  In optimal control problems, disturbances are typically dealt with using
robust solutions, such as H-infinity or tube model predictive control, that
plan control actions feasible for the worst-case disturbance. Yet, planning for
every contingency can lead to over-conservative, poorly performing solutions or
even, in extreme cases, to infeasibility. Resilience addresses these
shortcomings by adapting the underlying control problem, e.g., by relaxing its
specifications, to obtain a feasible, possibly still valuable trajectory.
Despite their different aspects, robustness and resilience are often conflated
in the context of dynamical systems and control. The goal of this paper is to
formalize, in the context of optimal control, the concept of resilience
understood as above, i.e., in terms of adaptation. To do so, we introduce a
resilient formulation of optimal control by allowing disruption-dependent
modifications of the requirements that induce the desired resilient behavior.
We then propose a framework to design these behaviors automatically by trading
off control performance and requirement violations. We analyze this
resilience-by-compromise method to obtain inverse optimality results and
quantify the effect of disturbances on the induced requirement relaxations. By
proving that robustness and resilience optimize different objectives, we show
that these are in fact distinct system properties. We conclude by illustrating
the effect of resilience in different control problems.
