Architecturally Constrained Monotone Functions as Lyapunov-Stable Performance Certificates: A Function-Space Approach

Abstract

Prescribed performance control (PPC) guarantees that tracking errors evolve within user-defined envelopes, yet the design of the performance certificate ρ(t) itself has been restricted to hand-tuned parametric forms since the theory’s inception. We establish that the standard exponential family Fexp constitutes a nowhere-dense, three-dimensional submanifold of the infinite-dimensional space of valid certificates, and we prove that any learnable function with architecturally enforced monotonicity and positivity constitutes a valid, stability-preserving performance certificate. Three theorems formalize this principle. Theorem 1 proves that a feedforward neural network with constrained non-positive time-input weights, non-negative cross-channel weights, and Softplus output activation generates a valid certificate for all conditioning inputs. Theorem 2 establishes a modularity property; closed-loop stability is preserved for any certificate in the family whenever the controller renders the transformed error uniformly ultimately bounded, decoupling certificate validity from certificate origin. Theorem 3 proves that every piecewise-monotone decreasing positive certificate can be uniformly approximated by a constrained monotone network for a suitable conditioning vector; this expressivity is achieved jointly over the network parameters θ and the conditioning vector c, with c acting as a continuous (but not dense) semantic-steering mechanism for fixed θ. The theoretical results are illustrated through an analytical comparison of certificate families and a numerical example on a Duffing-type oscillator, illustrating, in the considered example, improved tracking performance under identical Lyapunov stability guarantees.