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A finite horizon nonlinear optimal control problem is considered for which the associated Hamiltonian satisfies a uniform semiconcavity property with respect to its state and costate variables. It is shown that the value function for this optimal control problem is equivalent to the value of a min-max game, provided the time horizon considered is sufficiently short. This further reduces to maximization of a linear functional over a convex set. It is further proposed that the min-max game can be relaxed to a type of stat (stationary) game, in which no time horizon constraint is involved. 

Abstract Protein posttranslational modification (PTM) is a biochemical mechanism benefitting cellular adaptation to dynamic intracellular and environmental conditions. Recently, several acylation marks have been identified as new protein PTMs occurring on specific lysine residues in mammalian cells: isobutyrylation, methacrylation, benzoylation, isonicotinylation, and lactylation. These acylation marks were initially discovered to occur on nucleosomal histones, but they potentially occur as prevalent biomarkers on non‐histone proteins as well. The existence of these PTMs is a downstream consequence of metabolism and demonstrates the intimate crosstalk between active cellular metabolites and regulation of protein function. Emerging evidence indicates that these acylation marks on histones affect DNA transcription and are functionally distinct from the well‐studied lysine acetylation. Herein, we discuss enzymatic regulation and metabolic etiology of these acylations, 'reader' proteins that recognize different acylations, and their possible physiological and pathological functions. Several of these modifications correlate with other well‐studied acylations and fine‐tune the regulation of gene expression. Overall, findings of these acylation marks reveal new molecular links between metabolism and epigenetics and open up many questions for future investigation. © 2023 The Authors. Current Protocols published by Wiley Periodicals LLC. 

Kirigami metamaterials dramatically change their shape through a coordinated motion of nearly rigid panels and flexible slits. Here, we study a model system for mechanism-based planar kirigami featuring periodic patterns of quadrilateral panels and rhombi slits, with the goal of predicting their engineering scale response to a broad range of loads. We develop a generalized continuum model based on the kirigami’s effective (cell-averaged) nonlinear deformation, along with its slit actuation and gradients thereof. The model accounts for three sources of elasticity: a strong preference for the effective fields to match those of a local mechanism, inter-panel stresses arising from gradients in slit actuation, and distributed hinge bending. We provide a finite-element formulation of this model and implement it using the commercial software Abaqus. Simulations of the model agree quantitatively with experiments across designs and loading conditions.