Bo Li, Hongwei Lou
An optimal control problem governed by semilinear elliptic partial differential equations is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.
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Convergence analysis of the semismooth Newton method for sparse control problems governed by semilinear elliptic equations
本文证明在二阶充分条件和严格互补条件下，半光滑牛顿法求解半线性椭圆方程稀疏最优控制问题具有超线性收敛性，允许盒式约束和稀疏项。
An optimal control problem for a semilinear elliptic equation of divergence form is considered. Both the leading term and the semilinear term of the state equation contain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.
Optimal control problems for semilinear elliptic equations with control costs in the space of bounded variations are analysed. BV-based optimal controls favor piecewise constant, and hence 'simple' controls, with few jumps. Existence of optimal controls, necessary and suffcient optimality conditions of first and second order are analysed. Special attention is paid on the effect of the choice of the vector norm in the definition of the BV-seminorm for the optimal primal and adjoined variables.
In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using IPOPT algorithm are presented to verify the efficiency of our approach.
In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.
We investigate full Lipschitzian and full Hölderian stability for a class of control problems governed by semilinear elliptic partial differential equations, where all the cost functional, the state equation, and the admissible control set of the control problems undergo perturbations. We establish explicit characterizations of both Lipschitzian and Hölderian full stability for the class of control problems. We show that for this class of control problems the two full stability properties are equivalent. In particular, the two properties are always equivalent in general when the admissible con…
Bilinear control of semilinear elliptic PDEs: Convergence of a semismooth Newton method
本文针对半线性椭圆PDE的双线性控制问题，分析了半光滑牛顿方法，证明了状态方程解的存在唯一性、最优性条件，并证明了该方法在无间隙二阶充分条件和严格互补条件下的超线性收敛性。
Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations
本文针对受半线性椭圆偏微分方程约束的仿射最优控制问题，通过分析相关集值映射的度量次正则性研究其稳定性，在更弱假设下覆盖广义非线性扰动，并用Tikhonov正则化误差估计验证结果适用性。
This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are show…
Lipschitz continuity of solutions and corresponding multipliers to distributed and boundary semilinear elliptic optimal control problems with mixed pointwise control-state constraints
本文研究了由半线性椭圆方程支配、具有混合逐点控制-状态约束的最优控制问题，证明了极小化元及相应乘子的存在性，并通过bootstrapping技术和分数阶Sobolev空间工具证明了最优解与Lagrange乘子的Lipschitz连续性。
Optimal Control of Semilinear Elliptic Partial Differential Equations with Non-Lipschitzian Nonlinearities
针对含非利普希茨非线性项的半线性椭圆型偏微分方程最优控制问题，证明某类方程解映射的弗雷歇可微性，建立一阶必要最优性条件（耦合偏微分方程组形式），并提出最优控制正则性等相关问题。
Global convergence of $W^{1,\infty}$-steepest descent for PDE constrained shape optimisation with semilinear elliptic equations in function space
本文研究了PDE约束形状优化中半线性椭圆方程的$W^{1,\infty}$-最陡下降法的全局收敛性，证明了在函数空间中的收敛性及二维空间中的条件收敛结果。
Optimal control of a class of semilinear fractional elliptic equations
针对与谱分数阶 Dirichlet Laplace 算子相关的半线性椭圆方程，研究其最优控制问题，证明最优解存在性并给出一阶与二阶最优性条件。
In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In this paper we propose a deep neural network algorithm for solving such partial differential equations in high dimensions. The algorithm is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with random terminal time.
In this paper we study optimal control problems governed by a semilinear elliptic equation. The equation is nonmonotone due to the presence of a convection term, despite the monotonocity of the nonlinear term. The resulting operator is neither monotone nor coervive. However, by using conveniently a comparison principle we prove existence and uniqueness of solution for the state equation. In addition, we prove some regularity of the solution and differentiability of the relation control-to-state. This allows us to derive first and second order conditions for local optimality.
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. We devise two strategies of discretization to approximate a solution of the optimal control problem: a semidiscrete scheme where the control variable is not discretized -- the so-called variational discretization approach -- and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniqu…
Adaptive Approximations of Inclusions in a Semilinear Elliptic Problem Related to Cardiac Electrophysiology
本文针对心脏缺血建模中的半线性椭圆方程包含体重构问题，提出一种基于相场松弛的自适应有限元方法，并证明了离散解子序列的强收敛性，数值实验验证了其收敛性和效率。
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