Moreau's Sweeping Process Overview

• Moreau's sweeping process is a differential inclusion that constrains trajectories within a moving set through normal cone reactions.
• It exhibits dissipative and rate-independent behavior, ensuring robustness in phenomena such as plasticity and crowd motion.
• Modern variants extend the framework to prox-regular, stochastic, and controlled settings, enhancing numerical methods and practical implementations.

Moreau’s sweeping process is a first-order differential inclusion for a trajectory constrained by a moving set and driven by a normal-cone reaction. In its classical form, for a moving closed set $C(t)$, it reads

$-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$

and in perturbed form one writes

$$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$

Introduced by J.J. Moreau in the early 70's with motivation in plasticity theory, it is a prototypical example of a rate-independent evolution driven by unilateral constraints, and the modern theory includes convex, prox-regular, polyhedral, tame, stochastic, controlled, state-dependent, and manifold-valued variants (Bouach et al., 2021, Colombo et al., 2015, Garrido et al., 8 Jan 2025, Bernicot et al., 2015).

1. Classical formulation and geometric meaning

For a closed convex set $C\subset\mathbb R^n$, the normal cone at $x\in C$ is

$$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$

and $N_C(x)=\emptyset$ otherwise. In prox-regular settings, the normal cone is defined through projection geometry: for example,

$$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$

while under $r$-prox-regularity the Clarke normal cone and the proximal normal cone coincide. These formulations are equivalent ways to encode the fact that the trajectory is forced to remain in the moving constraint set and, at contact, can move only in directions compatible with the negative normal cone (Colombo et al., 2015, Bernicot et al., 2010, Bouach et al., 2021).

Geometrically, the condition says that the trajectory is “pushed” by the moving set from the interior toward its boundary. In the moving half-space example

$C(t)=\{x\mid \langle u(t),x\rangle\le b(t)\},$

the normal cone is $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$0 in the interior and $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$1 on the boundary. Thus the state is “stuck” while it remains strictly interior, and once the boundary overtakes it, it “sticks and slides” along it with velocity proportional to the outward normal. This is the archetypal “play operator” or “Moreau sweeping along a hyperplane” (Colombo et al., 2015).

Two qualitative properties recur throughout the theory. First, the process is dissipative: since $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$2, one shows for any fixed $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$3 that the function $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$4 is nonincreasing while $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$5. Second, it is rate independent: reparameterizing time by any increasing homeomorphism does not change the geometry of trajectories, a hallmark of hysteresis-type models (Colombo et al., 2015).

2. Well-posedness regimes and structural assumptions

The classical existence-and-uniqueness theory assumes that $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$6 is Lipschitz continuous in the Hausdorff metric and that the values $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$7 are nonempty, closed, and convex. Under these hypotheses one obtains a unique Lipschitz trajectory solving the Cauchy problem. This remains the reference framework for the standard catching-up construction and for many Hilbert-space extensions (Colombo et al., 2015, Adly et al., 2019).

A particularly important finite-dimensional regime is the moving polyhedral case

$-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$8

with $-\dot x(t)\in N_{C(t)}(x(t)),\qquad x(0)\in C(0),$9 and $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$0 absolutely continuous. Here the Linear Independence Constraint Qualification requires that at every active face the corresponding normals remain linearly independent. Under this LICQ hypothesis, and assuming the polyhedron is nonempty for each $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$1, there exists a unique absolutely continuous solution. The example in $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$2 with three active normals becoming linearly dependent shows that LICQ is essential: the set remains nonempty, but no absolutely continuous solution exists because the projection $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$3 has a jump at $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$4 (Colombo et al., 2015).

Beyond convexity, the modern theory replaces convexity by prox-regularity or related geometric regularity. For uniformly prox-regular moving sets, existence and uniqueness hold under Hausdorff-Lipschitz motion in Hilbert spaces, on Riemannian Hilbert manifolds, and for perturbed problems with Lipschitz fields. More recent work weakens the continuity requirements on the moving set. For excess-continuous $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$5-prox-regular sets, assuming a uniform interior cone condition, one obtains a unique solution $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$6, thereby replacing symmetric Hausdorff continuity by one-sided continuity with respect to the excess $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$7 and dropping continuity of $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$8. For locally bounded retraction, if each $$\dot x(t)\in -\,N\bigl(C(t);x(t)\bigr)+F(t,x(t)) \quad\text{or}\quad \dot q(t)+N_{Q(t)}\bigl(q(t)\bigr)\ni f(t,q(t)).$$9 is $C\subset\mathbb R^n$0-prox-regular and every jump size $C\subset\mathbb R^n$1 satisfies $C\subset\mathbb R^n$2, there exists a unique $C\subset\mathbb R^n$3 solution and the local bounded retraction case is reduced to the 1-Lipschitz case by a reparametrization technique (Bernicot et al., 2015, Recupero et al., 29 Jul 2025, Recupero, 2024).

