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1. A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints

   https://doi.org/10.1515/cmam-2022-0148  

   Brenner, Susanne C; Gedicke, Joscha; Sung, Li-Yeng (December 2022, Computational Methods in Applied Mathematics)

   Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains.The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm.Both theoretical analysis and corroborating numerical results are presented. 

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2. 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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3. A C1 virtual element method for an elliptic distributed optimal control problem with pointwise state constraints

   https://doi.org/10.1142/S0218202521500640  

   Brenner, Susanne C.; Sung, Li-Yeng; Tan, Zhiyu (December 2021, Mathematical Models and Methods in Applied Sciences)

   We design and analyze a [Formula: see text] virtual element method for an elliptic distributed optimal control problem with pointwise state constraints. Theoretical estimates and corroborating numerical results are presented. 

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4. A Circuit-Theoretic Approach to State Estimation

   https://doi.org/10.1109/ISGT-Europe47291.2020.9248872  

   Li, Shimiao; Pandey, Amritanshu; Kar, Soummya; Pileggi, Larry (October 2020, IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe))

   null (Ed.)

   Traditional state estimation (SE) methods that are based on nonlinear minimization of the sum of localized measurement error functionals are known to suffer from nonconvergence and large residual errors. In this paper we propose an equivalent circuit formulation (ECF)-based SE approach that inherently considers the complete network topology and associated physical constraints. We analyze the mathematical differences between the two approaches and show that our approach produces a linear state-estimator that is mathematically a quadratic programming (QP) problem with closed-form solution. Furthermore, this formulation imposes additional topology-based constraints that provably shrink the feasible region and promote convergence to a more physically meaningful solution. From a probabilistic viewpoint, we show that our method applies prior knowledge into the estimate, thus converging to a more physics-based estimate than the traditional observation-driven maximum likelihood estimator (MLE). Importantly, incorporation of the entire system topology and underlying physics, while being linear, makes ECF-based SE advantageous for large-scale systems. 

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5. A Galerkin Approach to the Generalized Karush–Kuhn–Tucker Conditions for the Solution of an Elliptic Distributed Optimal Control Problem with Pointwise State and Control Constraints

   https://doi.org/10.1515/cmam-2025-0007  

   Brenner, Susanne C; Sung, Li-Yeng (February 2025, Computational Methods in Applied Mathematics)

   Abstract We develop a convergence analysis for the simplest finite element method for a model linear-quadratic elliptic distributed optimal control problem with pointwise control and state constraints under minimal assumptions on the constraint functions.We then derive the generalized Karush–Kuhn–Tucker conditions for the solution of the optimal control problem from the convergence results of the finite element method and the Karush–Kuhn–Tucker conditions for the solutions of the discrete problems. 

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