Differential Inclusion Systems: Theory & Applications

Updated 16 November 2025

Differential inclusion systems are defined by set-valued maps that generalize classical ODEs, capturing phenomena where the dynamics can be discontinuous or ambiguous.
They rely on conditions like upper semicontinuity, convexity, and compactness to guarantee the existence of Carathéodory solutions and support multiflow analysis.
These systems are applied in diverse areas such as nonsmooth mechanics, hybrid control, chemical kinetics, and numerical verification to model uncertain and complex dynamical behaviors.

A differential inclusion system is a nonclassical dynamical system in which the ordinary differential equation (ODE) $\dot x = f(x)$ is generalized to an evolution law of the form

$\dot x(t) \;\in\; F(x(t)),$

where $F: X \to 2^{\mathbb{R}^n}$ is a set-valued map assigning to each state $x$ a possibly non-singleton, often convex subset of velocities. This generalization captures discontinuous, ambiguous, or control-driven phenomena, unifying diverse applications in nonsmooth mechanics, control theory, hybrid systems, partial differential equations, mass-action chemical kinetics, stochastic approximation, and beyond.

1. Foundations and Solution Concepts

A standard differential inclusion on an open set $X \subset \mathbb{R}^n$ involves a set-valued map

$F: X \longrightarrow \{ \text{closed, convex subsets of } \mathbb{R}^n \},$

with solutions defined as absolutely continuous curves $x: I \to X$ such that $\dot x(t) \in F(x(t))$ for almost every $t$ (Thieme, 2019). Maximal generality is often achieved by imposing:

Upper semicontinuity (u.s.c.) of $F$ (i.e., its graph is closed and values vary “nicely” with $x$ ),
Nonempty, convex, and compact values.

This structure guarantees classical existence of Carathéodory solutions via Peano-type compactness arguments (Kiseleva et al., 2018).

Flow-like properties in ODEs fail for general differential inclusions due to nonuniqueness, replaced by the multiflow construct: a closed set of possible solution pairs, forming a monoid under concatenation and enabling a topological theory of invariant sets, attractors, and recurrence (Thieme, 2019).

When the right-hand side is single-valued but discontinuous (e.g., switching or relay control), the Filippov solution replaces pointwise evaluation with a closed convex hull over neighborhood limits, ensuring convexity and existence in sliding and chattering phenomena (Kiseleva et al., 2018, Fominyh, 2022). Rigorous distinctions exist between Filippov, Aizerman–Pyatnitskiĭ (approximation by smoothings), and Gelig–Leonov–Yakubovich (direct solution of the inclusion) concepts, particularly relevant in control and mechanics.

2. Geometric and Analytical Structures

Inclusion systems extend the scope of classical ODE analysis by introducing geometric and set-valued differential topologies:

Piecewise constant and polyhedral inclusions form the basis for Filippov systems and polygonal differential inclusions, with the phase space partitioned into regions where the vector field is constant or polyhedral set-valued (Thieme, 2019, Sandler et al., 2017).
Convex analysis underpins solution existence (using upper semicontinuity and convexity), Lyapunov stability theory, and the construction of forward invariant sets via separation properties (Ivanov et al., 8 Nov 2025).
Toric and quasi-toric differential inclusions embed polynomial dynamical systems (e.g., mass-action systems) into geometrically structured, piecewise-constant cone-valued inclusions in logarithmic space, yielding polyhedral fans that systematically control persistence and permanence properties (Craciun, 2019, Craciun et al., 2019).

A tabulation of key geometric constructs is given below:

Structure	Definition	Application
Polyhedral fan	Finite collection of cones, closed under faces/intersections, covering $\mathbb{R}^n$	Toric/quasi-toric inclusions
Polar cone	$C^o = \{ u \mid u \cdot w \le 0, \forall w\in C \}$	Right-hand side in toric/quasi-toric inclus.
Multiflow	Closed, concatenable relation on $X$ , generalizing flows for nonunique systems	Topological dynamics of inclusions

The embedding of weakly reversible reaction networks into toric (and then quasi-toric) differential inclusions is achieved by constructing a fan of cones defined by the reaction graph cycles, with the mass-action vector field lying within the polar cones up to a small uncertainty determined by time-varying rates (Craciun, 2019, Craciun et al., 2019).