A different line of generalization relies on tameness rather than metric regularity. If the graph of the sweeping set is definable in an o-minimal structure and the map is locally bounded, then there exist piecewise absolutely continuous solutions; if the set-valued map is inner-semicontinuous, one gets a single absolutely continuous solution on the whole interval. Moreover, any bounded definable trajectory has finite length. The same paper emphasizes a limitation: when the current state enters the definition of the moving set, the finite-length conclusion can fail dramatically (Daniilidis et al., 2014).

3. Discretization, catching-up schemes, and convergence rates

Moreau’s original numerical idea is the catching-up algorithm. On a partition $C\subset\mathbb R^n$4, one sets $C\subset\mathbb R^n$5 and iteratively projects a drifted point onto the next constraint set: $C\subset\mathbb R^n$6 followed by piecewise interpolation. Under prox-regularity or convexity of $C\subset\mathbb R^n$7 and suitable assumptions on the perturbation, this scheme converges to a solution of the sweeping inclusion and, in the prox-regular monotone setting, to the unique solution (Garrido et al., 8 Jan 2025).

For moving sets defined by finitely many smooth inequality constraints,

$C\subset\mathbb R^n$8

an implementable time-stepping scheme uses a local linearized correction set

$C\subset\mathbb R^n$9

and the prediction-correction step

$x\in C$0

For $x\in C$1 small enough, each $x\in C$2, so the iterates remain feasible. The convergence proof combines compactness, a metric qualification condition between the linearized half-spaces,

$x\in C$3

and a quadratic Taylor defect estimate. The result is the global error bound

$x\in C$4

The paper interprets the $x\in C$5 order by noting that the time-step projection defect is quadratic in the step size while the driving term is only assumed Hölder-$x\in C$6 in time (Bernicot et al., 2010).

Recent work removes the requirement of exact projections. For a closed set $x\in C$7, an $x\in C$8–$x\in C$9 approximate projection is defined by

$$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$0

where $$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$1. Replacing the exact projection by such approximate projections yields an inexact catching-up algorithm. If $$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$2 and $$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$3, convergence holds in three frameworks: prox-regular moving sets, uniformly subsmooth moving sets, and fixed ball-compact closed sets. In the prox-regular case, the full sequence converges uniformly to the unique Lipschitz solution; in the subsmooth and merely closed settings one obtains subsequential convergence to a Lipschitz solution (Garrido et al., 8 Jan 2025, Garrido et al., 2023).

A further numerical development exploits equivalence with an extended projected dynamical system. Under suitable conditions, the perturbed sweeping process with time-varying set can be reformulated as an ePDS, then as a dynamic complementarity system under LICQ. This equivalence supports a high-order time-discretization based on Finite Elements with Switch Detection, designed to avoid order reduction at active-set changes and to recover the full order of the underlying Runge–Kutta method in the continuous trajectory even across arbitrary switch patterns (Pozharskiy et al., 2024).

4. Controlled sweeping processes and optimal control

In controlled form, the sweeping dynamics is typically written

$$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$4

or, in polyhedral controlled-set models,

$$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$5

This creates optimal control problems with intrinsic state constraints of inequality and equality types, highly non-Lipschitzian and unbounded differential inclusions, and cost functionals of Bolza or Mayer type. The control may act through additive perturbations, through the motion of the set, or through both (Colombo et al., 2015, Arroud et al., 2016, Cao et al., 2017).

A central technique is the method of discrete approximations. In the convex polyhedral setting, discrete approximations converge strongly in $$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$6 to a given local minimizer, and constructive necessary optimality conditions are derived entirely in terms of the problem data. For nonconvex prox-regular sweeping sets, strong convergence of discrete optimal solutions is also established, together with measure-valued adjoint systems and complementarity multipliers. With measurable controls acting only through additive perturbations, refined discrete approximations lead to continuous-time maximum conditions and adjoint inclusions for local minimizers (Colombo et al., 2015, Cao et al., 2017, Colombo et al., 2018).

Pontryagin-type conditions have also been derived directly for controlled Moreau processes with smooth sweeping sets. In the finite-dimensional Mayer problem with compact convex control set $$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$7, a global minimizer admits an adjoint vector of bounded variation, a finite signed Radon measure, and a maximum condition. The measure term captures the effect of the state constraint encoded by the normal-cone part. In a different direction, continuous interior approximations for prox-regular sets lead to strong $$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$8 convergence of state trajectories and refined necessary conditions stated through new $$N_C(x)=\{v\in\mathbb R^n\mid \langle v,y-x\rangle\le 0 \text{ for all } y\in C\},$$9-subdifferentials that are strictly smaller than the standard Clarke and Mordukhovich subdifferentials (Arroud et al., 2016, Nour et al., 2023).

