Worst-Case Finite-Horizon Analysis of Perturbed LQR via Performance Estimation Problem

We propose an exact formulation to quantify the finite-horizon robustness of a Linear Quadratic Regulator (LQR) subject to norm-bounded Linear Time-Invariant (LTI) parametric uncertainties, using the Performance Estimation Problem (PEP) framework. Originally developed for optimization, this method allows us to reformulate the worst-case performance search as a rank-constrained Semidefinite Program (SDP). By applying operator interpolation, this formulation retains the sequence variables, allowing the worst-case to be explicitly reconstructed when a rank-one solution is found. This approach also opens new possibilities for the finite-horizon certification of broader system classes, such as data-driven models or nonlinear components. Using this framework, we observe that finite-horizon LTI perturbations do not simply follow asymptotic strategies. Instead, they align slowly decaying invariant manifolds with specific initial conditions to drive transient growth, causing a bifurcation in the worst-case behavior as the horizon length changes. Finally, we show numerically that our performance bounds sit between the upper bounds of Integral Quadratic Constraint (IQC) relaxations and a lower bound derived from a pseudospectral heuristic.