Stable Baselines and Policy Response Capacity

Conceptual map

IV-11.01Response matrices
IV-11.02Transition accounting
IV-11.03Action menus
IV-11.04Mechanism separation
IV-11.05Value of instruments

1. Baseline stability and intervention capacity use different matrices

The baseline transition determines persistence. The intervention map determines which response directions policy can reach over a declared horizon.

Consider $x_{t+1}=Ax_{t}+Bu_{t}$ and target $y_{t}=Cx_{t}$ . Baseline stability in discrete time requires spectral radius $\rho(A)<1$ . Starting from a fixed state, the finite-horizon derivative of stacked future targets with respect to an intervention sequence is formed from $CB,CAB,\ldots,CA^{H-1}B$ .

Definition 1 · Linear response span

The algebraic response capacity through horizon $$H$$ is the column space of the corresponding finite-horizon response matrix. Its dimension is the matrix rank.

Proposition 1 · Finite-horizon reachability from zero

For the state recursion above with $x_{0}=0$ , the states reachable at date $$H$$ by unrestricted signed inputs form the column space of $[A^{H-1}B,\ldots,AB,B]$ . Applying $$C$$ gives the corresponding target-response span.

Proof. Repeated substitution gives $x_{H}=\sum _{j=0}^{H-1}A^{H-1-j}Bu_{j}$ . Stacking the input vectors turns this sum into the stated matrix times the stacked intervention vector. Its image is exactly its column space. Left multiplication by $$C$$ yields the target image. ∎

2. One input can generate two temporal directions

Let $A=\operatorname{diag}(0.5,0.2)$ , $C=I_{2}$ , and one intervention loading $B=(1,1)^{\top}$ . The two-date state-response matrix is

\[[B,AB]=\begin{bmatrix}1 & 0.5 \\ 1 & 0.2\end{bmatrix},\](1)

whose determinant is $$-0.3$$ and rank is two. With signed interventions at two dates, the algebraic span covers $\mathbb{R}^{2}$ . If interventions must be nonnegative with a total budget, attainable responses form a bounded convex hull or cone slice rather than the full span.

3. The same stable baseline can have different capacity

Holding $$A$$ fixed while replacing $$B$$ by zero removes all policy response without changing baseline dynamics. Replacing it by $e_{1}$ confines every response to the first coordinate. A stable or quiet observed path therefore supplies no capacity conclusion without the intervention map and action support.

Randomized mixtures generate convex combinations of arm-specific mean responses. They do not automatically create arbitrary signed linear combinations. Assignment probabilities, ethical exclusions, delivery failures, and dose limits determine the feasible set.

4. Horizon, stability, and feasibility boundaries

Failure case · Response rank reported as implementable policy set

Rank ignores sign, magnitude, budget, assignment, and delivery constraints. The rank-two matrix in (1) can coexist with a small one-sided feasible region.

At $$H=0$$ no future response block is present. At eigenvalues on the unit circle, baseline asymptotic stability fails even though a finite-horizon response remains defined. Nonlinear systems use derivatives only locally; finite interventions need remainder bounds and state-constraint checks.

5. Implementation, exercises, and sources

Fix timing conventions, build the horizon response matrix, compute rank under a declared metric and threshold, and separately propagate the feasible action set. Report stability eigenvalues, intervention support, delivery mapping, budget, hull or cone representation, and target values.

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Exercises

Verify the determinant and rank in (1).
Characterize responses when two intervention dates satisfy $u_{0},u_{1}\ge0$ and their sum is at most one.
Compare $$B=0$$ , $e_{1}$ , and $(1,1)^{\top}$ under the same $$A$$ .

Partial solutions

1. The columns are linearly independent because $$0.2-0.5=-0.3$$ . 2. The response set is the convex hull of zero and the two columns in (1). 3. Their two-date response ranks are zero, one, and two.

Thomas Kailath, Linear Systems, chapters on controllability and stability.Finite-horizon response spaces.
Roger A. Horn and Charles R. Johnson, Matrix Analysis.Spectral stability and rank.
Chae-Yeon Xon (2026), “Perfectly Still: Stability and Policy Response Capacity.”Related policy-history application.

6. Audit checkpoint

State transition timing, stability region, initial state, target, intervention loading, horizon, action signs and bounds, budget, assignment and delivery constraints, response metric, rank threshold, feasible hull, and nonlinear remainder domain.

7. Scope boundary

The chapter covers finite-horizon linear response capacity and feasible action images. General controllability design and nonlinear control require additional theory.