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Abstract This paper provides a (rigorous) theoretical framework for the numerical approximation of Riccati-based feedback control problems of hyperbolic-like dynamics over a finite-time horizon, with emphasis on genuine unbounded control action. Both continuous and approximation theories are illustrated by specific canonical hyperbolic-like equations with boundary control, where the abstract assumptions are actually sharp regularity properties of the hyperbolic dynamics under discussion. Assumptions are divided in two groups. A first group of dynamical assumptions (actually dynamic properties) imply some preliminary critical properties of the control problem, including the definition of the would-be Riccati operator, in terms of the original data. However, in order to guarantee that such an operator is moreover the unique solution (within a specific class) of the corresponding Differential/Integral Riccati Equation, additional smoothing assumptions on the operators defining the performance index are required. The ultimate goal is to show that the the discrete finite dimensional Riccati based feedback operator, when inserted into the original PDE dynamics, provides near optimal performance. 

Abstract Designing protein-binding proteins is critical for drug discovery. However, artificial-intelligence-based design of such proteins is challenging due to the complexity of protein–ligand interactions, the flexibility of ligand molecules and amino acid side chains, and sequence–structure dependencies. We introduce PocketGen, a deep generative model that produces residue sequence and atomic structure of the protein regions in which ligand interactions occur. PocketGen promotes consistency between protein sequence and structure by using a graph transformer for structural encoding and a sequence refinement module based on a protein language model. The graph transformer captures interactions at multiple scales, including atom, residue and ligand levels. For sequence refinement, PocketGen integrates a structural adapter into the protein language model, ensuring that structure-based predictions align with sequence-based predictions. PocketGen can generate high-fidelity protein pockets with enhanced binding affinity and structural validity. It operates ten times faster than physics-based methods and achieves a 97% success rate, defined as the percentage of generated pockets with higher binding affinity than reference pockets. Additionally, it attains an amino acid recovery rate exceeding 63%. 

The terminal alkyne C≡C stretch has a large Raman scattering cross section in the “silent” region for biomolecules. This has led to many Raman tag and probe studies using this moiety to study biomolecular systems. A computational investigation of these systems is vital to aid in the interpretation of these results. In this work, we develop a method for computing terminal alkyne vibrational frequencies and isotropic transition polarizabilities that can easily and accurately be applied to any terminal alkyne molecule. We apply the discrete variable representation method to a localized version of the C≡C stretch normal mode. The errors of (1) vibrational localization to the terminal alkyne moiety, (2) anharmonic normal mode isolation, and (3) discretization of the Born–Oppenheimer potential energy surface are quantified and found to be generally small and cancel each other. This results in a method with low error compared to other anharmonic vibrational methods like second-order vibrational perturbation theory and to experiments. Several density functionals are tested using the method, and TPSS-D3, an inexpensive nonempirical density functional with dispersion corrections, is found to perform surprisingly well. Diffuse basis functions are found to be important for the accuracy of computed frequencies. Finally, the computation of vibrational properties like isotropic transition polarizabilities and the universality of the localized normal mode for terminal alkynes are demonstrated. 

Abstract An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heat-structure interaction partial differential equation (PDE) models, with structure subject to high Kelvin-Voigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevantfluid-structure models, with wave- or plate-structures, possibly without Kelvin-Voigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDE-energy estimates dictate how to define the interface/boundary feedback control in each of the three cases. 

We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $$\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $$\end{document}, which, moreover, in the canonical case \begin{document}$$ \gamma = 0 $$\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$$ \gamma = 0 $$\end{document} or \begin{document}$$ 0 \neq \gamma \in L^{\infty}(\Omega) $$\end{document}, since \begin{document}$$ \gamma \neq 0 $$\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$$ g $$\end{document} "smoother" than \begin{document}$$ L^2(\Sigma) $$\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$$ L^2(0, T;L^2(\Gamma)) $$\end{document}, and [44] for control less regular in space than \begin{document}$$ L^2(\Gamma) $$\end{document}$. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2]. 

We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.