Radouen Ghanem
We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in $H^{2}$. We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions existence and give necessary optimality conditions.
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Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems
本文证明了单边和双边障碍问题的解映射在特定条件下是牛顿可微的，并给出显式导数公式，应用于半光滑牛顿法求解最优控制问题。
We consider an optimal control problem in which the state is governed by an unilateral obstacle problem (with obstacle from below) and restricted by a pointwise state constraint (from above). In the presence of control constraints, we prove, via regularization of the state constraints, that a system of C-stationarity is necessary for optimality. In the absence of control constraints, we show that local minimizers are even strongly stationary by a careful discussion of the primal first-order conditions of B-stationary type.
In the present paper, the maximum principle for finite horizon state constrained problems from the book by R. Vinter [\textit{Optimal Control}, Birkhäuser, Boston, 2000; Theorem~9.3.1] is analyzed via parametric examples. The latter has origin in a recent paper by V.~Basco, P.~Cannarsa, and H.~Frankowska, and resembles the optimal growth problem in mathematical economics. The solution existence of these parametric examples is established by invoking Filippov's existence theorem for Mayer problems. Since the maximum principle is only a necessary condition for local optimal processes, a large am…
Directional differentiability for the solution map of the bilateral parabolic obstacle problem
本文证明双侧抛物型障碍问题解映射在Lebesgue和Sobolev空间间方向可微，基于单侧结果和新不动点定理，可用于最优控制条件推导。
We consider optimal control problems for a wide class of bilateral obstacle problems where the control appears in a possibly nonlinear source term. The non-differentiability of the solution operator poses the main challenge for the application of efficient optimization methods and the characterization of Bouligand generalized derivatives of the solution operator is essential for their theoretical foundation and numerical realization. In this paper, we derive specific elements of the Bouligand generalized differential if the control operator satisfies natural monotonicity properties. We constru…
We consider an optimal control problem $\cQ$ governed by an elliptic quasivariational inequality with unilateral constraints. The existence of optimal pairs of the problem is a well known result, see \cite{SS}, for instance. We associate to $\cQ$ a new optimal control problem $\wQ$, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of Problem $\wQ$ to a solution of Problem $\cQ$. The proofs are based on convergence results…
In this work we consider the numerical resolution of the bilateral obstacle optimal control problem given in Bergounioux et al. Where the main feature of this problem is that the control and the obstacle are the same.
We study the optimal stopping problem for a monotonous dynamic risk measure induced by a BSDE with jumps in the Markovian case. We show that the value function is a viscosity solution of an obstacle problem for a partial integro-differential variational inequality, and we provide an uniqueness result for this obstacle problem.
In the present work, control of time-optimal trajectory for a Dubins airplane in presence of moving and fixed obstacles is obtained. We show that for a Dubins airplane with an initial position, the control variable can be obtained using the exact penalty function method so that the airplane reaches the end position in the shortest time in the presence of obstacles.
This paper is concerned with second-order optimality conditions for Tikhonov regularized optimal control problems governed by the obstacle problem. Using a simple observation that allows to characterize the structure of optimal controls on the active set, we derive various conditions that guarantee the local/global optimality of first-order stationary points and/or the local/global quadratic growth of the reduced objective function. Our analysis extends and refines existing results from the literature, and also covers those situations where the problem at hand involves additional box-constrain…
We address the optimal dynamic formation problem in mobile leader-follower networks where an optimal formation is generated to maximize a given objective function while continuously preserving connectivity. We show that in a convex mission space, the connectivity constraints can be satisfied by any feasible solution to a mixed integer nonlinear optimization problem. When the optimal formation objective is to maximize coverage in a mission space cluttered with obstacles, we separate the process into intervals with no obstacles detected and intervals where one or more obstacles are detected. In …
We consider the problem of time-optimal path planning for simple nonholonomic vehicles. In previous similar work, the vehicle has been simplified to a point mass and the obstacles have been stationary. Our formulation accounts for a rectangular vehicle, and involves the dynamic programming principle and a time-dependent Hamilton-Jacobi-Bellman (HJB) formulation which allows for moving obstacles. To our knowledge, this is the first HJB formulation of the problem which allows for moving obstacles. We design an upwind finite difference scheme to approximate the equation and demonstrate the effica…
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.
Optimality conditions in terms of Bouligand generalized differentials for a nonsmooth semilinear elliptic optimal control problem with distributed and boundary control pointwise constraints
针对含分布与边界逐点约束的非光滑半线性椭圆最优控制问题，利用Bouligand广义导数提出新最优性条件，并证明其与强平稳性等价。
A numerical study of an optimal control formulation for a shape optimization problem governed by an elliptic variational inequality is performed. The shape optimization problem is reformulated as a boundary control problem in a fixed domain. The discretized optimal control problem is a non-smooth and non-convex mathematical programing problem. The performance of the standard BFGS quasi-Newton method and the BFGS method with the inexact line search are tested.
