Koenig-D’Amico Reachable Set Solver

• Koenig-D’Amico Reachable Set Theory Solver is a computational framework for synthesizing and certifying reachable sets with a focus on geometric structure and limit shape convergence.
• It uses the Banach–Mazur distance to robustly compare shapes under linear feedback and coordinate transformations, ensuring invariance and canonical representation.
• The framework underpins practical algorithms for system verification and control synthesis, leveraging small-time approximations for efficient safety analysis.

The Koenig-D’Amico Reachable Set Theory Solver refers to a class of computational methods and theoretical results for synthesizing, approximating, and certifying reachable sets in control systems, especially with an emphasis on the geometric structure, convergence properties, and invariance of reachable sets under linear transformations and feedback. Central to this framework is the analysis of the limit shape of reachable sets for linear control systems under geometric input constraints, as initiated and rigorously developed in "Birth of the shape of a reachable set" (Goncharova et al., 2013). This theory underpins both the efficient approximation of reachable sets required for system verification and safety analysis, and the design and implementation of practical solvers that exploit invariance, robustness, and canonical forms.

1. Geometric Approach to Reachable Sets

The Koenig-D’Amico framework investigates the linear control system

$$\frac{dx}{dt} = A x + B u, \quad x \in \mathbb{R}^n, \quad u \in U \subset \mathbb{R}^m$$

where $U$ is a centrally symmetric convex body. For fixed terminal time $T>0$, the reachable set $D(T)$ consists of all end states achievable from the origin using admissible controls over $[0, T]$. Rather than focusing on $D(T)$ itself, the central object is its "shape": the equivalence class of $D(T)$ up to invertible linear transformations, allowing comparison of the geometry of different reachable sets independent of scaling and coordinates.

The metric for this space is the Banach–Mazur distance:

$$p(K_1, K_2) = \log \left( t(K_1, K_2) \cdot t(K_2, K_1) \right)$$

where $$t(K_1,K_2) = \inf\{ t \geq 1 : K_1 \subseteq t K_2 \}$$, for convex bodies $K_1, K_2$. The set of shapes $S$ under the general linear group $GL(\mathbb{R}^n)$ forms a metric space that encodes geometric equivalence.

2. Limit Shape Convergence: Autonomous Systems

For time-invariant $(A,B)$, provided the pair satisfies the Kalman controllability condition, the principal result (Theorem 3.1) demonstrates that as $T \to 0$, the shape of $D(T)$ converges at rate $O(T)$ to a canonical, time-independent convex body $S_0$:

$p( \text{Sh}(D(T)), S_0 ) = O(T)$

Here, $\text{Sh}(D(T))$ is the shape of the reachable set at time $T$, and $S_0$ is its limit as $T \to 0$.

Two technical lemmas underpin this result:

• Lemma 3.2: Adding constant linear feedback $A \to \widetilde{A} = A + B C$ (for some matrix $C$) changes the reachable set's shape by at most $O(T)$ in Banach–Mazur distance, showing robustness to feedback.
• Lemma 3.3: Under a gauge (coordinate) transformation $A \to C^{-1}AC$, $B \to C^{-1}B$, the shape is invariant: $\text{Sh}(D(T)) = \text{Sh}(\tilde{D}(T))$.

These lemmas allow the system to be brought into Brunovsky normal form, in which the reachable set's shape is independent of $T$. This canonicalization enables identification of the unique limit shape $S_0$ without requiring normalization by a rescaling of $T$.

3. Generalization to Non-Autonomous Systems

For time-varying $(A(t), B(t))$, and possibly time-dependent $U$, the analysis is extended via augmentation of the state space and careful use of Lie algebraic tools. In this context, the vector fields $(1, A(t)x)$ and $(0, B(t)u)$ generate a Lie algebra $\mathcal{L}$.

