Play Hysteresis Operator in Rate-Independent Systems

Updated 31 January 2026

The play hysteresis operator is a rate-independent memory operator that tracks input extrema via variational inequalities, ensuring robust stability in dynamic systems.
It exhibits key properties like boundedness, Lipschitz continuity, and a semigroup property, which facilitate accurate modeling of stick-slip behavior and hysteresis loops.
The operator is integral to modeling physical, engineering, and biological systems, enabling effective numerical approximations in ODEs, PDEs, and hybrid system frameworks.

The play hysteresis operator is a canonical rate-independent memory operator central to the mathematical modeling of systems where output lags and tracks the input in a manner governed only by the sequence of input extrema and not their rate of change. Originally introduced in the context of elasto-plasticity and magnetic hysteresis, the play operator and its generalizations underpin a wide spectrum of nonlinear, nonsmooth dynamical behaviors in ODEs, PDEs, and hybrid systems. Its modern theoretical formulation utilizes variational inequalities and differential inclusions, which capture the core idea of constrained evolution within moving sectors or strips determined by the input signal.

1. Mathematical Definition and Variational Formulation

The scalar play operator $w=P_\rho[u,w_0]$ for amplitude (or "gap") parameter $\rho>0$ is defined on an input $u:[0,T]\to\mathbb{R}$ and initial state $w_0$ as the unique mapping satisfying:

Sector constraint: $|u(t)-w(t)| \leq \rho$ for all $t$ .
Evolution law (variational inequality):

$\dot w(t) \cdot (u(t) - w(t) - v) \geq 0, \quad \forall v\in[-\rho, \rho],$

for almost every $t$ , with $w(0)=w_0$ .

Equivalent representations include the min–max formula and the differential inclusion:

$|\dot w(t)| \neq 0 \implies w(t) = u(t) \pm \rho, \qquad \dot w(t) = \dot u(t)\ \text{if}\ u(t) - w(t) = \pm\rho\ \text{and}\ \pm\dot u(t) \geq 0; \qquad \dot w(t)=0\ \text{otherwise}$

(Goatin et al., 24 Jan 2026, Bagagiolo et al., 2020, Bagagiolo et al., 2024).

The generalized play operator $S = P[f; L, U](t)$ extends this notion to input $f(t)$ with moving Lipschitz boundaries $L(f(t)), U(f(t))$ . It enforces $S(t) \in [L(f(t)), U(f(t))]$ and the evolution constraint as a sweeping process (differential inclusion):

$\dot S(t) \in -N_{[L(f(t)), U(f(t))]}(S(t)),$

where $N_{K}(x)$ is the normal cone at $x\in K$ (Fellner et al., 2018, Kuehn et al., 2017).

These variational or inclusion-based formulations encode precisely the classical stick-slip (or “sticking–sliding”) mechanism and guarantee uniqueness, causality, and robust stability properties.

2. Fundamental Properties

The play hysteresis operator exhibits a collection of fundamental features:

Rate-independence: The output path is invariant under time reparameterizations: for any strictly increasing bijection $\varphi:[0,T]\rightarrow[0,T]$ , $P_\rho[u\circ\varphi,w_0]=P_\rho[u,w_0]\circ\varphi$ (Goatin et al., 24 Jan 2026, Bagagiolo et al., 2020, Bagagiolo et al., 2024).
Memory (hysteresis): The output $w(t)$ depends on the full past of $u$ up to time $t$ ; the future output is independent of the future extension of $u$ .
Boundedness: $w(t)\in[u(t)-\rho,u(t)+\rho]$ .
Lipschitz continuity: The mapping $u\mapsto w$ is Lipschitz continuous from $C([0, T])\to C([0,T])$ (Goatin et al., 24 Jan 2026, Fellner et al., 2018).
Monotonicity: If $u_1 \le u_2$ and $w_{0,1}\le w_{0,2}$ , then $P_\rho[u_1,w_{0,1}] \le P_\rho[u_2,w_{0,2}]$ .
Semigroup property: For any $0\le\tau\le t\le T$ , $P_\rho[u,w_0](t) = P_\rho[u|_{[\tau,T]}, P_\rho[u,w_0](\tau)](t-\tau)$ (Bagagiolo et al., 2020).
Sector constraint and hysteresis loops: For cyclic $u(t)$ , the $(u,w)$ -trajectory forms a nontrivial loop in the $(u,w)$ -plane, reflecting irreversible “energy dissipation”.

These properties establish the mathematical backbone for the use of the play operator in physical, engineering, and biological models (Kuehn et al., 2017, Fellner et al., 2018).

3. Structural Role in Dynamical and PDE Systems

When coupled with ODEs or PDEs, the play operator models rate-independent memory effects, threshold phenomena, or internal friction:

Fast–slow reduction: In singular perturbation limits of systems with a fast variable constrained to an evolving sector $[F_-(y), F_+(y)]$ , the fast-slow system

$\varepsilon \dot x = f(x,y), \quad \dot y = g(x, y, t), \qquad \varepsilon\to 0$

reduces to $x(t)\in[F_-(y(t)),F_+(y(t))]$ with rate-independent play-law evolution (Kuehn et al., 2017, Fellner et al., 2018).

