Controlled and Conditional Invariance

• Controlled and conditional invariance refer to mathematical criteria ensuring a system’s state or probability measure remains within prescribed sets through appropriate control policies or symmetry conditions.
• Analytical methods such as score-based controllers, normal cone tests, and support function verification are used to derive necessary and sufficient conditions for invariance.
• These techniques underpin practical applications in safety verification, robust control, and stochastic process reduction in fields like engineering, finance, and machine learning.

Controlled and conditional invariance properties refer to the mathematical structures and criteria underlying the ability to keep system states or probability measures within prescribed sets—possibly under system controls, uncertainties, stochastic effects, or in conditional settings (such as group actions)—and the methods for certifying, testing, or enforcing such properties. These concepts span deterministic and stochastic control theory, dynamical systems, semimartingale calculus, measure theory, stochastic processes, machine learning, statistics, and applications in engineering, finance, and the sciences.

1. Foundational Definitions and Classes of Invariance

Controlled invariance is defined for dynamical or stochastic systems as the property that, given an initial state within a prescribed set, there exists a control policy (feedback, open-loop, or Markovian, depending on context) such that the system's trajectory remains within the set for all future times, possibly with probability one or up to a stopping time. Conditional invariance may refer to symmetry properties conditional on structural group actions (as in equivariance of conditional distributions (Chiu et al., 18 Dec 2024)) or to regularity and positivity properties of measure-valued solutions, conditioned on initial data belonging to certain function spaces (Hille et al., 29 Jan 2025).

The classes of invariance include:

• Controlled Set Invariance for Diffusions: For a controlled Itô diffusion $$\mathrm{d}x_t^u = [f(t, x_t^u) + G(t, x_t^u)u(t, x_t^u)]\mathrm{d}t + \sigma(t, x_t^u)\mathrm{d}w_t$$, almost-sure invariance requires a Markovian controller such that $x_t^u$ remains in the set $\mathcal{X}$ for all $t$ with probability one (Wang et al., 30 Jul 2025).
• Conditional Invariance/Equivariance of Conditional Distributions: A conditional distribution $P_{Y|X}$ is invariant under a group $G$ if $P_{Y|X}(B|x) = P_{Y|X}(gB|gx)$ for all $g\in G$, Borel sets $B$ and almost every $x$ (Chiu et al., 18 Dec 2024).
• Invariance of Measure-Valued Evolution Equations: For semilinear transport equations, invariance refers to the preservation of positivity, absolute continuity, and $L^p$-regularity of measures under the solution operator, conditional on the corresponding properties of the initial data (Hille et al., 29 Jan 2025).

These notions generalize to hybrid (hybrid inclusions), switched, or infinite-dimensional settings, and are foundational in safety verification, viability theory, and robust control under uncertainty.

2. Analytical and Geometric Criteria for Invariance

Mathematically, controlled and conditional invariance properties are often established via differential, variational, or geometric criteria, which provide necessary and sufficient conditions:

• Score-Based Necessary and Sufficient Conditions (Diffusions): For controlled diffusions, invariance is certified by computing a score vector field $s(t,x) = \Sigma(t,x)\nabla_x\log h_T(t,x)$, where $h_T$ solves a Dirichlet boundary value problem (e.g., exit probability or principal eigenfunction for asymptotic cases). Invariance with probability one is possible if and only if $s(t,x) \in \mathrm{Range}(G(t,x))$ for all $(t,x)$; this provides a constructive test and characterizes all Markovian controllers achieving invariance (Wang et al., 30 Jul 2025).
• First-Order Normal Cone and Viability: For controlled piecewise deterministic Markov processes (PDMPs), viability and invariance of a closed set are characterized via the first-order normal cone to the set using viscosity solutions to Hamilton–Jacobi integro-differential equations (HJIDEs). Necessary and/or sufficient conditions are obtained by testing the Hamiltonian in directions of the normal cone; the resulting geometric–analytic bridge is rooted in nonlinear analysis (Goreac, 2010).
• Sub-Tangentiality and Polyhedral Verification: In linear uncertain systems, especially for positive $\mathcal{D}$-invariance, it suffices to verify that the closed-loop velocity at each vertex of the constraint set does not point outside—formalized as $F(q)v + E(q)\delta \in T_\mathcal{S}(v)$ for all relevant $v$, $q$, and $\delta$, where $T_\mathcal{S}(v)$ is the tangent cone (Morbidi et al., 2010).
• Support Functions and Geometric Control: For continuous-time algebraic systems with convex invariant candidates, invariance conditions reduce to algebraic inequalities involving the support function $\sigma_C(z)$, namely $\langle \nabla \sigma_C(z), f(x, u)\rangle \leq 0$ for $x \in \partial C$ (Legat et al., 2021).

