nucleation

control-theory

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17 items with this tag.

Nov 02, 2026
CLAIM-533: The sensitivity that formal chaos gives a will's dynamics is not control over them: in Kane's self-forming actions, chaotic amplification keeps the outcome open, but it magnifies the agent's efforts, uncontrolled fluctuations and outside inputs alike, so what would make the outcome the agent's is how its own control couples to the amplifying directions, relative to noise and to other parties' inputs and with what feedback, which sensitivity does not supply; and under weak noise what amplification touches lasts only where it reaches the slow modes
Nov 02, 2026
CLAIM-541: Wu-wei is better read as control placed and timed where a system's own dynamics do most of the work than as the absence of control: Ott–Grebogi–Yorke control of chaos is a formal instance, in which small perturbations timed by the system's own wandering hold it on an unstable orbit, and the control, though small, continues for as long as the orbit is held; so reported effortlessness has at least three distinct sources, no conflict to override, conflict settled cheaply by learned accommodation, and conflict met as it arises by small well-timed feedback, which may feel alike and differ in mechanism
Nov 02, 2026
CLAIM-542: Sensitivity does not fix the allocation or the authorship of control: linearized, every input reaches the final state through the same transition operator, ∂x_T/∂w_t = Φ(T, t)B_w(t), so systems with the same Lyapunov spectrum can differ arbitrarily in what an agent's own channel, an outside party's channel and noise each reach and at what least energy; and along one unstable direction a larger exponent λ lowers the least effort to reach a displacement, since W_T = b²(e^{2λT} − 1)/2λ grows with λ, while a constant open-loop input against white noise entering the same direction has terminal signal-to-noise ratio tending to (bu/σ)√(2/λ), which falls as λ grows, so only feedback that observes the state turns cheaper control into reliable direction
Nov 02, 2026
LIT-1972: Using chaos to direct trajectories to targets
- control-theory
- complex-systems
Nov 02, 2026
LIT-1986: A Survey of Viability Theory
- control-theory
- mathematics
Nov 02, 2026
LIT-1997: Controlling chaos
- control-theory
- complex-systems
Oct 31, 2026
LIT-1880: When Can Safe Controllers Adapt? Information before Commitment
Oct 31, 2026
LIT-1893: Input-to-State Safety With Control Barrier Functions
Oct 31, 2026
THEORY-1254: A controller that must stay safe under every model it has not ruled out cannot use the evidence its committing action produces: if the information it can gather before its first action that forecloses a safe continuation under an alternative is bounded by β, every such controller forfeits at least about ½e^{−β} of the value reachable only by committing, so a linear oracle gap gives linear regret, and acting on an exclusion at risk α needs about log(1/(2α)) nats gathered beforehand
Oct 31, 2026
THEORY-1256: When a conservative experiment is safe under every plausible model, can be repeated from a common state and keeps informing, a controller that acts on the lower end of an anytime confidence sequence stays safe with probability 1 − δ and approaches the safe oracle, at O(√(T log T)) regret in a braking example; a certainty-equivalent controller in the same example is unsafe with probability ½ in every episode however much data it has
Oct 31, 2026
THEORY-1267: A safe set that a nominal barrier-function controller keeps invariant can be left under an arbitrarily small input disturbance, but if the barrier inequality holds with a disturbance penalty, ḣ ≥ −α(h) − ι(‖d‖), the state stays in a larger set {h + γ(‖d‖∞) ≥ 0} with γ class-K, and adding a damping term along the barrier's input direction to any barrier controller gives this with an enlargement of ‖d‖²/(4ε) in units of h
Oct 20, 2026
LIT-1292: A contraction based, singular perturbation approach to near-decomposability in complex systems
Oct 20, 2026
THEORY-771: In a singularly perturbed system, fast states x̃ with dx̃/dt = f̃(x̃, y) + ε·δf and slow states y with dy/dt = ε·ḡ(y) + ε·δg, if f̃ is contracting in x̃ at rate β_f with x̃ = 0 an equilibrium for every y, ḡ is contracting at rate β_g, and |δf| ≤ d_f + α_f,x|x̃| + α_f,y|y|, |δg| ≤ d_g + α_g,x|x̃|, then for every ε below ε_c = β_f / [χ_f(α_f,x + χ_g α_f,y α_g,x / β_g)] the fast states stay within O(ε) of zero and, when d_g = 0, the slow states within O(ε) of the reduced trajectory, uniformly in time after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε)
Oct 20, 2026
THEORY-773: In a singularly perturbed system ẋ = f(x, z, t), εż = g(x, z, ε), if g is contracting in z at rate λ_z for every x and ε, the reduced system ẋ = f(x, γ(x), t) on the slow manifold z = γ(x) is contracting at rate λ_x, f is α-Lipschitz in z, |∂γ/∂x · f| ≤ d and g is K-Lipschitz in ε, then for all t ≥ 0 the fast state is within χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z of the manifold and the slow state within χ_x|x(0) − x_γ(0)|e^{−λ_x t} + ε[C₁(e^{−λ_x t} − e^{−λ_z t/ε}) + C₂(1 − e^{−λ_x t})] of the reduced trajectory; partial contraction of f in x alone is not enough
Oct 05, 2026
LIT-714: The Method of Characteristics Revisited: A Viability Approach
Oct 05, 2026
LIT-738: Viability Theory
Oct 05, 2026
THEORY-134: Whether a system can persist is a question about a constraint set fixed independently of its dynamics: a set is viable when from each of its states some evolution can stay in it, and when it is not, its viability kernel is the largest part that is, from the rest of which every evolution leaves in finite time

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