Differential-Inclusion Argument

• Differential-Inclusion Argument is a framework using set-valued mappings to model and analyze systems with uncertainty, discontinuity, and control constraints.
• It employs relaxation methods, over-approximation techniques, and Lyapunov-based stability analysis to guarantee analytical rigor and computational accuracy.
• Its applications span control theory, PDE analysis, and stochastic dynamics, providing insights into weak solutions and multivalued dynamics.

A differential-inclusion argument is a method or framework wherein the behavior of a system is modeled, analyzed, or approximated by employing differential inclusions—relations of the form $\dot{x}(t) \in F(x(t))$, where $F$ is a set-valued mapping. This approach is central in the analysis of dynamics under uncertainty, non-smoothness, control constraints, or discontinuities. Differential-inclusion arguments are foundational in the paper of weak solutions, relaxation phenomena, reachability analysis, and variational rigor across applied mathematics, control, partial differential equations, and stochastic dynamics.

1. Foundational Principles of Differential-Inclusion Arguments

At its core, a differential inclusion generalizes ordinary differential equations by allowing the right-hand side to be set-valued. In a typical case,

$\dot{x}(t) \in F(x(t)), \quad x(0) \in X_0,$

with $F : \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ a closed-valued (often upper semicontinuous and convex-valued) map. Practical and theoretical motivations include:

• Uncertain or adversarial disturbances (e.g., uncertain $v(t)$ with $v(t) \in V$),
• Discontinuous vector fields (modeling e.g., sliding-mode control, dry friction, switching),
• Modeling attainable sets in control systems and reachability under bounded controls,
• Variational and relaxation approaches where direct minimization or solution is intractable.

In stochastic extensions, one considers $x_{k+1} \in F(x_k, v_k)$, with $v_k$ a random process, or inclusions arising as mean-limit dynamical laws in stochastic approximation algorithms (Majewski et al., 2018, Borkar et al., 2023).

A characteristic feature is the interplay of analytical (e.g., existence, uniqueness, regularity, instability) properties of the solutions with algebraic, geometric, or variational structure of the set-valued map $F$.

2. Hierarchy and Structure of Differential-Inclusion Arguments

The class of differential-inclusion arguments encompasses several interrelated methodologies:

• Relaxation and Convex Integration: Non-attainable (rigid) constraints are replaced by inclusions into relaxed or convexified sets, capturing microstructure or oscillation phenomena (see the use of laminate convex hulls and the Baire category method (Barroso et al., 2017, Miao et al., 8 Sep 2025)).
• Over-Approximation and Reachability: Computational and analytical over-approximation of reachable sets via finite-dimensional reduction and rigorous error control (e.g., Taylor model methods and Fliess expansions (Zivanovic et al., 2012)).
• Regularity and Rigidity: Analysis connecting the structure of constraint sets (e.g., existence of rank-1 connections, ellipticity) to regularity/stability or the possibility of wild solutions (as in non-elliptic inclusion regularity (Lamy et al., 2019), or compensated compactness (Philippis et al., 2018)).
• Selection and Approximate Solutions: Solutions may be constructed via measurable selections with appropriate regularity (e.g., finite $p$-variation in Young and rough inclusion settings (Bailleul et al., 2018)).
• Lyapunov, Barrier, and Certificate Methods: Use of Lyapunov or barrier functions to prove stability, safety, or boundedness in inclusive, possibly stochastic, dynamical frameworks (Ghanbarpour et al., 27 Aug 2025, Ganesh et al., 2023).

3. Implementation Schemes and Error Control

Numerical and analytic arguments for differential inclusions must account for input uncertainty or set-valued disturbances:

Over-approximation via Finite-Dimensional Families:

• For input-affine inclusions:

$$\dot{x}(t) = f(x(t)) + \sum_{i=1}^m g_i(x(t)) v_i(t), \quad v(t) \in V,$$

the full set of possible functions $v(\cdot)$ is replaced by parameterized families of functions $w(\cdot)$, such as affine functions in time. The reachable set propagation relies on step-wise computation of the image of an approximating model plus a uniform error bound derived from the mismatch between $v(\cdot)$ and $w(\cdot)$. The error is controlled by high-order bounds, using Lipschitz constants, higher derivatives, and Grönwall-type inequalities via the logarithmic norm (Zivanovic et al., 2012).

Computability and Verification:

• When $F$ is upper semicontinuous with convex, compact values, upper-semicomputability results guarantee that reachable sets over finite intervals can be covered to arbitrary accuracy by finite unions of rational boxes, as described algorithmically (Gonzalez et al., 2020).

Variational Reduction for Nonsmooth Inclusions:

• For inclusions with support functions involving maximum and minimum over smooth functions, the argument proceeds by introducing a penalty functional (possibly superdifferentiable or quasidifferentiable) whose global minimum corresponds to a solution. Modified steepest descent algorithms (using superdifferential or quasidifferential information) are employed for numerical solution (Fominyh, 2023, Fominyh, 20 Apr 2024).

Stability via Lyapunov and Barrier Functions:

• For safety verification or stochastic inclusions $x_{k+1} \in F(x_k, v_k)$, one uses nonnegative barrier functions $B$ and shows that, under suitable convexity and regularity assumptions on $F$ and a concavity condition on $B$, worst-case probability bounds can be certified by checking a simplified barrier condition (e.g., $E[\sup_{y \in F(x, v)} B(y) | x] \leq \rho \Delta$ reduces to $\sup_{y \in F(x, E[v])} B(y) \leq \rho \Delta$) (Ghanbarpour et al., 27 Aug 2025).

