Robust control of linear systems under small parameter variations

In this article, we investigate first-order parameter insensitization for linear controlled evolutions: given a family A(σ) of generators on a Hilbert space H and a control operator B, does there exist a single control, independent of σ, that both steers the nominal system to a prescribed final state and cancels the first order sensitivity at the nominal system of the endpoint σ → yσ(T ), where yσ is the state of the system? We start from an abstract framework that does not depend on the dimension or on the type of system (ODE or PDE), and we derive the associated sensitivity cascade system. Insensitization is then recast as a controllability problem for an extended state (y, z). We illustrate this approach with two significant examples: in finite dimension, we prove the equivalence between four natural formulations and provide a necessary and sufficient algebraic condition of Kalman-type rank. We also treat infinite-dimensional examples, namely: the heat equation with uncertain diffusion coefficient, and show that the natural null and mixed approximate/exact insensitization problems are solvable; the wave equation with uncertain potential under appropriate geometric conditions on the control region allowing to derive approximate insensitization properties.