State-Dependent Sweeping Process

• State-dependent sweeping processes are dynamical systems defined by differential inclusions with constraints that explicitly depend on the state and control variables.
• They involve nonsmooth, discontinuous, and unbounded dynamics that challenge traditional analytical methods and require specialized numerical approximations.
• These processes have practical applications in elastoplasticity, crowd dynamics, robotics, and traffic management through advanced optimal control formulations.

A state-dependent sweeping process is a dynamical system governed by a differential inclusion in which the state evolution is constrained by a moving set whose geometry depends explicitly on the current state, control actions, or both. The mathematical structure and analysis of such processes involve nonsmooth, discontinuous, and often unbounded dynamics. State-dependent sweeping processes appear in diverse areas, including elastoplasticity, crowd and traffic modeling, robotics, optimal control, and contact mechanics. Below is a detailed exposition of their key mathematical formulations, control-theoretic frameworks, analytical challenges, major optimality theories, and representative applications.

1. Mathematical Definition and Prototypical Models

A state-dependent sweeping process generalizes the classical Moreau sweeping process by allowing the moving constraint set $C(t)$ to depend on the state $x$ and/or on control functions. The evolution is described via a differential inclusion of the form

$$\dot{x}(t) \in -N_{C(t, x(t), u(t))}(x(t)) + f(x(t), u(t)),\quad x(0) \in C(0, x(0), u(0)),$$

where $N_K(x)$ denotes the normal cone (often Mordukhovich/basic normal cone in nonconvex settings) to the closed set $K$ at $x$, and $f$ is a perturbation (e.g., representing drift or control action). Canonical instances include:

• Polyhedral controlled sets: $$C(t) = \{ x \in \mathbb{R}^n \mid \langle u_i(t), x\rangle \leq b_i(t),\, i=1,\ldots,m \}$$, with controls $(u, b)$ determining the moving set geometry (Colombo et al., 2015).
• State or control-dependent translation: $C(t, x, u) = K + v(x, u)$, for some convex $K$ and Lipschitz shift $v$ (Adly et al., 2023).
• Additive state-dependent terms: $C(t, x) = A + a(t) + c(x)$ with $a(\cdot)$ of BV and $c(x)$ Lipschitz (Kamenskii et al., 2018).

The state-dependence fundamentally alters the analytical landscape, introducing quasi-variational inequalities and rendering the right-hand side non-Lipschitz, often nonmonotone, and possibly multivalued.

2. Optimal Control Formulation and Structural Features

Optimal control problems governed by state-dependent sweeping processes are formulated by minimizing a cost functional, typically of Bolza-type,

$$J[x, u] = p(x(T)) + \int_0^T \ell(t, x(t), u(t), x'(t), u'(t))\,dt,$$

subject to the sweeping process dynamics,

$$x'(t) \in -N_{C(t, x(t), u(t))}(x(t)) + f(x(t), u(t)),\quad x(0) \in C(0, x(0), u(0)),$$

and possible endpoint or integral constraints. The controls act:

• Directly, by influencing the moving set $C$ (i.e., “control of constraints” as in (Colombo et al., 2015, Hoang et al., 2018)),
• Indirectly, via perturbations to the vector field, or
• Simultaneously in both roles (Cao et al., 2015, Colombo et al., 14 May 2024).

These models entail state and mixed (state-control) constraints, often of the form $h(x(t), u(t)) \leq 0$ in addition to the implicit constraint $x(t) \in C(t, x(t), u(t))$ (Cortez et al., 2023).

3. Analytical and Numerical Challenges

State-dependent sweeping processes pose several challenges:

• The normal cone mapping is unbounded, discontinuous, and "totally non-Lipschitz" (Colombo et al., 2015). Classical existence and uniqueness results generally fail without prox-regularity and other structural assumptions.
• The evolution depends both on the current state and controls in a nonclassical way, rendering the inclusion degenerate and susceptible to nonuniqueness, especially in models with loss of monotonicity (plasticity with softening) (Gudoshnikov, 22 Aug 2025).
• The presence of mixed and nonregular constraints requires measures and even finitely additive set functions (“charges”) in place of classical Lagrange multipliers in necessary conditions (Cortez et al., 2023).
• Analysis and numerics must account for possible jumps in the moving set, demanding prescribed state-dependent jump rules (Recupero et al., 2017), and well-posed discretizations that capture switchings and discontinuities effectively (Pozharskiy et al., 15 Dec 2024).

Consequently, standard relaxation or regularization techniques are typically inadequate; specialized methodologies are necessary to guarantee the existence, uniqueness, or approximate computation of solutions.

