The Hidden Geometry of Control: Mapping Every State a System Can Reach

Imagine you are navigating a spacecraft. You have thrusters, but they don't point in every direction — some degrees of freedom are simply beyond your reach. The question isn't just "where can I go?" but "exactly how much of the future is open to me, and which parts are locked shut forever?" This is not an abstract philosophical puzzle. It is a precise mathematical question, and until now it lacked a precise mathematical answer for a large and practically important class of systems.

A new paper by Fengjiao Liu, Yixiao Zhang, and Panagiotis Tsiotras (Liu et al., 2026) provides that answer. Working with linear time-varying (LTV) systems — dynamical systems whose governing equations change over time — they have characterized the exact set of states that can be reached, and the exact set of probability distributions that can be shaped, using state feedback control. The results hold even when the system is only partially controllable. This resolves questions that have lingered since the early study of dynamical systems and opens new pathways for spacecraft guidance, swarm robotics, and the control of uncertain, stochastic processes.

The Science

To understand what the paper accomplishes, we need to unpack a few key concepts.

A linear time-varying system is one where the relationship between a system's state and its rate of change is linear — roughly, double the state and you double the rate — but the coefficients governing that relationship shift with time. The dynamics of a satellite on an elliptical orbit, or a flexible robot arm moving through a task, are well modeled this way. The system is written compactly as $\dot{x}(t)=A(t)x(t)+B(t)u(t)$, where $x(t)$ is the system's state, $u(t)$ is the control input, and the matrices $A(t)$ and $B(t)$ encode how everything interacts at each moment (Liu et al., 2026).

When a state feedback control of the form $u(t)=K(t)x(t)$ is applied — meaning the control at each moment is a linear function of the current state — the system's evolution is completely encoded in a single matrix called the state transition matrix, $\Phi (T,0)$. This matrix is a kind of time-travel summary: it tells you exactly how any initial state $x(0)$ gets mapped to the terminal state $x(T)$, compactly capturing the entire history of the system's motion. The question "what terminal states can I reach?" is therefore equivalent to asking: "what terminal state transition matrices $\Phi (T,0)$ are achievable by choosing different feedback gains $K(t)$?"

There is a second, equally important object: the state covariance matrix $\Sigma (t)$. When the initial state of a system is uncertain — described by a probability distribution rather than a single point — the covariance matrix tracks how that uncertainty evolves and spreads over time. Steering the covariance is the central problem in stochastic control, distribution steering, and optimal transport. The two objects are linked by the elegant relation $\Sigma (T)=\Phi (T,0)\Sigma (0)\Phi (T,0{)}^{\mathsf{T}}$, so understanding the reachable set of $\Phi$ directly illuminates the reachable set of $\Sigma$ (Liu et al., 2026).

The paper's central tool is the matrix Riccati differential equation (RDE) — a nonlinear ordinary differential equation for a matrix-valued function $\Pi (t)$, given by

$$\dot{\Pi }=-A(t{)}^{\mathsf{T}}\Pi -\Pi A(t)+\Pi B(t)B(t{)}^{\mathsf{T}}\Pi .$$

Riccati equations are workhorses of control theory, appearing in everything from optimal filtering (the Kalman filter) to optimal control (the LQR problem). Here, Liu et al. deploy them in a new way: as a constructive device for generating the feedback gain $K(t)=-B(t{)}^{\mathsf{T}}\Pi (t)$ that steers the system to a prescribed target. A remarkable closed-form formula then connects the solution $\Pi (t)$ directly to the resulting state transition matrix, making the entire construction explicit and computable.

The research team — based at the FAMU-FSU College of Engineering and Georgia Tech's School of Aerospace Engineering — builds on a line of recent work [4, 5, 1] that had established controllability of the state transition matrix for linear time-invariant (LTI) systems. Their contribution is to extend these results to the much broader and more realistic LTV setting, and to handle the case of partial controllability with mathematical precision.

What They Found

The paper's three main theorems together paint a complete picture.

