Continuity and uniqueness of regularized output least squares optimal estimators (Q1910045)

The regularized output least squares estimation problem is formulated as: \[ \text{Find } a_0 \in Q_{ad} \subseteq Q \text{ such that } J(a_0) = \inf \{J(a) : a \in Q_{ad}\}.\tag{1} \] The authors study the continuity properties of the solution set of (1) with respect to perturbations of the data \(z\) for different regularizations. The paper contains 1. results for weak regularizations in which upper semicontinuity of the set-valued mapping \(z \to Q(z)\) is established, 2. results for stronger regularizations: stability in stronger topologies and the finiteness of the set of optimal estimators, 3. conditions that imply uniqueness of optimal estimators and Lipschitz continuity with respect to data.