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1. Mean field type control problems, some Hilbert-space-valued FBSDES, and related equations

   https://doi.org/10.1051/cocv/2025022  

   Bensoussan, Alain; Tai, Ho Man; Yam, Sheung_Chi Phillip (January 2025, ESAIM: Control, Optimisation and Calculus of Variations)

   In this article, we provide an original systematic global-in-time analysis of mean field type control problems on ℝⁿwith generic cost functions allowing quadratic growth by a novel “lifting” approach which is not the same as the traditional lifting. As an alternative to the recent popular analytical method of tackling master equations, we resolve the control problem in a proper Hilbert subspace of the whole space ofL²random variables, it can be regarded as a tangent space attached at the initial probability measure. The problem is linked to the global solvability of the Hilbert-space-valued forward–backward stochastic differential equation (FBSDE), which is solved by variational techniques here. We also rely on the Jacobian flow of the solution to this FBSDE to establish the regularity of the value function, including its linearly functional differentiability, which leads to the classical wellposedness of the Bellman equation. Together with the linear functional derivatives and the gradient of the linear functional derivatives of the solution to the FBSDE, we also obtain the classical wellposedness of the master equation. Our current approach imposes structural conditions directly on the cost functions. The contributions of adopting this framework in our study are twofold: (i) compared with imposing conditions on Hamiltonian, the structural conditions imposed in this work are easily verified, and less demanding on the cost functions while solving the master equation; and (ii) when the cost functions are not convex in the state variable or there is a lack of monotonicity of cost functions, an accurate lifespan can be provided for the local existence, which may not be that small in many cases. 

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2. Optimal Control Strategies for Systems with Input Delay using the PIE Framework

   https://doi.org/10.1016/j.ifacol.2022.11.339  

   Peet, Matthew M. (January 2022, IFAC-PapersOnLine)

   The Partial Integral Equation (PIE) framework provides a unified algebraic representation for use in analysis, control, and estimation of infinite-dimensional systems. However, the presence of input delays results in a PIE representation with dependence on the derivative of the control input, u˙. This dependence complicates the problem of optimal state-feedback control for systems with input delay – resulting in a bilinear optimization problem. In this paper, we present two strategies for convexification of the H∞-optimal state-feedback control problem for systems with input delay. In the first strategy, we use a generalization of Young's inequality to formulate a convex optimization problem, albeit with some conservatism. In the second strategy, we filter the actuator signal – introducing additional dynamics, but resulting in a convex optimization problem without conservatism. We compare these two optimal control strategies on four example problems, solving the optimization problem using the latest release of the PIETOOLS software package for analysis, control and simulation of PIEs. 

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3. Strong Stationarity for Optimal Control Problems with Non-smooth Integral Equation Constraints: Application to a Continuous DNN

   https://doi.org/10.1007/s00245-023-10059-5  

   Antil, Harbir; Betz, Livia; Wachsmuth, Daniel (December 2023, Applied Mathematics & Optimization)

   Abstract Motivated by the residual type neural networks (ResNet), this paper studies optimal control problems constrained by a non-smooth integral equation associated to a fractional differential equation. Such non-smooth equations, for instance, arise in the continuous representation of fractional deep neural networks (DNNs). Here the underlying non-differentiable function is the ReLU or max function. The control enters in a nonlinear and multiplicative manner and we additionally impose control constraints. Because of the presence of the non-differentiable mapping, the application of standard adjoint calculus is excluded. We derive strong stationary conditions by relying on the limited differentiability properties of the non-smooth map. While traditional approaches smoothen the non-differentiable function, no such smoothness is retained in our final strong stationarity system. Thus, this work also closes a gap which currently exists in continuous neural networks with ReLU type activation function. 

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4. A comparison principle for semilinear Hamilton–Jacobi–Bellman equations in the Wasserstein space

   https://doi.org/10.1007/s00526-024-02718-4  

   Daudin, Samuel; Seeger, Benjamin (April 2024, Calculus of Variations and Partial Differential Equations)

   Abstract The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution. 

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5. EikoNet: Solving the Eikonal Equation With Deep Neural Networks

   https://doi.org/10.1109/tgrs.2020.3039165  

   Smith, Jonathan D.; Azizzadenesheli, Kamyar; Ross, Zachary E. (December 2020, IEEE Transactions on Geoscience and Remote Sensing)

   null (Ed.)

   The recent deep learning revolution has created enormous opportunities for accelerating compute capabilities in the context of physics-based simulations. In this article, we propose EikoNet, a deep learning approach to solving the Eikonal equation, which characterizes the first-arrival-time field in heterogeneous 3-D velocity structures. Our grid-free approach allows for rapid determination of the travel time between any two points within a continuous 3-D domain. These travel time solutions are allowed to violate the differential equation--which casts the problem as one of optimization--with the goal of finding network parameters that minimize the degree to which the equation is violated. In doing so, the method exploits the differentiability of neural networks to calculate the spatial gradients analytically, meaning that the network can be trained on its own without ever needing solutions from a finite-difference algorithm. EikoNet is rigorously tested on several velocity models and sampling methods to demonstrate robustness and versatility. Training and inference are highly parallelized, making the approach well-suited for GPUs. EikoNet has low memory overhead and further avoids the need for travel-time lookup tables. The developed approach has important applications to earthquake hypocenter inversion, ray multipathing, and tomographic modeling, as well as to other fields beyond seismology where ray tracing is essential. 

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