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This paper introduces a novel $\mathcal{H}_{\infty}$-optimal intervalobserver synthesis for bounded-error/uncertain locally Lipschitz nonlinearcontinuous-time (CT) and discrete-time (DT) systems with noisy nonlinearobservations. Specifically, using mixed-monotone decompositions, the proposedobserver is correct by construction, i.e., the interval estimates readily framethe true states without additional constraints or procedures. In addition, weprovide sufficient conditions for input-to-state (ISS) stability of theproposed observer and for minimizing the $\mathcal{H}_{\infty}$ gain of theframer error system in the form of semi-definite programs (SDPs) with LinearMatrix Inequalities (LMIs) constraints. Finally, we compare the performance ofthe proposed $\mathcal{H}_{\infty}$-optimal interval observers with somebenchmark CT and DT interval observers.

$\mathcal{H}_{\infty}$-optimal Interval Observer Synthesis for Uncertain Nonlinear Dynamical Systems via Mixed-Monotone Decompositions