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1. Discrete Approximations and Optimality Conditions for Controlled Free-Time Sweeping Processes

   https://doi.org/10.1007/s00245-024-10108-7  

   Colombo, Giovanni; Mordukhovich, Boris S; Nguyen, Dao; Nguyen, Trang (April 2024, Applied Mathematics & Optimization)

   The paper is devoted to the study of a new class of optimal control problems governed by discontinuous constrained differential inclusions of the sweeping type involving the duration of the dynamic process into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical values with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of first-order and second-order variational analysis and generalized differentiation allows us to derive new necessary conditions for optimal solutions of the discrete-time problems and then, by passing to the limit in the discretization procedure, for designated local minimizers in the original problem of sweeping optimal control. The obtained results are illustrated by a numerical example 

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2. Constrained optimization problems governed by PDE models of grain boundary motions

   https://doi.org/10.1515/anona-2022-0242  

   Antil, Harbir; Kubota, Shodai; Shirakawa, Ken; Yamazaki, Noriaki (August 2021, Advances in Nonlinear Analysis)

   Abstract In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness. 

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3. Modified Legendre-Gauss Collocation Method for Optimal Control Problems with Nonsmooth Solutions

   Abadia-Doyle, G; Rao, A V (December 2024, IEEE)

   A modified form of Legendre-Gauss orthogonal direct collocation is developed for solving optimal control problems whose solutions are nonsmooth due to control discon- tinuities. This new method adds switch time variables, control variables, and collocation conditions at both endpoints of a mesh interval, whereas these new variables and collocation con- ditions are not included in standard Legendre-Gauss orthogonal collocation. The modified Legendre-Gauss collocation method alters the search space of the resulting nonlinear programming problem and optimizes the switch point of the control solution. The transformed adjoint system of the modified Legendre- Gauss collocation method is then derived and shown to satisfy the necessary conditions for optimality. Finally, an example is provided where the optimal control is bang-bang and contains multiple switches. This method is shown to be capable of solving complex optimal control problems with nonsmooth solutions. 

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4. Minimum-entanglement protocols for function estimation

   https://doi.org/10.1103/PhysRevResearch.5.033228  

   Ehrenberg, Adam; Bringewatt, Jacob; Gorshkov, Alexey V. (September 2023, Physical Review Research)

   We derive a family of optimal protocols, in the sense of saturating the quantum Cramér-Rao bound, for measuring a linear combination of d field amplitudes with quantum sensor networks, a key subprotocol of general quantum sensor network applications. We demonstrate how to select different protocols from this family under various constraints. Focusing primarily on entanglement-based constraints, we prove the surprising result that highly entangled states are not necessary to achieve optimality in many cases. Specifically, we prove necessary and sufficient conditions for the existence of optimal protocols using at most k-partite entanglement. We prove that the protocols which satisfy these conditions use the minimum amount of entanglement possible, even when given access to arbitrary controls and ancillas. Our protocols require some amount of time-dependent control, and we show that a related class of time-independent protocols fail to achieve optimal scaling for generic functions. 

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5. Structure Detection Method for Solving State-Variable Inequality Path Constrained Optimal Control Problems

   Byczkowski, Cale A.; Rao, Anil V. (January 2023, American Astronautical Society)

   A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc. 

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