LEVI Systems: Linear Dynamics & Convex Constraints

• LEVI systems are linear evolution models defined by differential equations constrained within convex sets, capturing complex dynamic phenomena.
• Under natural monotonicity and passivity conditions, LEVI systems guarantee unique, stable solutions validated by Lyapunov and LMI criteria.
• Applications span frictional mechanics, power converters, and gradient constraints, effectively linking convex analysis with control theory.

A Linear Evolution Variational Inequality (LEVI) system is a mathematical framework for modeling the dynamics of state variables constrained to evolve within convex sets, where the evolution combines linear differential equations with variational inequalities. LEVI systems serve as foundational models for a wide class of dynamic phenomena in systems with state constraints, switching, frictional or contact mechanics, and control under polyhedral or cone constraints. Their study draws on monotone operator theory, convex analysis, control theory, and recent advances in cone-stable dynamics and Lyapunov stability on convex cones.

1. Definition and Core Formulation

LEVI systems arise as a specialization of evolution variational inequalities (EVIs). Classical EVIs are of the form

$$\dot x(t) + A x(t) + \partial\Phi(x(t)) \ni F(x(t)),\qquad x(0)=x_0 \in K,\ x(t)\in K\ \forall t\ge0,$$

where $A$ is a linear operator on $\mathbb{R}^n$, $K$ is a closed convex set, and $\partial\Phi$ encodes the state constraint (typically $\Phi=I_K$, the indicator function, so that $\partial\Phi=N_K$, the normal cone to $K$) (Dey, 24 Dec 2025).

The LEVI case is characterized by: $$\dot x(t) + A x(t) \in -N_K(x(t)),\qquad x(0)=x_0 \in K,\ x(t)\in K,$$ or in variational inequality form, for a.e. $t$ and all $v\in K$: $\langle \dot x(t)+A x(t),\ v-x(t)\rangle \ge 0.$ This framework extends to formulations involving ODEs coupled to VIs on auxiliary variables: $$\begin{aligned} &\dot x(t) = f(t,x(t)) + G\,\eta(t),\ &v(t)=H\,x(t)+J\,\eta(t),\ v(t)\in S(t),\ &\langle v'-v(t),\ \eta(t)\rangle \ge 0\quad\forall v'\in S(t), \end{aligned}$$ with $S(\cdot)$ a time-varying, nonempty, closed convex set. The linear specialization (“LEVI” in the sense of (Tanwani et al., 2016)) is given by linear $f$, $S(t)$ a polyhedral or cone constraint, and all operators $A, G, H, J$ constant.

2. Well-Posedness: Existence and Uniqueness

Under natural regularity and monotonicity assumptions, LEVI systems admit unique solutions. Critical assumptions include: positive semidefiniteness and passivity properties of system matrices; invariance and regularity (absolute continuity in Hausdorff distance) of moving constraint sets; and constraint qualification conditions ensuring the compatibility of the linear operators and the constraint geometry (Tanwani et al., 2016, Adly et al., 2018). The main results can be summarized as:

• Well-posedness holds for any initial condition in $K$, provided the evolution set $S(t)$ varies absolutely continuously and the matrix pencils associated to the system (e.g., the LMI in Theorem 4.1 of (Tanwani et al., 2016)) meet appropriate passivity or monotonicity conditions.
• For the abstract Hilbert-space model $-\dot u(t) \in N_{C(t)}(A\dot u(t) + Bu(t))$, existence and uniqueness is implied if $A$ is coercive and symmetric, $B$ is positive semidefinite, and $C(t)$ varies absolutely continuously (Adly et al., 2018).
• Regularity of solutions follows the regularity of the data: absolutely continuous constraint trajectories yield locally absolutely continuous solutions, while bounded variation gives rise to solutions of bounded variation with computable jump rules.

3. Monotone Operator and Lyapunov Stability Criteria

A central advance in recent work is the development of Lyapunov criteria for LEVI systems using cone-stable quadratic forms, inspired by the theory of $\mathcal{K}$-Lorentzian polynomials (Dey, 24 Dec 2025). A real symmetric matrix $A$ is $K$-copositive if $x^\top A x \ge 0$ for all $x \in K$, and strictly $K$-copositive if the inequality is strict for all nonzero $x\in K$.

• If $q(x) = x^\top A x$ is (strictly) $K$-Lorentzian, then $A$ is (strictly) $K$-copositive, and $V(x)=q(x)$ serves as a (strict) Lyapunov function for the LEVI system.
• Under these conditions, solutions with $x(0)\in K$ remain bounded and converge asymptotically to the origin when $A$ is strictly $K$-copositive.
• The equivalence between quadratic Lorentzian property and Rayleigh monotonicity ($M_q(x)\succeq 0$ on $K$) provides a bridge from convex-geometric analysis to the global dynamics of cone-constrained systems.

