Differential Inclusions: Theory & Applications

Updated 26 March 2026

Differential inclusions are dynamic models where the state derivative belongs to a set-valued map that can be nonconvex and nonsmooth, capturing discontinuous behaviors and nonunique evolutions.
Analytical and numerical methods, such as Euler polygonal approximations, Filippov solutions, and quasi-Lyapunov techniques, establish existence, stability, and invariant properties despite the system’s inherent nonuniqueness.
These models are applied in mechanical systems, control design, and reaction network analysis, providing robust frameworks for handling friction, sliding modes, and uncertainties in dynamic phenomena.

A differential inclusion is a dynamical system in which the derivative of the state is constrained to lie in a set-valued map, i.e., $\dot x(t) \in F(x(t))$ , with $F$ a multifunction that may be set-valued, possibly nonconvex and nonsmooth. This non-single-valued generalization of ODEs arises in a broad range of settings: discontinuous control, friction models, mechanical nonsmoothness, systems with uncertain, nonunique, or state-constrained evolutions. Analytical, topological, and computational frameworks for differential inclusions extend and generalize the classical theory of ODEs, but the loss of uniqueness and the emergence of sliding, relaxation, and measure-valued phenomena require fundamentally different methods.

1. Fundamental Concepts and Existence Theorems

A classical ODE $\dot x(t) = f(x(t))$ enforces deterministic flow via a single vector field $f$ . In contrast, a differential inclusion $\dot x(t) \in F(x(t))$ admits all absolutely continuous trajectories with a.e. derivatives in $F(x(t))$ . Under upper semicontinuity and compact convex values, classical Peano/Filippov–Aubin–Cellina theory guarantees existence (but rarely uniqueness) of local solutions through any initial point (Thieme, 2019).

For $F$ upper semicontinuous with nonempty, compact, convex values, an Euler polygonal approximation yields existence by Arzelà–Ascoli and measurable selection. More general results incorporate nonconvex right-hand sides via barrier/Lyapunov constructions and fine-grained invariance properties, e.g., for a bounded, closed-valued $F$ on a locally-closed domain, with an exhaustion by invariant convex-valued approximations and continuous quasi-Lyapunov functions, solutions exist from all points outside descending "bad sets" determined by Lyapunov barriers (Ivanov et al., 8 Nov 2025).

In distributional and infinite-dimensional contexts, analogous existence holds for measure-driven, non-autonomous, or operator-valued inclusions, provided maximal monotonicity and appropriate continuity/commutator identities are satisfied (Trostorff, 2018).

2. Solution Concepts and Regularizations

Three central solution concepts address discontinuous or nonsmooth systems:

Filippov solutions: For a piecewise-continuous $f$ , the Filippov set $K[f](x)$ convexifies the set of limit values around $x$ (excluding a null set), ensuring that absolutely continuous solutions to $\dot x \in K[f](x)$ capture sliding and nonunique evolution across switching surfaces (Kiseleva et al., 2018).
Aizerman–Pyatnitskiĭ (AP) regularization: Regularize $f$ to a family $f_\varepsilon$ , take ODE solutions, and extract uniform limits as $\varepsilon \to 0$ . This may not coincide with Filippov at rigid stick-slip loci (static friction exceeds dynamic).
Set-valued (GLY) approach: Directly postulate a set-valued $F$ encoding all physically admissible instantaneous velocities, e.g., in friction or control systems, and analyze inclusions at this primitive level.

Generalized (Filippov) solutions for higher-order inclusions and discontinuities in velocity variables invoke closed convex hulls on vertical slices in $(x, \dot x)$ , leading to set-valued Newton–Euler inclusions (Zubelevich, 2020).

3. Lyapunov, Invariance, and Barrier Methods

Lyapunov methods for inclusions must generalize the derivative concept and handle the nonuniqueness of solutions:

Set-valued and Clarke generalized derivatives: Given a locally Lipschitz $V$ , consider Clarke’s generalized derivative $\dot{\overline{V}}(x,t)$ or Bacciotti–Ceragioli constructions, which take minimum/maximium across $F(x)$ and generalized gradients.
Reduction and auxiliary functions: By identifying regular functions $U$ that separate "inadmissible" directions and reduce the set $F(x)$ to a smaller subset $G_U^F(x)$ compatible with $U$ , one can tighten Lyapunov bounds and secure less conservative stability/invariance conclusions (Kamalapurkar et al., 2017).
Quasi-Lyapunov functions and barrier arguments: For nonconvex or weakly continuous $F$ , families of nested quasi-Lyapunov functions $(w_k)$ associated with invariant partitions can be used to sustain trajectory evolution away from "bad" regions where Lyapunov increase fails (Ivanov et al., 8 Nov 2025). This machinery unifies upper/lower semicontinuous cases and recovers Olech–Łojasiewicz-type results.
Trapping domains and global boundedness: If one constructs a $C^4$ barrier $F$ with sublevel set $D_c$ compact, and $F(x)$ points strictly inside $D_c$ at the boundary, then all solutions from $D_c$ remain globally in $D_c$ for all time, ensuring forward-backward boundedness (Zubelevich, 2020).

