021).
• Compositionality and tooling: The hybrid program approach supports compositional reasoning and tool-supported deductive proofs (e.g., KeYmaera X), obtaining complete or sound verification results for a broad class of safety specifications.

4. Analytic and Numerical Formulations: SIE-PDE, Least-Squares, and Occupation Measures

Distinct analytic frameworks address boundary-value, PDE, and integral-equation hybrid problems:

• Hybrid SIE–PDE formulations: In electromagnetic analysis, overlapping surface integral equation (SIE) and PDE subdomains allow simultaneous modeling of highly conductive inclusions (via SIE and equivalent electric current densities) and arbitrarily inhomogeneous backgrounds (via PDE, inhomogeneous Helmholtz equations). The interface is treated by embedding the equivalent SIE current as an excitation in the PDE, with matched basis expansions ensuring seamless Galerkin discretization. The method eliminates explicit boundary condition enforcement and achieves high computational efficiency and accuracy (Sun et al., 2021, Sun et al., 2022).
• Least-squares with functional connections: Hybrid ODE/PDE BVPs over partitioned domains are solved via analytic expressions embedding all boundary/junction continuity constraints. The spatial domain is partitioned, constrained expressions are built per segment, and the problem reduces to a global parameter least-squares or Gauss–Newton solve, achieving machine precision for hybrid sequences of arbitrary length and type (Johnston et al., 2019).
• Occupation measure and SDP relaxations: For discrete-time polynomial hybrid systems, feedback controller synthesis is encoded as a measure-theoretic linear program, with infinite-dimensional occupation measures and Liouville balance equations. Moment-SOS hierarchies yield polynomial-size SDPs amenable to convex solvers, producing convergent approximations to otherwise intractable hybrid optimal control and verification problems (Han et al., 2018).

5. Geometric and Categorical Formulations

The geometric, topological, and categorical perspectives yield global controllability tests, compositionality, and hierarchical modeling:

• Geometric controllability: Hybrid control systems defined on mode-indexed manifolds admit global controllability tests via the geometry and topology of the “jump set.” The possibility of “teleportation” across state-space via jumps and the structure of the reset map is exploited to overcome controllability failures present in the continuous subsystems. Sufficient and necessary conditions are formalized using transversality and tangent space spanning arguments (Liñán et al., 2019).
• Formal composition: Hybrid systems are objects in a category with semiconjugacy as morphisms. Three interacting structured compositions are defined:
  • Hierarchical (template-anchor via pullback of spans),
  • Sequential (directed system composition via graph pushout and flow mapping),
  • Independent parallel composition (categorical product).
• The resulting double category framework allows modular construction, reduction, and synthesis while preserving hybrid-system semantics (Culbertson et al., 2019).

6. Hybrid Systems in Physical Sciences and Multi-Physics Domains

Hybrid system formulations extend naturally to multi-physics problems at the intersection of analysis, field theory, and statistical mechanics:

• Variational hybrid Vlasov–fluid (MHD) models: Euler–Poincaré reduction is used to derive several classes (current-coupling, first and second pressure-coupling) of hybrid Vlasov-MHD equations, with distinct reduced Lagrangians and associated Kelvin–Noether theorems, Ertel’s relations (potential vorticity), and new cross-helicity invariants. The formal variational structure systematically yields the equations for mixed-fluid/hot-particle interaction, with comparative tables of conservation laws and dynamical consequences (Holm et al., 2010).
• Quantum–classical hybrids: Statistical ensembles on hybrid classical–quantum phase-space lead to Liouville equations that irreducibly couple mixed quantum–classical states. Evolution of the reduced quantum density operator is in general not autonomous, but depends on the full ensemble, with implications for backaction and operational predictions in hybrid quantum–classical systems (Buric et al., 2012).

7. Advanced Hybrid Planning, Scheduling, and Real-Time Implementation

Recent work operationalizes hybrid system formulations for optimal planning and embedded implementation:

• Hybrid zonotope planning and MIQP/ADMM heuristics: Piecewise-affine (PWA) hybrid systems are encoded in the hybrid zonotope set representation (continuous and binary generators plus affine constraints). Closed-form operations (affine maps, Minkowski sum, intersection) enable reachability and optimal plan construction. Mixed-integer MIQP planning is efficiently solved by an ADMM-based feasibility pump exploiting the zonotope structure, yielding lower memory and tighter relaxations compared to traditional MLD approaches, with demonstrated real-time performance for autonomous driving scenarios on embedded platforms (Robbins et al., 19 Feb 2026).
• Strong MIQP formulations: For $n$-period hybrid control problems, convex-hull and “tight perspective” cuts in $(x,u,\delta)$ space dramatically strengthen the continuous relaxation. Feasibility and gradient cuts generalize to multi-mode, multi-variable settings, yielding order-of-magnitude speedups in branch-and-bound search and widespread reductions of root-node gap and CPU time in large-scale MIQP instances (Lee et al., 2024).

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Hybrid system formulation synthesizes continuous and discrete models into a unified mathematical, analytic, and algorithmic structure. Research advances enable tractable analysis, control and verification, robust numerical solution, and high-level compositional modeling, supporting a wide range of disciplines and application domains (Holm et al., 2010, Westenbroek et al., 2015, Culbertson et al., 2019, Tan et al., 2021, Sun et al., 2021, Lee et al., 2024, Heemels et al., 16 Jan 2026, Robbins et al., 19 Feb 2026).