ARVIS: M1: Formal System Definition

Objective

This document provides a fully explicit mathematical definition of the ARVIS core system.
It defines: - the exact state space - the system dynamics - the switching structure - the disturbance model - the output structure

This definition is the reference model for all subsequent proofs.

1. Notation

$t \in \mathbb{N}$: discrete time
$\|\cdot\|$: Euclidean norm unless specified
$\mathcal{Q}$: finite set of modes

Time Semantics

Each time index $t$ corresponds to a logical system transition.

A transition $(t \to t+1)$ is defined as a fully completed system update, regardless of the internal execution mechanism.

2. State Space

We define the global state:

$$ s_t = (x_t, z_t, q_t) $$

with:

2.1 Fast State

$$ x_t \in \mathbb{R}^n $$

Represents: - risk signals - stability indicators - instantaneous cognitive metrics

2.2 Slow State

$$ z_t \in \mathbb{R}^m $$

Represents: - memory accumulation - long-term adaptation - contextual integration

2.3 Switching State

$$ q_t \in \mathcal{Q} $$

finite set of modes
each mode corresponds to a regime of operation

3. Disturbance Model

We define:

$$ w_t \in \mathcal{W} $$

where: - $\mathcal{W} \subset \mathbb{R}^d$ is bounded - $\|w_t\| \leq W_{\max}$

Disturbances represent: - input noise - model uncertainty - cognitive perturbations (approximate)

4. System Dynamics

The following equations define the abstract state transition between two consecutive logical time steps.

Each transition $(t \to t+1)$ is assumed to be atomic at the model level, even if its computation is performed incrementally in an implementation.

4.1 Fast Dynamics

$$ x_{t+1} = f_{q_t}(x_t, z_t, w_t) $$

where:

$$ f_q : \mathbb{R}^n \times \mathbb{R}^m \times \mathcal{W} \to \mathbb{R}^n $$

4.2 Slow Dynamics

$$ z_{t+1} = z_t + \eta \, g_{q_t}(x_t, z_t, w_t) $$

where: - $\eta > 0$ is a small parameter (time-scale separation) - $g_q : \mathbb{R}^n \times \mathbb{R}^m \times \mathcal{W} \to \mathbb{R}^m$

4.3 Switching Dynamics

$$ q_{t+1} = \sigma(q_t, \xi_t) $$

where: - $\sigma : \mathcal{Q} \times \Xi \to \mathcal{Q}$ - $\xi_t$ : switching trigger signal

5. Switching Constraints

5.1 Average Dwell-Time

There exist constants: - $\tau_d > 0$ - $N_0 \geq 0$

such that for any interval:

$$ N_\sigma(t_0, t) \leq N_0 + \frac{t - t_0}{\tau_d} $$

5.2 Jump Condition

There exists $J \geq 1$ such that:

$$ W_{q'}(x, z) \leq J \cdot W_q(x, z) $$

for all admissible mode transitions $q \to q'$

6. Target Map

For each mode $q$, define:

$$ T_q : \mathbb{R}^n \to \mathbb{R}^m $$

Interpretation: - desired slow state given fast state

7. Lyapunov Functions

7.1 Fast Lyapunov

For each $q$:

$$ V_q : \mathbb{R}^n \to \mathbb{R}_{\geq 0} $$

7.2 Composite Lyapunov

$$ W_q(x, z) = V_q(x) + \lambda |z - T_q(x)|^2 $$

with $\lambda > 0$

8. Output Structure

We define observable output:

$$ y_t = h(x_t, z_t) $$

where:

$$ h : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p $$

This represents: - runtime signals - monitoring outputs

9. Initial Conditions

$$ x_0 \in \mathbb{R}^n, \quad z_0 \in \mathbb{R}^m, \quad q_0 \in \mathcal{Q} $$

10. Well-Posedness Requirements

The system is well-defined if: 1. $f_q$, $g_q$ are measurable 2. solutions exist for all $t$ 3. trajectories remain bounded under assumptions

Additionally, it is assumed that each transition $(t \to t+1)$ corresponds to a fully evaluated system update, ensuring consistency between the mathematical model and any staged or incremental execution.

11. Separation of Time Scales

We assume:

$$ 0 < \eta \ll 1 $$

ensuring: - fast dynamics dominate locally - slow dynamics evolve gradually

12. Summary

The ARVIS core system is: - a discrete-time switched nonlinear system - with two time scales - subject to bounded disturbances - governed by a mode-dependent Lyapunov function

13. Next Step

The next document defines: - regularity assumptions - contraction properties - bounds on disturbances - Lipschitz conditions

→ see M1_assumptions.md