3. Variational, Nonsmooth, and High-Order Analytical Methods

Differential inclusion systems often arise in nonsmooth analysis and variational formulations, particularly where ODE right-hand sides involve max/min, set-valued gradients, or discontinuous feedbacks.

Variational reduction represents state and derivative by variables $(x(\cdot), z(\cdot))$ and encodes the inclusion, boundary, and surface constraints via penalty functionals. For instance, when the support function of $F_i(x)$ involves maxima (or combined max–min), the resulting functional is superdifferentiable (or quasidifferentiable), and descent methods (superdifferential/quasidifferential) can be implemented for solution synthesis (Fominyh, 2023, Fominyh, 20 Apr 2024).
Superdifferential/quasidifferential calculus enables correct handling of nonsmooth penalties, with directional derivatives computed using partitioning of the time interval into regions of constant active set, followed by convex analysis of the integrand (Fominyh, 2023, Fominyh, 20 Apr 2024).
High-order validated integration for differential inclusions with uncertainty, as realized in tools like Ariadne, leverages stepwise approximation of input uncertainties, Taylor model propagation, and validated error bounds that guarantee enclosure of all possible solutions up to the desired order (Gonzalez et al., 2020).
Lyapunov functions and quasi-Lyapunov conditions play a critical role both for existence and for qualitative properties (invariance, attractivity), including for inclusions with only closed, possibly nonconvex values, relying on monotonicity of "patchy" partitions and forward invariance of locally defined "good" sets (Ivanov et al., 8 Nov 2025).

4. Applications and Modeling Frameworks

Differential inclusions are the mathematical backbone for modeling a variety of real-world systems with inherent nonsmoothness, ambiguity, or hybrid features:

Nonsmooth mechanics and control: Mechanical systems with dry friction, backlash, or Coulomb friction, variable-structure (sliding-mode) control, relay or signum feedbacks, and friction-induced stick–slip, are modeled via inclusions defined by the convex hulls of discontinuous right-hand sides (Kiseleva et al., 2018, Fominyh, 2022). These models admit global existence, stability proofs via Lyapunov functions, and encode both classical and sliding motions.
Hybrid and piecewise-linear/hybrid automata: Planar polygonal differential inclusion systems (SPDIs) partition the plane into convex polygons, each with constant or bounded-cone dynamics, yielding a decidable reachability theory and enabling high-performance analysis through tools such as ParaPlan (Sandler et al., 2017).
Chemical and biological networks: Polynomial dynamical systems and reaction networks admit embedding into toric or quasi-toric inclusions, allowing persistent/non-extinction theorems and the systematic construction of invariant sets, barrier certificates, and global attractivity results (Craciun, 2019, Craciun et al., 2019).
Stochastic approximation and learning: Nonsmooth stochastic approximation algorithms, including subgradient and proximal stochastic gradient descent, are analyzed via mean-limit differential inclusions capturing the set-valued drift in the limit of vanishing step size, providing convergence guarantees even in nonconvex, high-dimensional regimes (Majewski et al., 2018).
Fractional and nonlocal PDEs: Double-phase, nonlocal, or fractional systems with inclusion right-hand sides model complex elliptic/parabolic systems under mixed or Dirichlet boundary conditions, with generalized solution schemes based on Galerkin approximation and surjectivity theorems for multifunctions (Cen et al., 11 Feb 2025, Cen et al., 10 May 2025).