4. Computational Methods, Discretization, and Optimality Conditions

Discretization and coderivative calculus: The reachable map for a discretized inclusion admits coderivative estimates, and discrete Euler–Lagrange and transversality conditions can be derived, passing to the limit to produce continuous necessary optimality conditions for inclusions (Pang, 2011).
High-order and parameterized numerical methods: Over-approximation of reachable sets, via stepwise Taylor-model flowpipe integration and moment-matching of uncertain inputs, allows validation and rigorous error bounds for inclusions in high-consequence systems. Fine-grained error estimation and parameter control address the curse of dimensionality and parameter growth (Zivanovic et al., 2012, Gonzalez et al., 2020).
Quasidifferentiable variational reduction: Differential inclusions with support functions involving sums of max and min of $C^1$ functions can be reformulated as unconstrained minimization of functionals that are quasidifferentiable. The descent direction and necessary/sufficient conditions are expressible in terms of sub- and superdifferentials, with convergence guarantees for the constructed algorithms (Fominyh, 2024).

5. Advanced Structures: Wasserstein Space and Rough Paths

Differential inclusions in Wasserstein and metric measure spaces: By lifting the inclusion to the space of probability measures (Wasserstein space), one describes mean-field and continuity equation dynamics via set-valued velocity fields $V(t,\mu)$ taking values in $C^0(\mathbb R^d,\mathbb R^d)$ , with Filippov, relaxation, and compactness theorems extending to this infinite-dimensional setting (Bonnet et al., 2020, Bonnet et al., 2020). Convexity and sublinear growth ensure tightness and compactness of measure-valued trajectories, enabling control-theoretic and mean-field analyses.
Young and rough differential inclusions: For stochastic and highly-irregular driving signals ( $\alpha$ -Hölder, rough paths), existence theory for inclusions $dz_t\in F(z_t)\,dx_t$ (deterministic) or $dz_t\in F(z_t)\,dt + G(z_t)dX_t$ (rough) requires compactness, H\"older continuity, and semi-continuity of $F$ , with solutions constructed via dyadic piecewise-constant approximations, sewing lemmas, and compactness in variation/sewing topologies (Bailleul et al., 2018).

6. Applications: Mechanical Systems, Reaction Networks, and Control

Differential inclusions model rich nonsmooth phenomena in:

Mechanical systems: Systems with dry or Coulomb friction, drilling dynamics, Watt governor, and Chua circuits are modeled via set-valued maps encoding stick-slip, asymmetrical torque, or hidden attractors. Appropriate inclusion formulation (Filippov, GLY) combined with Lyapunov or variational methods ensures correct modeling and accurate simulation (Kiseleva et al., 2018, Fominyh, 2022).
Reaction network theory: Mass-action kinetics with uncertain, variable, or nonautonomous rate constants are encoded as vertexical families of differential inclusions, supporting structural induction via coordinate projections and category-theoretic functors. This allows recursive proofs of persistence, permanence, and, under repulsion/persistence at faces/vertices, addresses the Global Attractor Conjecture (Gopalkrishnan et al., 2012).
Control and sliding modes: Discontinuous/fault-tolerant control is formalized via inclusions; sliding-mode design and analysis use barrier functionals, variational minimization, or smoothing strategies (exponential or polynomial-root laws) to achieve robust performance under uncertainty and discontinuity (Fominyh, 2022, Ivanov et al., 8 Nov 2025, Fominyh, 2024).

7. Higher-Order, Variational, and Dual Formulations

Higher-order and variational inclusions: Problems involving higher derivatives ( $k$ th order) have dual formulations via locally adjoint mappings, inducing generalized Euler–Lagrange conditions and transversality in necessary optimality (Mahmudov, 2019). Polyhedral, semilinear, and measure-valued inclusions (e.g., in continuum mechanics) are encompassed by this duality framework, connecting primal and dual trajectories via explicit ODE/PDE systems and convex analysis.
Nonsmooth, oscillatory, and hemivariational problems: In PDEs with oscillatory or locally nonsmooth right-hand sides (subdifferential-driven), existence and multiplicity results depend on detailed variational/nonsmooth critical point theory, leveraging Clarke generalized derivatives and localization arguments to address genuinely nonregular forces arising in mechanical and engineering models (Kristály et al., 2020).

Table: Key Analytical Regimes in Differential Inclusions

Theory/Setting	Existence/Compactness Tools	Solution/Trajectory Properties
Upper-semicontinuous, convex	Euler polygonal/Peano–Filippov	Existence, nonuniqueness; sliding modes
Nonconvex, closed, bounded	Quasi-Lyapunov/barrier invariance	Existence via approximations/barriers
Measure/Wasserstein-valued	Ascoli–Arzelà in Wasserstein, trajectory-selection	Solution compactness, relaxation, control equivalence
Nonsmooth/oscillatory subdifferential	Clarke critical point, localization	Infinitely many solutions, hemivariational phenomena
High-order, optimal control	Locally adjoint mapping, conjugate support function	Euler–Lagrange inclusion, duality

The theory of differential inclusions unifies classical deterministic dynamics, nonuniqueness/relaxation phenomena, and nonsmooth or discontinuous effects, providing a robust framework for modeling, stability analysis, algorithmic integration, and optimal control across pure and applied mathematics. The ongoing development of analytic, topological, and computational methods for these systems is central to the mathematical treatment of friction, nonsmooth mechanics, control, hybrid systems, and network dynamics. Recent advances include rigorous reachability computations, measure differential inclusions for large-scale mean-field interactions, and effective variational and Lyapunov-based methods for both theoretical and applied problems (Ivanov et al., 8 Nov 2025, Bonnet et al., 2020, Fominyh, 2024).