3. Transfer and Reduction Properties in Stochastic Settings

In filtered probability spaces, controlled and conditional invariance can enable the transfer of structural properties between different filtrations and measure spaces:

• Invariance Times and the Semimartingale Dictionary: An invariance time $\tau$ is a stopping time with a compensator such that local martingales in a reduced filtration $(\mathbb{F},\mathbb{P})$, stopped before $\tau$, remain local martingales in an enlarged filtration $(\mathbb{G},\mathbb{Q})$. This yields a comprehensive "dictionary" by which conditional expectations, martingale representation properties, stochastic integrals, semimartingale characteristics, Markov properties, transition semigroups, and infinitesimal generators can be coherently translated and reduced between $(\mathbb{G},\mathbb{Q})$ and $(\mathbb{F},\mathbb{P})$, a fact that is central in credit risk modeling, counterparty risk, and related applications (Crépey, 22 Jul 2024).
• Conditional Invariance in Randomization Tests: In the context of hypothesis testing for symmetry of conditional distributions under group actions, randomization tests can exploit conditional invariance to provide finite-sample Type I error guarantees and non-asymptotic power bounds, using MMD statistics and Monte Carlo resampling exploiting group transformations (Chiu et al., 18 Dec 2024).

4. Constructive Synthesis and Testing Methods

The analysis of controlled and conditional invariance leads naturally to constructive methods for controller synthesis and statistical testing:

• Score-Based Controller Computation: In score-based criteria, the controller is obtained by solving $G(t, x) u(t, x) = s(t, x)$; the set of all such controllers is fully characterized when $s$ lies in the range of $G$ everywhere (Wang et al., 30 Jul 2025).
• Partition-Based and Interval Analysis Controllers: For switched nonlinear systems, interval analysis provides inner and outer approximations of maximal controlled invariant sets; a partition-based invariance controller is then synthesized by recording admissible switching actions on subregions during the computation (Li et al., 2016).
• Robust Control Lyapunov Functions and Regulation Maps: In hybrid dynamical systems, a robust control Lyapunov function (RCLF) enables the construction of state feedback laws guaranteeing forward invariance by making all directional derivatives (flows) and post-jump values (jumps) nonincreasing within the level set (Chai et al., 2020).
• Randomization and Kernel Methods for Testing Conditional Symmetry: For conditional group invariance testing, randomization (e.g., using group orbits and inversion kernels) combined with kernel MMD statistics yields exact or asymptotically valid p-values, and power analysis can be explicitly related to the MMD gap (Chiu et al., 18 Dec 2024).
• Data-Based and Model-Free Controller Design: Set invariance and input constraints can be guaranteed using data-based linear program synthesis in discrete-time systems, requiring only persistently exciting open-loop experiment data and bypassing explicit model identification (Bisoffi et al., 2019).

5. Preservation and Propagation of Structural Properties

Many controlled and conditional invariance properties are inherently about preserving desirable structure under system evolution—positivity, regularity, or symmetry:

• Preservation of Measure Theoretic Properties: For measure-valued transport equations, the solution operator preserves positivity, absolute continuity with respect to Lebesgue, and $L^p$ density regularity under appropriate assumptions on the velocity field and reaction term; this is established via careful analysis of the mild solution representation and compactness properties (e.g., Fortet–Mourier topology) (Hille et al., 29 Jan 2025).
• Lyapunov and Weak Invariance: In hybrid and nonlinear dynamics, sublevel sets of Lyapunov or barrier functions—possibly under robust disturbance conditions—are preserved by the system, yielding forward invariance, and, with appropriate test function regularity, (almost) strong invariance (Chai et al., 2020).

6. Practical Applications and Contextual Impact

Applications of controlled and conditional invariance properties pervade diverse domains:

• Safety Verification and Control: Rigorous certification and synthesis of controllers ensure that stochastic or deterministic systems remain within safety constraints, with implications for automotive systems, robotics, cyber-physical infrastructure, and biological networks (Wang et al., 30 Jul 2025, Chai et al., 2020).
• Stochastic Process Reduction in Finance: The invariance-time framework permits reduction of BSDEs, semimartingale representation, and pricing models in credit risk to more tractable forms by leveraging stopping-times and associated filtrations (Crépey, 22 Jul 2024).
• Statistical Symmetry and Model Validation: Randomization tests for conditional group symmetry are used in high-energy physics (e.g., checking for rotational or Lorentz invariance in particle collider data) and for validating or diagnosing the invariance of learned representations in machine learning models (Chiu et al., 18 Dec 2024).
• Preservation of Physical Meaning: Guaranteeing the preservation of mass, positivity, or regularity in measure-theoretic evolutions under nonlinear or nonlocal terms is fundamental for modeling populations, cell migration, or material flows (Hille et al., 29 Jan 2025).

7. Broader Mathematical and Practical Connections

Controlled and conditional invariance properties are deeply connected to concepts such as viability theory, reachability, entropy and information constraints (notably stabilization entropy in stochastic systems under communication limitations (Kawan et al., 2019)), representation theory, optimal transport, and duality in optimization and nonlinear analysis.

These connections enable the extension of classical deterministic control and symmetry methods to stochastic, measure-valued, hybrid, and high-dimensional statistical learning settings, providing a unified mathematical lens for ensuring system integrity, robust performance, and faithful modeling in the presence of uncertainty, latent structure, or imposed symmetries.