5. Modern Theoretical Developments

Several influential lines of recent research address advanced aspects of differential inclusion systems:

Infinite-dimensional and parameterized limits: Vanishing viscosity and doubly nonlinear differential inclusions under nonconvex energies, as in rate-independent damage in elastic solids, are analyzed with energy-dissipation inequalities, parameterized solutions, and variational convergence (Knees et al., 2013).
Rough and Young differential inclusions: Differential inclusions driven by Hölder-continuous or rough signals generalize the classic ODE and inclusion framework to systems such as

$dz_t \in F(z_t)\,dx_t$

with $x$ an $\alpha$ -Hölder control ( $\alpha > 1/2$ ) or a rough path ( $\alpha\in(1/3,1/2]$ ), and $F$ possibly non-convex or non-smooth. Existence is established under Hölder- or upper/lower semicontinuity and compact-valuedness of $F$ , using pathwise selection, sewing-lemma estimates, and fixed-point theorems (Bailleul et al., 2018).

Contraction, Lyapunov, and ISS theory for inclusions: Systematic frameworks for input-to-state stability (ISS) under differential inclusions leverage nonpathological, locally Lipschitz Lyapunov functions, generalized Lie derivatives, and interconnection/small-gain theorems, with explicit extension to switched, hybrid, and observer-based control systems (Rossa et al., 2021, Harapanahalli et al., 18 Nov 2024).
Nonuniqueness and wild solutions: Convex integration and Baire category methods applied to systems such as high-dimensional ideal MHD exploit the inherent flexibility of inclusion constraints; via wave-cone directions and geometric convexification of the constraint set $K$ , one constructs compactly supported weak solutions with non-conserved quantities (e.g., total energy), establishing nonuniqueness and the richness of solutions for these PDEs (Miao et al., 8 Sep 2025).

6. Computational Aspects and Algorithmic Methods

Numerical solution and verification of differential inclusion systems require set-based, validated, and sometimes parallelized approaches:

Set-valued integrators: Validated solvers propagate enclosures of reachable sets through integration of an auxiliary ODE with representative input uncertainty, using Taylor models and higher-order error bounds. Analytical control of enclosure growth and error allows global guarantees (Gonzalez et al., 2020).
Superdifferential/quasidifferential descent: For inclusions specified via max/min in their support functions, descent algorithms (with explicit sub/superdifferential computation) are used to minimize penalty functionals constructed to encode the original inclusion and constraints, converging under weak or strong stationarity conditions (Fominyh, 2023, Fominyh, 20 Apr 2024).
Parallel reachability and verification tools: Decidable classes such as planar SPDIs admit efficient symbolic propagation of edge-interval images, with reachability reduced to enumerating feasible edge-signature types; parallelization exploits branch decomposition across multicore architectures for significant empirical speedup (Sandler et al., 2017).
Construction of geometric invariants: For toric and quasi-toric inclusions, the explicit construction of forward-invariant compact sets, zero-separating hypersurfaces, and barriers relies on the fan and cone structure, with all computations carried out in logarithmic coordinates to exploit the geometry of the positive orthant (Craciun, 2019, Craciun et al., 2019).

7. Open Directions and Research Trends

Recent work extends the theory and application of differential inclusion systems to new settings:

Extension to fractional and nonlocal operators, and the design of inclusion systems for nonstandard double-phase or mixed operator regimes (Cen et al., 11 Feb 2025, Cen et al., 10 May 2025).
Enhanced regularity and selection theory for inclusions with nonconvex or low-regularity drift, leveraging quasi-Lyapunov functions and patchwise construction (Ivanov et al., 8 Nov 2025).
Algorithmic advances in scalable, high-order, and set-valued numerical methods for validation and safety verification.
Nonuniqueness, turbulence, and wild solutions in nonlinear PDEs via convex integration and set-valued relaxation frameworks, raising foundational questions about determinacy and physical relevance (Miao et al., 8 Sep 2025).
Further development of rough and Young-driven inclusion theories, bridging stochastic analysis, control, and nonsmooth dynamics (Bailleul et al., 2018).

Overall, the theory of differential inclusion systems supplies a unifying framework for modeling, analysis, and computation in nonsmooth, uncertain, or multifaceted dynamical phenomena across mathematical, engineering, and scientific domains.