We introduce an alternative approach for the analysis and numerical approximation of the optimal feedback control mapping. It consists in looking at a typical optimal control problem in such a way that feasible controls are mappings depending both in time and space. In this way, the feedback form of the problem is built-in from the very beginning. Optimality conditions are derived for one such optimal mapping, which by construction is the optimal feedback mapping of the problem. In formulating optimality conditions, costates in feedback form are solutions of linear, first-order transport syste…
Control in the Coefficients of an Obstacle Problem
本文研究系数受控的障碍型最优控制问题，利用正则化与 H-收敛结合 bootstrapping 论证，证明控制的强 L^p 收敛并导出极限最优性条件。
We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control constraints. A comparison shows that this system is stronger than the known system of C-stationarity.
Optimal control problems without control costs in general do not possess solutions due to the lack of coercivity. However, unilateral constraints together with the assumption of existence of strictly positive solutions of a pre-adjoint state equation, are sufficient to obtain existence of optimal solutions in the space of Radon measures. Optimality conditions for these generalized minimizers can be obtained using Fenchel duality, which requires a non-standard perturbation approach if the control-to-observation mapping is not continuous (e.g., for Neumann boundary control in three dimensions). …
Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control
本文证明障碍型变分不等式解算子的 Newton 可微性，并据此构造半光滑 Newton 方法求解带 H1 代价和控制约束的最优控制问题，验证其超线性收敛与网格无关性。
This paper mainly focuses on solving the dynamic optimization of the planar controlled crowd motion models with obstacles which is an application of a class of optimal control problems governed by a general perturbed nonconvex sweeping process. This can be considered as a significant extension of the previous work regarding the controlled crowd motion models, where the obstacles have not been considered. The necessary optimality conditions for the problem under consideration are established and illustrated by a nontrivial example of practical importance.
A Path Planning Model for Intercepting a Moving Target with Finite Obstacle Avoidance
本文针对无人机以恒速拦截移动目标问题，提出一种结合障碍规避与拦截约束的最优曲率直线路径规划模型，并通过数值实验和真实地理数据案例验证其有效性。
The standard Dynamic Programming (DP) formulation can be used to solve Multi-Stage Optimization Problems (MSOP's) with additively separable objective functions. In this paper we consider a larger class of MSOP's with monotonically backward separable objective functions; additively separable functions being a special case of monotonically backward separable functions. We propose a necessary and sufficient condition, utilizing a generalization of Bellman's equation, for a solution of a MSOP, with a monotonically backward separable cost function, to be optimal. Moreover, we show that this propose…
Shape and topology optimization for variational inequalities with pointwise boundary observation
本文提出了一种基于哈密顿方法的正则化和惩罚技术，用于解决变分不等式中带有点观测的形状和拓扑优化问题，并通过数值示例验证了方法的有效性。
This work develops feasible path trajectories for a coordinated strike with multiple aircraft in a constrained environment. Using direct orthogonal collocation methods, the two-point boundary value optimal control problem is transcribed into a nonlinear programming problem. A coordinate transformation is performed on the state variables to leverage the benefits of a simplex discretization of the search domain. Applying these techniques allows each path constraint to be removed from the feasible search space, eliminating computationally expensive, nonlinear constraint equations and problem spec…
Optimization problems governed by Allen-Cahn systems including elastic effects are formulated and first-order necessary optimality conditions are presented. Smooth as well as obstacle potentials are considered, where the latter leads to an MPEC. Numerically, for smooth potential the problem is solved efficiently by the Trust-Region-Newton-Steihaug-cg method. In case of an obstacle potential first numerical results are presented.
This paper studies a system of $m$ variational inequalities with interconnected obstacles in infinite horizon associated to optimal multi-modes switching problems. Our main result is the existence and uniqueness of a continuous solution in viscosity sense, for that system. The proof of the main result strongly relies on the connection between the systems of variational inequalities and reflected backward stochastic differential equations (RBSDEs) with oblique reflection, which will be characterized through a Feynman-Kac's formula. The main feature of our system of infinite horizon RBSDEs is th…
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^{\infty}-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.
We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we prove that the singularities of the distance function propagate, in the sense that each singular point belongs to a nontrivial singular continuum. Finally, we investigate the lack of differentiability of the distance function when …
In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors fo…
We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map $A:\mathbb{R}^N\rightarrow2^{\mathbb{R}^N}$. We do not assume that $D(A)=\mathbb{R}^N$, incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.
