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1. Optimal control of fractional semilinear PDEs

   https://doi.org/10.1051/cocv/2019003  

   Antil, Harbir; Warma, Mahamadi (January 2020, ESAIM: Control, Optimisation and Calculus of Variations)

   In this paper, we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order 2 s with s ∈ (0, 1). We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condition on the nonlinearity to show the Lipschitz continuity of the solution map for the semilinear elliptic equations with respect to the data. We further apply our ideas to show existence of solutions to optimal control problems with semilinear fractional equations as constraints. Under the standard assumptions on the nonlinearity (twice continuously differentiable) we derive the first and second order optimality conditions. 

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2. 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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3. The regular free boundary in the thin obstacle problem for degenerate parabolic equations

   Banerjee, A.; Danielli, D.; Garofalo, N.; Petrosyan, A. (June 2020, Algebra i analiz)

   In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial_t - \Delta_x)^s$$ for $$s \in (0,1)$$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $$H^{1+\gamma,\frac{1+\gamma}{2}}$$ regularity of the free boundary near such regular points. 

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4. A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

   https://doi.org/10.3934/dcdss.2022012  

   Antil, Harbir; Gal, Ciprian G.; Warma, Mahamadi (January 2022, Discrete and Continuous Dynamical Systems - S)

   We consider optimal control of fractional in time (subdiffusive, i.e., for \begin{document}$$ 0<\gamma <1 $$\end{document}) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we \begin{document}$$\mathsf{first\;show}$$\end{document} the existence and regularity of solutions to the forward and the associated \begin{document}$$\mathsf{backward\;(adjoint)}$$\end{document} problems. In the second part, we prove existence of optimal \begin{document}$$\mathsf{controls }$$\end{document} and characterize the associated \begin{document}$$\mathsf{first\;order}$$\end{document} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces. 

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5. Optimal control for the Paneitz obstacle problem

   https://doi.org/10.1051/cocv/2023036  

   Ndiaye, Cheikh Birahim (January 2023, ESAIM: Control, Optimisation and Calculus of Variations)

   In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q -curvature. We show also C ∞ -regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls. 

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