Theorem 1 establishes that, for an LTV system satisfying a mild technical condition (Assumption 1 — essentially that the system's controllability doesn't degenerate on any short subinterval) and with a positive definite controllability Gramian $H(T,0)\succ 0$, the state transition matrix is fully controllable. The controllability Gramian, defined as

$$H(T,0)={\int }_0^T{\Phi }_A(t,\tau )B(\tau )B(\tau {)}^{\mathsf{T}}{\Phi }_A(t,\tau {)}^{\mathsf{T}}d\tau ,$$

is a standard measure of how much the control inputs can influence the system's state over time. When it is positive definite, the control inputs can — in principle — reach every direction in state space. The theorem confirms that, under these conditions, every matrix ${\Phi }_f$ with positive determinant is reachable.

Theorem 2 handles the far more delicate case where $H(T,0)$ is only positive semidefinite — meaning some directions are uncontrollable. Here the reachable set is no longer all of ${\text{GL}}^{+}(n,\mathbb{R})$ (the set of all invertible matrices with positive determinant). Instead, it is a structured subset ${\mathcal{R}}_T$. If the controllability Gramian has rank $r<n$, we can decompose the space into a controllable $r$-dimensional subspace and an uncontrollable $(n-r)$-dimensional subspace. In the controllable directions, any invertible transformation is achievable. In the uncontrollable directions, the state transition matrix is frozen at the identity — the system simply cannot reach anything else there. The reachable set has the precise block structure

$${\mathcal{R}}_T=\left\{{\Phi }_f:{\Phi }_A(0,T){\Phi }_f=U\left[\begin{array}{cc}{\bar{\Phi }}_f& {\tilde{\Phi }}_f\\ 0& I_{n-r}\end{array}\right]U^{\mathsf{T}},\det ({\bar{\Phi }}_f)>0\right\},$$

where $U$ is an orthogonal matrix that aligns with the structure of $H(T,0)$, and ${\bar{\Phi }}_f\in {\mathbb{R}}^{r\times r}$ can be any $r\times r$ invertible matrix with positive determinant (Liu et al., 2026). This is a mathematically clean and satisfying answer: the reachable set is determined entirely by the rank structure of the controllability Gramian.

Theorem 3 delivers the covariance result, and it is arguably the most practically consequential finding. It characterizes the exact set ${\mathcal{S}}_T$ of terminal covariance matrices reachable from any given initial covariance ${\Sigma }_0\succ 0$ over $[0,T]$. The condition is elegant: a positive definite matrix ${\Sigma }_f$ is reachable if and only if, when you project both ${\Sigma }_f$ (propagated backward through the uncontrolled dynamics) and ${\Sigma }_0$ onto the kernel of the reachability Gramian — the subspace the control inputs cannot touch — they agree. In other words, within the uncontrollable subspace, the covariance is immovable; it evolves exactly as if no control were applied. But within the controllable subspace, any positive definite target is achievable.

Crucially, Theorem 3 requires no analog of Assumption 1 — it holds in full generality. And the construction is not merely existential: for every reachable ${\Sigma }_f\in {\mathcal{S}}_T$, there is an explicit feedback control $K(t)$ that achieves it, and that control is continuous in both the initial covariance ${\Sigma }_0$ and the target covariance ${\Sigma }_f$. This continuity property is critical for applications — it means that small changes in the desired target produce only small changes in the control law, a prerequisite for robust, practical implementation.

Corollary 2 closes the loop cleanly: the covariance is fully controllable — meaning any positive definite ${\Sigma }_f$ is reachable from any positive definite ${\Sigma }_0$ — if and only if $H(T,0)\succ 0$. The controllability Gramian is the single deciding quantity.

The paper also identifies and resolves a subtle topological obstruction to continuity. For systems with $n\ge 2$ state dimensions, there cannot exist a single feedback gain $K(t)$ that is simultaneously continuous in ${\Phi }_f$ for all targets ${\Phi }_f\in {\text{GL}}^{+}(n,\mathbb{R})$. The reason lies in algebraic topology: the space ${\text{GL}}^{+}(n,\mathbb{R})$ is not contractible for $n\ge 2$, meaning it has a nontrivial "hole" that prevents any global continuous selection. This is not a failure of cleverness — it is a fundamental geometric fact. The paper's remedy is to identify large, well-characterized subsets of ${\text{GL}}^{+}(n,\mathbb{R})$ where continuity does hold, using three propositions that give explicit sufficient conditions (Liu et al., 2026).

