More Like this

1. Conversion of a Class of Stochastic Control Problems to Fundamental-Solution Deterministic Control Problems

   McEneaney, William M; Dower, Peter M. (July 2020, Proceedings of the American Control Conference)

   A class of nonlinear, stochastic staticization control problems (including minimization problems with smooth, convex, coercive payoffs) driven by diffusion dynamics and constant diffusion coefficient is considered. Using dynamic programming and tools from static duality, a fundamental solution form is obtained where the same solution can be used for a variety of terminal costs without re-solution of the problem. Further, this fundamental solution takes the form of a deterministic control problem rather than a stochastic control problem. 

   more » « less
2. 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. 

   more » « less
3. A min-plus fundamental solution semigroup for a class of approximate infinite dimensional optimal control problems

   McEneaney, William M; Dower, Peter M. (July 2020, Proceedings of the American Control Conference)

   By exploiting min-plus linearity, semiconcavity, and semigroup properties of dynamic programming, a fundamental solution semigroup for a class of approximate finite horizon linear infinite dimensional optimal control problems is constructed. Elements of this fundamental solution semigroup are parameterized by the time horizon, and can be used to approximate the solution of the corresponding finite horizon optimal control problem for any terminal cost. They can also be composed to compute approximations on longer horizons. The value function approximation provided takes the form of a min-plus convolution of a kernel with the terminal cost. A general construction for this kernel is provided, along with a spectral representation for a restricted class of sub-problems. 

   more » « less
4. A Computational Framework for the Numerical Solution of Optimal Control Problems Governed by Partial Differential Equations

   Davies, A M; Dennis, M E; Rao, A V (July 2024, American Control Conference)

   A computational framework for the solution of op- timal control problems with time-dependent partial differential equations (PDEs) is presented. The optimal control problem is transformed from a continuous time and space optimal control problem to a sparse nonlinear programming problem through state parameterization with Lagrange polynomials and discrete controls defined at Legendre-Gauss-Radau (LGR) points. The standard LGR collocation method is coupled with a modified Radau method to produce a collocation point on the typically noncollocated boundary. The newly collocated endpoint allows for a representation of the state derivative and control on the originally noncollocated boundary such that Neumann boundary conditions may be satisfied. Finally, the method developed in this paper is demonstrated on a viscous Burgers’ tracking problem and the results are compared to an existing solution. 

   more » « less
5. Elliptic Inverse Problems of Identifying Nonlinear Parameters

   Jadamba, B; Khan, A. A; Kahler, R; Sama, M. (January 2018, Pure and Applied Functional Analysis)

   Inverse problems of identifying parameters in partial differential equations (PDEs) is an important class of problems with many real-world applications. Inverse problems are commonly studied in optimization setting with various known approaches having their advantages and disadvantages. Although a non-convex output least-squares (OLS) objective has often been used, a convex modified output least-squares (MOLS) attracted quite an attention in recent years. However, the convexity of the MOLS has only been established for parameters appearing linearly in the PDEs. The primary objective of this work is to introduce and analyze a variant of the MOLS for the inverse problem of identifying parameters that appear nonlinearly in variational problems. Besides giving an existence result for the inverse problem, we derive the first-order and second-order derivative formulas for the new functional and use them to identify the conditions under which the new functional is convex. We give a discretization scheme for the continuous inverse problem and prove its convergence. We also obtain discrete formulas for the new MOLS functional and present detailed numerical examples. 

   more » « less