4. Variational and Approximation Techniques

A range of advanced methodologies for rigorous analysis and computation of state-dependent sweeping processes has been developed:

• Discrete approximation: The continuous-time inclusion is approximated via difference inclusions on refined partitions, yielding piecewise linear trajectories with strong $W^{1,2}$ convergence to continuous minimizers (Colombo et al., 2015, Cao et al., 2015, Hoang et al., 2018, Henrion et al., 2021).
• Generalized differentiation: Coderivatives and generalized subdifferentials (Mordukhovich, Clarke) are used to handle the non-Lipschitz, set-valued normal cone and compute limiting optimality conditions (Colombo et al., 2015, Hoang et al., 2018).
• Moreau–Yosida and exponential penalization: Nonregular and state-dependent sets are regularized via Moreau-Yosida approximations (Narváez et al., 2021), or penalized using exponential terms to approximate sweeping dynamics by smooth ODEs (Chamoun et al., 12 Sep 2024).
• Fixed-point arguments and enhanced Gronwall lemmas: Existence and well-posedness proofs in the presence of history dependence or nonconvex sets are obtained via compactness, continuity of auxiliary maps, and a priori exponential estimates (Godoy et al., 26 Dec 2024).
• Complementarity and extended projected dynamical systems: For time-dependent or moving sets, equivalence is established with extended projected dynamical systems and dynamic complementarity systems, facilitating accurate discretization strategies (Pozharskiy et al., 15 Dec 2024).

In particular, the Banach contraction principle and Schauder’s fixed point theorem have been adapted for sweeping processes in infinite-dimensional or history-dependent contexts (Krejci et al., 2020, Godoy et al., 26 Dec 2024).

5. Necessary Optimality Conditions, Maximum Principles, and Structure of Solutions

Necessary optimality conditions for state-dependent sweeping processes have been derived under both convex and nonconvex structures. Key features include:

• Nondegenerate Euler–Lagrange and Maximum Principle conditions: Adjoints, multipliers (including BV functions and Radon measures), and transversality conditions encapsulate both the dynamic constraints and state-dependent phenomena (Colombo et al., 2015, Hoang et al., 2018, Cortez et al., 2023, Colombo et al., 14 May 2024).
• Extended Hamiltonian formulations: The Pontryagin Maximum Principle is modified to accommodate the measure-valued (and possibly charge-valued) nature of multipliers arising from sweeping and mixed constraints (Hoang et al., 2018, Arroud et al., 2016, Cortez et al., 2023).
• Complementary slackness, nontriviality, and stationarity: Conditions ensure that multipliers vanish unless the corresponding state or control constraint is active, and that the system does not degenerate to triviality (Colombo et al., 2015, Colombo et al., 14 May 2024).
• Discrete-time optimality conditions: First-order stationarity for discrete approximations employs primal-dual recursions involving coderivatives, with limit passage securing necessary conditions for continuous systems (Colombo et al., 2015, Henrion et al., 2021).

Nonuniqueness arises in degenerate and nonmonotone cases (notably plasticity with softening), with fixed-point iterations displaying bifurcation phenomena and the emergence of shear bands (Gudoshnikov, 22 Aug 2025).

6. Applications and Representative Examples

State-dependent sweeping processes underpin a range of real-world and engineered systems:

• Elasto-plasticity and hysteresis: Evolution of plastic strains, yield conditions, and hardening/softening behaviors are modeled via sweeping sets determined by stress or strain constraints (Colombo et al., 2015, Hoang et al., 2018, Gudoshnikov, 22 Aug 2025).
• Crowd motion and robotics: Nonoverlapping constraints for colliding agents or robots manifest naturally as polyhedral state-dependent constraints, with optimal trajectories and contact times computable from necessary conditions (Cao et al., 2015, Colombo et al., 2018, Colombo et al., 14 May 2024).
• Traffic flow: Vehicle or pedestrian dynamics with density and position constraints are managed using control over moving boundaries (Cao et al., 2015, Colombo et al., 2018).
• Rate-independent and memory effects: History-dependent perturbations produce evolution with long memory, prevalent in viscoelasticity, and resolved via fixed-point frameworks (Godoy et al., 26 Dec 2024, Adly et al., 2019).

Certain works offer explicit algorithms and numerical methods, such as the implicit catching-up scheme, finite element discretizations with switch detection, and implementations in libraries such as GEKKO (Kamenskii et al., 2018, Colombo et al., 14 May 2024, Pozharskiy et al., 15 Dec 2024).

7. Theoretical and Algorithmic Advancements

Recent research has advanced both the theoretical and computational understanding of state-dependent sweeping processes.

• Convergence theory: Strong $W^{1,2}$ and $BV$ convergence of discrete approximations reliably passes optimality conditions from discretized to continuous settings (Colombo et al., 2015, Hoang et al., 2018, Henrion et al., 2021).
• Algorithmic developments: Linear-rate, derivative-free methods for the associated quasi-variational inequalities, robust to nonsmooth and nonconvex data, have been established (Adly et al., 2023).
• Sensitivity and stability results: Quantitative estimates describe how solutions react to variations in controls or set parameters, crucial for robust control and practical design (Henrion et al., 2021).
• Handling nonuniqueness: The interplay between iterative/numerical stability (in implicit catch-up maps) and physical (energetic or mechanical) stability hints at a deeper variational characterization of solution robustness and selection in nonmonotone regimes (Gudoshnikov, 22 Aug 2025).

A plausible implication is that the selection of stable fixed points via state-dependent sweeping process iterations aligns with local minima of corresponding energy functionals, at least in elastoplastic lattice models.

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State-dependent sweeping processes thus represent a rich mathematical framework capturing asymmetric, nonmonotone, and history-dependent evolution in systems with state and control-dependent constraints. Recent developments in variational analysis, optimal control theory, numerical algorithms, and applications to materials science and engineering have significantly expanded both theoretical underpinnings and computational methodologies for these challenging dynamical systems.