Why This Changes Things

The significance of this work operates on several levels.

At the most immediate level, it closes a gap in the theory of LTV systems that has existed for decades. Previous results either assumed full controllability, or provided only an upper bound on the reachable covariance set without confirming that bound was tight (Liu et al., 2026). The new characterization is exact — not an outer approximation, but the true reachable set.

For spacecraft and satellite control, LTV systems are the natural model. A satellite in orbit experiences time-varying gravitational gradients, atmospheric drag, and solar pressure. Its attitude control system may have actuators that can only push in certain directions — a partial controllability scenario exactly captured by Theorem 2 and Theorem 3. Knowing the precise reachable set tells mission designers not just what trajectories are theoretically achievable, but also which terminal states require more actuators, longer time windows, or different mission architectures.

For multi-agent systems and swarm robotics, the connection is direct: steering the state transition matrix is mathematically equivalent to steering a collection of $n$ agents with identical linear dynamics from one configuration to another. The controllability results here tell engineers exactly which collective formations are achievable through linear feedback, and the constructive control laws tell them how to get there (Liu et al., 2026).

For stochastic control and distribution steering, the covariance results are perhaps the most immediately impactful. Distribution steering — the problem of moving a probability distribution from one shape to another using a controlled dynamical system — has seen explosive interest in recent years, driven by applications in autonomous systems, robotics, and machine learning. The Theorem 3 result tells practitioners the precise set of target distributions achievable from any given starting distribution, under linear feedback. The continuity of the control law in the problem data is a practical guarantee that these solutions can be implemented robustly.

There is also a deep connection to optimal transport — a mathematical framework for finding the most efficient way to move mass from one distribution to another, with applications ranging from image processing to economics. The authors note that optimal transport theory can, in general, produce nonlinear feedback laws for distribution steering. Their work characterizes what is achievable with the more restrictive class of linear feedback laws, providing a precise boundary between the linear and nonlinear regimes.

The key novelty over prior work is the shift from LTI to LTV systems — a jump of enormous practical importance. Real physical systems are almost never truly time-invariant. Treating them as such is an approximation, sometimes a very good one, but often not. The LTV framework embraced here is strictly more general, and the results hold without approximation.

What's Next

The paper leaves several threads productively open.

The mild Assumption 1 required by Theorems 1 and 2 — that the rank of the controllability Gramian stays constant across five consecutive subintervals — is automatically satisfied for constant matrix pairs but needs verification for general LTV systems. A natural next step is to either relax this assumption further or to characterize the LTV systems for which it fails, and understand what happens to the reachable set in those edge cases.

The paper's constructive control laws are based on solving a Riccati differential equation, which is numerically well-conditioned in most practical settings. But the computational cost of doing this in real time, for high-dimensional systems or tight time windows, remains an open engineering challenge. Future work on fast numerical methods for RDEs in this context would substantially broaden the applicability of these results.

The connection to nonlinear feedback and optimal transport is sketched but not fully developed. The authors note that when the map between initial and terminal states is a general diffeomorphism (a smooth, invertible transformation), nonlinear feedback laws can achieve configurations that linear feedback cannot. A precise characterization of the gap between linear and nonlinear reachable sets — building on the exact linear reachable set now available — would be a significant theoretical advance.

Finally, the results here treat the noise-free case: the state covariance evolves purely through the controlled dynamics, without additive stochastic disturbances. The authors note compatibility with a more general result that includes additive noise (Liu et al., 2026), but a full LTV treatment of the noisy case, with the same level of precision for the reachable set, remains an open problem.

What the paper achieves is something rare in applied mathematics: not just a new technique, but a complete answer to a well-posed question. The reachable set of a partially controllable LTV system is no longer a mystery wrapped in upper bounds and sufficient conditions. It has a precise shape, determined by a single computable matrix. And for every point inside that shape, there is an explicit, robust control law that gets you there. In a field defined by the gap between what is theoretically possible and what is practically achievable, that kind of completeness matters.