A Duality Approach to Set Invariance and Safety for Nonlinear Systems

This paper proposes a duality approach to guarantee set invariance for nonlinear dynamical systems. Building from the so-called mirror descent algorithm from the optimization literature, we develop a new version of the given nonlinear system such that the desired set is forward (pre-)invariant. Such new version of the model is constructed using duality between the given system — called the primal system — and a new system — called the dual system. By appropriately mapping the dual system back to the original space, the resulting system — called the modified primal system — has the desired set forward pre-invariant. The power of the approach is illustrated in several applications pertaining to constrained optimization and feedback control under constraints. Academic examples are provided to illustrate the approach and utility of the results.