# ⚛  L1 Principle — Optimal Control (LQR)

**ID:** `L1-432` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Control Theory — *Linear quadratic control*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** control gain matrix
> **📡 Carrier:** N/A · **🌫 Noise:** deterministic
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555257

---

## 🧠 1. Introduction

**Optimal Control (LQR)** is a **parameter-estimation problem** whose unknown lives in **control gain matrix** space, within the **Linear quadratic control** sub-domain of **Control Theory**.

Measurements consist of N/A via a **full state feedback lqr** sensing mechanism.

The forward operator applies, in order: computes eigen-pairs of a linear operator; S · lqr · gain computation operator; O · lyapunov · closed loop operator.

Observations are corrupted by no stochastic noise (deterministic measurement). Existence of the recovered control_gain_matrix is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 20); model_uncertainty_A_B dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → S · lqr · gain computation → **y** (detector).

```
y = `S.lqr.gain_computation` x    (deterministic)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.lqr.gain_computation` | S · lqr · gain computation operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `E.eigensolve` | Computes eigen-pairs of a linear operator |

**🛡 Analytical properties** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.lyapunov.closed_loop` | O · lyapunov · closed loop operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Control Theory |
| Sub domain | Linear quadratic control |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | control_gain_matrix |
| Noise model | deterministic |
| Integration axis | time_horizon |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered control_gain_matrix is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 20); model_uncertainty_A_B dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | closed_loop_performance_index |
| Secondary | stability_margin_dB |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `full_state_feedback_lqr`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Q r ratio | — | 1 |
| N inputs m | m | 1 |
| System order n | N | 4 |
| Spectral radius a | — | 0.9 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Q r ratio | — | 0.01 – 100 |
| N inputs m | m | 1 – 10 |
| System order n | N | 1 – 50 |
| Spectral radius a | — | 0.0 – 0.999 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 1.05 closed_loop_performance_index

| Metric | Range |
|---|---|
| Closed loop performance index | 0.9 – 2.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **closed_loop_performance_index**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xec453bc6b303c57f8ed7c38e68dc69f7f37ac381200764f902b1645782b13e78`
- **Chain tx hash:** `0x2a18c6e029f0d084e0113108cf9d25d0b316b60d89c51e8a94c126a64707b504`
- **Chain block:** `41555257`

---

## File Mapping

This bundle consists of: `L1-432.md`, `L1-432.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-432.md` | Source of truth — edit this | Human or LLM |
| `L1-432.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
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> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-432`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._