These developments connect variational stability, negative dependence, and cone-restricted log-concavity, allowing unified treatment of stability and robustness in evolution systems constrained by convex cones or polyhedra (Dey, 24 Dec 2025).

4. Control Design and Regulation under Constraints

Output regulation for LEVI systems exploits the passivity structure and is achieved by feedback interconnections satisfying regulator equations. For the plant

$$\dot x = A x + B u + F x_r + G \eta,\quad v = H x + J \eta,\ \eta \in -N_{S(t)}(v),$$

and an exosystem (reference generator) of analogous LEVI form, regulation is addressed by two main approaches (Tanwani et al., 2016):

• Static Full-State Feedback: Design $u(t) = K x(t) + (M - K\Pi)x_r(t)$ with $(K, M, \Pi)$ solving the Francis–Wonham-type regulator equations under the constraint that the closed-loop system is strictly passive (verifiable by an LMI).
• Dynamic Error Feedback (Observer-Based): Construct an EVI-structured observer for the unmeasured state and implement dynamic error feedback using a compensator whose parameters are chosen to maintain strict passivity.
• Both cases exploit the separation principle when the relevant LMIs (for plant and observer loops) are feasible. Output regulation (error convergence to zero) is achieved while ensuring admissibility and invariance relative to the constraint set.

This methodology enables robust control of nonsmooth or switching systems, including power converters with diodes/switches, and systems with moving polyhedral constraints (Tanwani et al., 2016).

5. Connections to Sweeping Processes and Quasistatic Evolution

LEVI systems subsume and generalize Moreau's sweeping processes. The implicit sweeping process with velocity constraint

$$-\dot u(t) \in N_{C(t)}(A \dot u(t) + B u(t)),\qquad u(0) \in C(0),$$

is rigorously equivalent to a quasistatic evolution variational inequality, after suitable variable transformations (Adly et al., 2018). Applications include frictional or unilateral contact mechanics, where the moving set $C(t)$ encodes kinematic or frictional limits.

Discretization schemes such as the catching-up algorithm (implicit Euler) inherit convergence from monotonicity and coercivity, providing a natural numerical framework for time-discrete simulation of LEVI flows. The precise analytic connection between the sweeping and variational inequality formulations enables transfer of well-posedness, stability, and numerical analysis across both perspectives (Adly et al., 2018).

6. Generalizations: Evolutionary Quasi-Variational Inequalities

The LEVI paradigm extends to problems with dynamic or state-dependent constraints, described as evolutionary quasi-variational inequalities (QVI). The general abstract formulation encompasses systems of the form

$$u(t) \in K_{G[u]}(t),\quad \langle u'(t), v-u(t) \rangle + \langle A(t,u(t)), v-u(t) \rangle \ge 0,\ \forall v \in K_{G[u]}(t),$$

where the constraint set $K_{G[u]}(t)$ depends functionally or nonlocally on the current state. Existence of weak QVI solutions is obtained via a double penalization/regularization scheme, leveraging monotonicity but without requiring strong coercivity (Miranda et al., 2018). For time-dependent but state-independent convex sets, uniqueness and continuous dependence hold for strong solutions. Examples include gradient-constrained diffusion, sandpile evolution, thick-fluid models, and problems with nonlocal or PDE-coupled constraints.

7. Examples and Applications

LEVI systems and their extensions have been successfully applied to:

• Polyhedral and cone-constrained regulation: Enforcement of state constraints in the positive orthant or with general convex cones, for viability and tracking under hard bounds (Tanwani et al., 2016, Dey, 24 Dec 2025).
• Frictional/contact mechanics: Modeling of stick–slip laws and constraints in contact dynamics, using the LEVI/sweeping process correspondence (Adly et al., 2018).
• Power-converter control: Dynamic regulation in circuits with nonsmooth elements (diodes, switches) formulated as LEVI problems with complementarity constraints (Tanwani et al., 2016).
• Gradient and Laplacian constraints: Evolution of systems under pointwise gradient or derivative bounds, including sandpile dynamics and superconductivity models (Miranda et al., 2018).

Notably, the interplay of quadratic K–Lorentzian Lyapunov functions and cone-copositivity allows for stabilizing otherwise unstable linear flows purely by constraint geometry (e.g., (Dey, 24 Dec 2025), Goeleven–Brogliato examples).

---

LEVI systems represent a rigorous and rich framework for evolution dynamics under linear flow and convex constraints, with well-established existence, uniqueness, Lyapunov theory, and control design methods. The ongoing synthesis of operator-theoretic, convex-analytical, and combinatorial-geometric perspectives continues to broaden the reach and applicability of this paradigm in mathematical systems theory and engineering (Tanwani et al., 2016, Dey, 24 Dec 2025, Adly et al., 2018, Miranda et al., 2018).