PDE coupling: In coupled PDE–ODE models, e.g., reaction–diffusion with fast buffer/stock dynamics, the fast ODE collapses to a generalized play law for the slow variable, producing nontrivial spatially inhomogeneous patterns via hysteresis–diffusion-driven instabilities (Fellner et al., 2018).

Example: In population–resource PDE–ODE models,

$\varepsilon \dot S = F(t) - N(t) c(S, N, F), \quad \text{as}\ \varepsilon\rightarrow 0,$

the stock $S(t)$ converges to the solution of a generalized play operator tied to resource consumption thresholds (Fellner et al., 2018).

In conservation laws, embedding $w = P_\rho[u]$ as a memory term in models such as

$\partial_t u + \partial_t w + \partial_x f(u) = 0$

produces entropy-admissible, memory-driven solution behavior, and modifies Riemann problem structure and admissible shock speeds (Goatin et al., 24 Jan 2026, Bagagiolo et al., 2024).

4. Discrete and Numerical Realizations

Algorithmic construction of the play operator leverages explicit projection schemes and implicit variational steps:

Time stepping: Backward Euler for the differential inclusion yields at each step $V^n = \min\{\max\{V^{n-1}, U^n-\rho\}, U^n+\rho\}$ (Peszynska et al., 2020).
Finite volume schemes: In Godunov-type PDE discretizations, the splitting

$u^{n+1}_i + w^{n+1}_i = u^n_i + w^n_i - \frac{\Delta t}{\Delta x} (f_{i+1/2}^n - f_{i-1/2}^n)$

is used with interface fluxes to respect the intrinsic memory and shock admissibility induced by the play constraint (Goatin et al., 24 Jan 2026).

Preisach and generalized play models: Superpositions of elementary relays or “hysterons” approximate arbitrary play or generalized play operators and admit efficient calibration from data (Peszynska et al., 2020). Discrete nonlinear play models combine ramp/hysteron functions and time-grid evolution, recovering rich primary and secondary loop structures.

For vector-valued or energy-based extensions (e.g., magnetic hysteresis), convex optimization problems with nonsmooth energy and pinning terms are solved iteratively, respecting play-like incremental update rules (Egger et al., 2024).

5. Control and Controllability of Hysteretic Systems

The play operator’s presence in control systems introduces nontrivial delays and sticking behavior in actuated responses (Bagagiolo et al., 2020):

Controllability preservation: If a driftless affine system is controllable in the absence of hysteresis, then approximate controllability can be preserved under play-operator-induced hysteresis, via suitable approximation of piecewise constant controls by continuous inputs processed through play (Bagagiolo et al., 2020).
Semigroup and Lipschitz properties: These are instrumental in establishing boundedness and convergence of approximants in control trajectory spaces.
Approximation technique: Piecewise-constant desired controls are “smoothed” into admissible controls that, when passed through the play operator, converge in $L^1$ to the desired outputs, leveraging the operator’s sector and projection properties.

A direct corollary is the ability to design robust feedback and open-loop inputs in networks or mechanical systems with known memory-induced inertia.

6. Generalizations and Physical Modeling

The play operator extends naturally:

Generalized/boundary play: Replace constant thresholds by arbitrary Lipschitz upper and lower curves $L(y),U(y)$ , allowing path-dependent dependence on the input (Fellner et al., 2018, Kuehn et al., 2017).
Extended play for porous media: To model capillarity-driven hysteresis with nonvertical scanning curves, extended play operators alter the classical law $p\in[P_{\mathrm{im}}(S),P_{\mathrm{dr}}(S)]$ to allow physically realistic spreading during dynamic scans, with the ODE/PDE system remaining well-posed via regularization (Mitra, 2020).
Vector/energy-consistent play: In magnetics, convex-analytic operator representations yield play-like vectorial hysteresis via subdifferential inclusions tied to energy/pinning functionals (Egger et al., 2024).

Such generalizations capture complex rate-independent effects in adsorption–desorption, elasto-plasticity, ferromagnetics, and unsaturated porous flow, ensuring both physical soundness (via maximum principles and sector bounds) and mathematical well-posedness.

7. Existence, Uniqueness, and Regularity in Applications

Well-posedness of ODE/PDE systems incorporating play operators crucially relies on:

Lipschitz regularity: Ensures existence and uniqueness in $W^{1,\infty}$ or $BV$ spaces for the play variable, and permits coupling to PDEs using standard techniques (semigroup theory, Rothe method) (Fellner et al., 2018, Goatin et al., 24 Jan 2026, Bagagiolo et al., 2024).
Entropy-admissibility: In conservation laws, entropy inequalities for $(u,w)$ guarantee uniqueness and $L^1$ contraction (Bagagiolo et al., 2024, Goatin et al., 24 Jan 2026).
Strong maximum principles: For extended play in porous media, maximum principles confine physically relevant variables to allowable ranges, even in degenerate limits (e.g., vanishing permeability or zero diffusion) (Mitra, 2020).

Theoretical frameworks built on differential inclusions, monotone operator theory, and accretive mappings furnish the analytic foundation for subsequent well-posedness, regularity, and numerical stability in the broad class of play-augmented dynamical systems.

Relevant references: (Fellner et al., 2018, Kuehn et al., 2017, Goatin et al., 24 Jan 2026, Bagagiolo et al., 2020, Peszynska et al., 2020, Bagagiolo et al., 2024, Mitra, 2020, Egger et al., 2024)