The minimum-time problem for the controlled Moreau sweeping process introduces another control-theoretic layer. Given

$N_C(x)=\emptyset$0

the minimum time function $N_C(x)=\emptyset$1 is characterized through Hamilton-Jacobi inequalities derived from weak and strong invariance of the epigraph and hypograph of $N_C(x)=\emptyset$2 under an augmented dynamics. Under a Petrov-type controllability condition, $N_C(x)=\emptyset$3 is finite and continuous, and the resulting Hamiltonians explicitly contain both the perturbation $N_C(x)=\emptyset$4 and the normal-cone term $N_C(x)=\emptyset$5 (Michele et al., 2020).

5. Variants, extensions, and reformulations

A stochastic version replaces the deterministic perturbation by drift and diffusion terms: $N_C(x)=\emptyset$6 with $N_C(x)=\emptyset$7 supported on $N_C(x)=\emptyset$8 and $N_C(x)=\emptyset$9. Under admissibility, regularity, absolute continuity of the moving set, and bounded Lipschitz coefficients $$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$0, there is a unique adapted process in $$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$1 satisfying

$$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$2

An Euler–projection scheme converges almost surely in the uniform norm (Bernicot et al., 2010).

A Volterra-type integro-differential extension studies

$$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$3

for uniformly $$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$4-prox-regular moving sets with absolutely continuous variation. Under Carathéodory, growth, and Lipschitz assumptions on $$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$5 and $$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$6, there is a unique absolutely continuous solution. The proof uses a semi-discretization method, hypomonotonicity of the prox-regular normal cone, Mazur’s lemma, and a new Gronwall-like differential inequality adapted to the Volterra term (Bouach et al., 2021).

Another family of generalizations constrains the velocity rather than the state alone. The implicit sweeping process

$$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$7

is shown to be equivalent to a quasistatic evolution variational inequality, with well-posedness obtained via an adapted catching-up algorithm. Related Hilbert-space models with velocity constraints, strongly monotone operators, and history-dependent terms provide unique weak solvability results for contact-mechanics problems involving frictionless contact, Signorini conditions, and Coulomb friction (Adly et al., 2018, Adly et al., 2019).

State-dependent and manifold-valued sweeping processes further enlarge the class. Degenerate state-dependent problems of the form

$$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$8

have been treated via Moreau–Yosida regularization under truncated Hausdorff-Lipschitz dependence in $$N_{Q(t)}(q):=\Bigl\{v\in\mathbb R^d:\exists\alpha>0\;\text{s.t.}\;q\in P_{Q(t)}\bigl(q+\alpha v\bigr)\Bigr\},$$9 and $r$0, positive $r$1-farness, equi-uniform subsmoothness, and a compactness condition. On Riemannian Hilbert manifolds, the notions of proximal normal cone and local prox-regularity admit intrinsic analogues, and under Hausdorff-Lipschitz motion and bounded Lipschitz perturbations one obtains a unique absolutely continuous solution $r$2 of

$r$3

These results show that the sweeping framework is not tied to Euclidean linear geometry (Narváez et al., 2021, Bernicot et al., 2015).

6. Applications, limitations, and recurring themes

The range of applications in the cited literature is broad. Crowd motion is a standard example: $r$4 rigid disks in the plane must satisfy the no-overlap constraint

$r$5

and the true velocity is the projection of a desired velocity onto the cone of admissible velocities. The same sweeping structure appears in nonlinear integro-differential complementarity systems, electrical circuits with ideal diodes and time-varying capacitors, quasistatic frictional contact for viscoelastic materials, and numerical optimal control examples such as a wave-rider problem and path planning with moving obstacles (Bernicot et al., 2010, Bouach et al., 2021, Adly et al., 2019, Pozharskiy et al., 2024).

Several themes cut across these applications. One is that the normal-cone term encodes the state constraint intrinsically: if $r$6, then the normal cone is empty, while in the interior the dynamics reduces to the perturbation. Another is that numerical feasibility is not automatic; this motivates projection-based and approximate-projection-based schemes. A third is that uniqueness is strongly geometry-dependent. LICQ is crucial in moving polyhedra, prox-regularity or subsmoothness governs projection behavior, and ball-compactness or compactness estimates often replace monotonicity when uniqueness is unavailable (Colombo et al., 2015, Garrido et al., 8 Jan 2025, Garrido et al., 2023).

The literature also corrects several common oversimplifications. The sweeping process is not restricted to convex sets; prox-regular, subsmooth, tame, and even state-dependent moving sets have been analyzed. Hausdorff continuity is not the only natural continuity notion: one-sided continuity with respect to the excess may suffice, and Moreau already observed that the excess provides the natural topological framework for sweeping process. At the same time, not every desirable asymptotic property survives generalization: bounded definable trajectories have finite length, but when the current state enters the set definition, finite-length conclusions may fail dramatically. These results place the sweeping process at the intersection of variational analysis, nonsmooth dynamics, complementarity theory, and numerical time-stepping, with the normal-cone geometry remaining the common dynamical core (Recupero et al., 29 Jul 2025, Daniilidis et al., 2014).