We study an optimal control problem associated to the conformal Laplacian obstacle problem on closed n-dimensional Riemannian manifolds with n >2. When the Yamabe invariant of the Riemannian manifold is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal metric with constant scalar curvature. For the standard sphere, we prove that the standard bubbles -- namely conformal factor of metrics conformal to the standard one with constant positive scalar curvature -- are the only optimal …
A Discontinuous Galerkin Method for Optimal Control of the Obstacle Problem
本文研究椭圆障碍问题约束的最优控制问题，采用对称内罚间断Galerkin方法做有限元离散，基于改进$L^2$误差估计等给出拟最优先验误差估计，无需标准网格限制性假设。
We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings, continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for mult…
In this paper, we investigate optimal boundary control problems for Cahn-Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the follo…
A Single-Loop Bilevel Deep Learning Method for Optimal Control of Obstacle Problems
提出一种无网格、可扩展的单环双层深度学习方法，通过约束嵌入神经网络和单环随机一阶双层算法高效求解障碍问题的最优控制。
Altitude-Loss Optimal Glides in Engine Failure Emergencies -- Accounting for Ground Obstacles and Wind
本文研究发动机失效后无推力飞机在存在地面障碍物和风影响下的高度损失最优滑翔轨迹规划，将最优解刻画为障碍物极值点间的直线滑翔段，并通过缩减可见性图和A*搜索求解，在Cessna 172模型上验证了轨迹最优性与跑道可达性判定。
Optimal Control of Incompressible Ideal Flows with Obstacle Avoidance
本文提出一种变分障碍回避形式化方法，通过引入屏障势函数修改欧拉方程，研究障碍回避对流体动力学的影响，展示屏障项对流体局部变形的作用。
Global minima for optimal control of the obstacle problem
本文研究了受椭圆障碍问题约束的最优控制问题，通过Moreau-Yosida型惩罚方法离散化得到的PDE进行数值近似，并提供判断必要一阶条件解是否为全局极小值的条件，同时证明该结果可推广到极限问题，最后给出了具有唯一全局解的数值示例。
In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phas…
The paper arXiv:1804.11290 contains well-posedness and regularity results for a system of evolutionary operator equations having the structure of a Cahn-Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn-Hilliard system could be of a very general type that inc…
Safety-Critical Planning and Control for Dynamic Obstacle Avoidance Using Control Barrier Functions
本文提出一种结合MPC与离散时间高阶控制屏障函数的框架，通过网格映射生成凸多面体构建安全约束，无需已知障碍物边界方程，并集成路径规划保证全局最优性，在动态狭窄环境中实现独轮机器人的高效安全避障。
We adapt the Stochastic Perron's method in Bayraktar and Sirbu (ArXiv: 1103.0538) to the case of double obstacle problems associated to Dynkin games. We construct, symmetrically, a viscosity sub-solution which dominates the upper value of the game and a viscosity super-solution lying below the lower value of the game. If the double obstacle problem satisfies the viscosity comparison property, then the game has a value which is equal to the unique and continuous viscosity solution. In addition, the optimal strategies of the two players are equal to the first hitting times of the two stopping re…
Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability and optimal control
本文研究障碍型拟变分不等式最小与最大解映射的Lipschitz稳定性、方向可微性，证明最优控制存在性，并利用Moreau-Yosida惩罚逼近推导C-平稳性条件。
Optimal Stopping of BSDEs with Constrained Jumps and Related Double Obstacle PDEs
本文研究带约束跳的BSDE最优停止问题与双障碍PDE的关系，通过Feynman-Kac表示和压缩论证证明非局部情形下粘性解的存在唯一性，并建立新型非线性Snell包络。
In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace-Beltr…
Finite-Horizon Optimal Consumption and Investment with Time-Varying Job-Switching Costs
本文研究了在时间变化就业转换成本下，经济代理人在有限时间 horizon 内的最优消费、投资及就业转换策略问题，通过PDE理论分析了双障碍问题的解及其自由边界性质。
In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraints for the cost function where the cost function is described by the solution of one reflected backward stochastic differential equations. We will give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton-Jacobi-Bellman equations.
We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub- and supersolutions as bounds for Perron's method. Our motivation stems from so called optimal switching problems on bounded domains.
In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of no…
This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors; the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interv…
This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories and subsequently classified, with local uniqueness results obtained in both cases.
In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the conse…
We obtain sharp local $C^{1,α}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ Δ_p u=γ(u-φ)^{γ-1}\,\text{ in }\,\{u>φ\}, $$ for $0<γ<1$ and $p\ge2$. At the free boundary $\partial\{u>φ\}$, we prove optimal $C^{1,τ}$ regularity of solutions, with $τ$ given explicitly in terms of $p$, $γ$ and smoothness of $φ$, which is new even in the linear setting.
We present a novel penalty approach for a class of quasi-variational inequalities (QVIs) involving monotone systems and interconnected obstacles. We show that for any given positive switching cost, the solutions of the penalized equations converge monotonically to those of the QVIs. We estimate the penalization errors and are able to deduce that the optimal switching regions are constructed exactly. We further demonstrate that as the switching cost tends to zero, the QVI degenerates into an equation of HJB type, which is approximated by the penalized equation at the same order (up to a log fac…
In modern production facilities industrial robots and humans are supposed to interact sharing a common working area. In order to avoid collisions, the distances between objects need to be measured conservatively which can be done by a camera network. To estimate the acquired distance, unmodelled objects, e.g., an interacting human, need to be modelled and distinguished from premodelled objects like workbenches or robots by image processing such as the background subtraction method. The quality of such an approach massively depends on the settings of the camera network, that is the positions an…
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