Assuming the generic Kalman-type condition

$\dim \mathcal{L}(T,x) = n+1 \tag{4.1}$

for every point $(T,x)$ (i.e., the evaluation of the Lie algebra on the tangent space), Theorem 4.1 establishes that

$p( \text{Sh}(D(T)), S_0 ) = O(T)$

with the same geometric meaning as before. The proof involves reduction to the $A=0$ case and the analysis of a natural filtration

$$\mathcal{F}^k = \{ \xi \in V^* : \langle B(t)^* \xi, \ldots \rangle = O(t^k) \}$$

in the dual space, determining the fine structure of the reachable set's boundary and asymptotic properties.

4. Robustness, Invariance, and Canonicalization

These convergence results provide a theoretical basis for several key engineering desiderata:

• Canonical representation: Every controllable linear system possesses a unique "microscopic" limit shape for its small-time reachable set, facilitating comparison and classification.
• Robustness: The limit shape and its convergence rate are robust to feedback and coordinate changes, ensuring consistency across transformations that frequently arise in controller synthesis and observer design.
• Reduction and simplification: Reduction to Brunovsky normal form transforms high-dimensional analysis to lower-dimensional, structured problems, making algebraic and computational methods tractable. Transformations preserve the essential geometry and allow standardization across systems.

5. Implications for Reachable Set Algorithms and Solvers

The theoretical framework developed in (Goncharova et al., 2013) is instrumental for designing practical algorithms, such as those in the Koenig–D’Amico Reachable Set Theory Solver, for control, safety verification, and estimation:

• Small-time approximations: The existence of an explicit $O(T)$ approximation rate to the canonical shape means that for small $T$, computational routines can use precomputed (or precharacterized) limit shapes as tight overapproximations, avoiding explicit integration.
• Optimal control and system verification: The tight geometry bound allows for precise constraint and safety verification using geometric invariants, with implications for controller synthesis, safety envelope computation, and abstraction of system behavior.
• Feedback invariance: The robustness of the shape to linear feedback adapts naturally to iterative design or adaptive control, where feedback law changes arise dynamically.

These insights are crucial for the implementation of algorithms requiring precise geometric modeling of reachable sets, particularly for tasks involving verification, control synthesis, and real-time safety analysis, as in high-dimensional or time-critical engineering systems. The explicit quantification of convergence and invariance properties enables both efficient simulation and enduring correctness guarantees.

6. Key Mathematical Results and Theorems

The theoretical pillars supporting this framework can be summarized as follows:

Result/Theorem	Statement	Context
Theorem 3.1	In autonomous controllable linear systems, reachable set shapes converge as $T\to 0$ to $S_0$ with Banach–Mazur rate $O(T)$.	Geometric structure/autonomous case
Theorem 4.1	Extension of Theorem 3.1 to non-autonomous systems under generic Lie algebraic controllability, with same convergence rate.	Non-autonomous time-varying systems
Lemma 3.2	Shape of reachable set is robust to constant linear feedback addition ($A \to A + BC$).	Feedback invariance
Lemma 3.3	Shape is invariant under invertible coordinate (gauge) transformations.	Coordinate invariance
Banach–Mazur metric	$p(K_1, K_2) = \log(t(K_1,K_2)t(K_2,K_1))$ for shapes.	Comparison of convex bodies up to $GL(\mathbb{R}^n)$ action

Explicit use of these results is foundational for theory-driven design and analysis of reachable set algorithms.

7. Influence and Context

The convergence of shapes and the explicit control over their approximation are now recognized as central in geometric control theory and computational reachable set methods. These developments have influenced subsequent extensions in nonlinear settings, data-driven estimation, and the integration of geometric invariants into formal verification and controller synthesis software frameworks.

In particular, the Koenig–D’Amico Reachable Set Theory Solver leverages these geometric insights, providing algorithmic routines for highly accurate reachable set approximations by utilizing limit shape information, invariance under feedback and coordinate changes, and sharp error control, as required for rigorous computational verification in engineering and scientific applications.