Pseudocode for Over-Approximate Reachable Set Computation:

R[0] = initial_set for k in range(0, N): # For each x in current set, choose parameterized input w(a_k, .) images = [] for x in R[k]: for a_k in parameter_space_A: y = flow_map(x, a_k) # solve ODE with w(a_k, .) images.append(y) # Error estimation based on step size, Lipschitz constants, derivatives error_box = compute_error_box(...) # Next reachable set is Minkowski sum of image set and error box R[k+1] = minksowski_sum(images, error_box)

4. Error Estimates, Regularity, and Propagation

Quantitative error control within single and multiple time steps is central. In input-affine, smooth systems, local error is bounded by constants involving Lipschitz constants ($L$), bounds on $f$ and $g_i$ ($K, K'$), and derivatives up to the required order, as in:

$$\left| x(t_{k+1}) - y(t_{k+1}) \right| \leq h_k \cdot K' \cdot \frac{e^{\Lambda h_k} - 1}{\Lambda h_k},$$

or

$$\|x(t_{k+1}) - y(t_{k+1})\| \leq h_k^2 [ ((K + K') L')/3 + 2K'(L + L') \frac{e^{\Lambda h_k}-1}{\Lambda h_k}],$$

where $\Lambda$ is the logarithmic norm bound (Zivanovic et al., 2012). For higher orders, additional moment constraints and smoothness are imposed for sharper error reduction.

Global error over $[0,T]$ is addressed via bootstrapping local errors with exponential inflation controlled by the logarithmic norm, e.g.,

$$\|y(t) - x(t)\| \leq \exp\left(\int_{t_0}^t l(s)ds\right) \left[\rho + \int_{t_0}^t \exp\left(-\int_{t_0}^s l(r)dr\right)\delta(s)ds\right].$$

5. Theoretical Insights: Rigidity, Flexibility, and Structural Results

Differential-inclusion arguments are deeply informed by the algebraic and geometric structure of the constraint set. For elliptic sets (without rank-1 connections), rigidity results ensure that weak or approximate solutions are forced to be smooth or even analytic, as in subspaces with no rank-1 connections (Lorent, 2012), or the minimal graphs context (Tione, 2020). When non-ellipticity or the presence of rank-1 connections exists, “flexibility” prevails—a differential inclusion may have wild, highly oscillatory, or even nonunique solutions (via convex integration), as for high-dimensional ideal MHD (Miao et al., 8 Sep 2025) or non-elliptic Aviles-Giga settings (Lamy et al., 2019).

A prototypical rigidity result is as follows: If the (generalized) gradient $Du(x)$ of a solution is constrained almost everywhere to a set $S$ that does not admit rank-1 connections, then under mild integrability $u$ is smooth (or even real analytic after elliptic bootstrapping). If $S$ is not elliptic, weak solutions may be extremely irregular unless additional structural entropy constraints exist.

6. Applications in Analysis, Control, and Computation

Differential-inclusion arguments are now standard in:

• Computational Reachability and Verification: Used in rigorous over-approximation algorithms for hybrid/continuous systems (Taylor and Fliess-based methods) (Zivanovic et al., 2012, Gonzalez et al., 2020).
• Convergence of Nonsmooth and Stochastic Algorithms: For stochastic approximation and machine learning, convergence proofs to the set of stationary points are established via mean-limit differential inclusions (Majewski et al., 2018, Borkar et al., 2023).
• Weak Solutions, Nonuniqueness, and Microstructure: For nonunique weak solutions in fluid dynamics and MHD, and the existence of microstructure in materials and phase transitions, via convex integration and Baire category methods (Barroso et al., 2017, Miao et al., 8 Sep 2025).
• Safety and Robustness Under Uncertainty: For the quantitative analysis of robustness, safety, or stabilization in systems with unmodeled dynamics, adversarial or hybrid uncertainties, and stochastic inputs (Ghanbarpour et al., 27 Aug 2025, Ganesh et al., 2023).
• Analysis of Systems with Discontinuous Right-Hand Side: Including control with dry friction, relay systems, sliding modes, and measure-driven systems (Zubelevich, 2020).

7. Limitations and Open Challenges

While differential-inclusion arguments are powerful, they have intrinsic limitations:

• Nonuniqueness and Regularity: For nontrivial inclusions (especially non-elliptic or with highly oscillatory/microstructured right-hand sides), uniqueness and regularity cannot be expected. Rectifying or quantifying the extent of nonuniqueness is an open direction.
• Numerical Analysis: Efficient, high-order, and sound over-approximation of reachable sets (especially in high dimensions or nonsmooth regimes) remains computationally demanding.
• Selection Principles: Existence of measurable, regular selections from set-valued maps (e.g., with finite $p$-variation) is guaranteed in specific settings, but explicit construction may be complex (Bailleul et al., 2018).
• Extension to Infinite Dimensions and Nonautonomous Systems: Many arguments hinge on Hilbert or Banach space settings, and additional care is required for time-dependent, measure-driven, or infinite-dimensional inclusions (Trostorff, 2012, Azzam-Laouir, 2021).

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A differential-inclusion argument provides a systematic, variational, and geometric framework for analyzing the behavior, stability, and computational approximation of systems with uncertainty, discontinuity, or multivalued dynamics, with rigorous technical control via set-valued mappings, structure-based regularity/rigidity, and algorithmic error quantification. The framework unifies purely analytical, variational, and computational methodologies across several domains of contemporary applied mathematics.