Delfim F. M. Torres, Alexander Yu. Plakhov
We address Newton-type problems of minimal resistance from an optimal control perspective. It is proven that for Newton-type problems the Pontryagin maximum principle is a necessary and sufficient condition. Solutions are then computed for concrete situations, including the new case when the flux of particles is non-parallel.
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New numerical solutions to Newton's problem of least resistance via a convex hull approach
提出基于凸包方法的数值方法求解凸函数类牛顿最小阻力问题，通过利用解的对称及极值点曲线分布特性，得到高精度解。
We consider Newton's problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial symmetry of solutions. Moreover, we show that certain conical parts contained in the boundary of a convex body inhibit the optimality in the classical Newton's problem with finite height. This result is applied to certain bodies considered in the literature, which are conjectured to be optimal for the classical Newton's problem, and we show that they are not.
We consider the following problem: minimize the functional $\int_Ωf(\nabla u(x))\, dx$ in the class of concave functions $u: Ω\to [0,M]$, where $Ω\subset \mathbb{R}^2$ is a convex body and $M > 0$. If $f(x) = 1/(1 + |x|^2)$ and $Ω$ is a circle, the problem is called Newton's problem of least resistance. It is known that the problem admits at least one solution. We prove that if all points of $\partialΩ$ are regular and ${|x|f(x)}/(|y|f(y)) \to +\infty$ as $|x|/|y| \to 0$ then a solution $u$ to the problem satisfies $u\rfloor_{\partialΩ} = 0$. This result proves the conjecture stated in 1993 fo…
We study the Newton-like problem of minimal resistance for a two-dimensional body moving with constant velocity in a homogeneous rarefied medium of moving particles. The distribution of the particles over velocities is centrally symmetric. The problem is solved analytically; the minimizers are shown to be of four different types. Numerical results are obtained for the physically significant case of gaussian circular distribution of velocities, which corresponds to a homogeneous ideal gas of positive temperature.
Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of {\it convex} axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but {\it generally nonconvex} bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the valu…
We consider the following problem stated in 1993 by Buttazzo and Kawohl: minimize the functional $\int\!\!\int_Ω(1 + |\nabla u(x,y)|^2)^{-1} dx\, dy$ in the class of concave functions $u: Ω\to [0,M]$, where $Ω\subset \mathbb{R}^2$ is a convex domain and $M > 0$. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resista…
Let $u$ minimize the functional $F(u) = \int_Ωf(\nabla u(x))\, dx$ in the class of convex functions $u : Ω\to {\mathbb R}$ satisfying $0 \le u \le M$, where $Ω\subset {\mathbb R}^2$ is a compact convex domain with nonempty interior and $M > 0$, and $f : {\mathbb R}^2 \to {\mathbb R}$ is a $C^2$ function, with $\{ ξ: \, \text{the smallest eigenvalue of} \, f"(ξ) \, \text{is zero} \}$ being a closed nowhere dense set in ${\mathbb R}^2$. Let epi$(u)$ denote the epigraph of $u$. Then any extremal point $(x, u(x))$ of epi$(u)$ is contained in the closure of the set of singular points of epi$(u)$. A…
We characterize the solution to the Newton minimal resistance problem in a class of radial q-concave profiles. We also give the corresponding result for one-dimensional profiles. Moreover, we provide a numerical optimization algorithm for the general nonradial case.
We consider the problem $\inf\big\{ \int\!\!\int_Ω(1 + |\nabla u(x,y)|^2)^{-1} dx dy : \text{ the function } u : Ω\to \mathbb{R} \text{ is concave and } 0 \le u(x,y) \le M \text{ for all } (x,y) \in Ω=\{ (x,y): x^2 + y^2 \le 1 \} \, \big\}$ (Newton's problem) and its generalizations. In the paper \cite{BrFK} it is proved that if a solution $u$ is $C^2$ in an open set $\mathcal{U} \subset Ω$ then $\det D^2u = 0$ in $\mathcal{U}$. It follows that graph$(u)\rfloor_\mathcal{U}$ does not contain extreme points of the subgraph of $u$. In this paper we prove a somewhat stronger result. Namely, there …
Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into n…
We study the problem of minimal resistance for a body moving with constant velocity in a rarefied medium of chaotically moving point particles, in Euclidean space R^d. The particles distribution over velocities is radially symmetric. Under some additional assumptions on the distribution function, the complete classification of bodies of least resistance is made. In the case of three and more dimensions there are two kinds of solutions: a body similar to the solution of classical Newton's problem and a union of two such bodies ``glued together'' by rear parts of their surfaces. In the two-dimen…
The problem of minimal resistance, old and new
本文综述了变分法中最小阻力问题的历史进展与理论成果，并提出了未来的新研究方向。
The method of Hessian measures is used to find the differential equation that defines the optimal shape of nonrotationally symmetric bodies with minimal resistance moving in a rare medium. The synthesis of optimal solutions is described. A theorem on the optimality of the obtained solutions is proved.
A body moves in a medium composed of noninteracting point particles; interaction of particles with the body is absolutely elastic. It is required to find the body's shape minimizing or maximizing resistance of the medium to its motion. This is the general setting of optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary $B \subset \mathbb{R}^2$ we assign a measure $ν_B$ on $\partial(…
A dimension reduction procedure for the selection of Two-spring lattice-spring topologies with minimal fabrication cost and required weighted force-resistance performance
本文提出了一种降维方法，用于选择在最小制造成本和所需加权力-阻力性能下最优的双弹簧晶格-弹簧拓扑结构。
Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance
该研究针对牛顿空气动力学中固定面积二维凸体的最小阻力问题，将其归约为拟芬斯勒度量等周问题，引入振荡微小性准则，揭示最优物体边界奇点条件并构造均匀振荡下的精确最优形状。
Quasi-Newton Method for Set Optimization Problems with Set-Valued Mapping Given by Finitely Many Vector-Valued Functions
本文提出一种求解无约束集合优化问题弱极小解的拟牛顿法，通过划分构造向量优化问题族并随迭代切换，证明了算法的良定性、收敛性及局部超线性收敛。
Local minima in Newton's aerodynamical problem and inequalities between norms of partial derivatives
针对Newton最小阻力曲面问题，本文将平面曲面的局部极小性归结为边界消失二元凹函数偏导数$L_2$范数的不等式，并给出依赖区域几何的完备判据。
Newton Method for Set Optimization Problems with Set-Valued Mapping of Finitely Many Vector-Valued Functions
本文针对由有限多个强凸二次连续可微向量值函数构成的无约束集优化问题，提出一种基于划分集和Gerstewitz标量化函数的牛顿法，用于求弱极小解，并证明了该方法在相应连续性条件下的局部超线性与二次收敛性。
A Newton-Kantorovich Inverse Function Theorem in Quasi-Metric Spaces
本文研究（拟）度量空间上的求根问题，提出Newton可微性概念，并证明Newton型方法的超线性收敛及Newton-Kantorovich反函数定理。
The calculus of variations is a field of mathematical analysis born in 1687 with Newton's problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler-Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton's approach to calculus. As in many applications negative values of admissible functions are not physic…
We present a novel communication-efficient Newton-type algorithm for finite-sum optimization over a distributed computing environment. Our method, named DINO, overcomes both theoretical and practical shortcomings of similar existing methods. Under minimal assumptions, we guarantee global sub-linear convergence of DINO to a first-order stationary point for general non-convex functions and arbitrary data distribution over the network. Furthermore, for functions satisfying Polyak-Lojasiewicz (PL) inequality, we show that DINO enjoys a linear convergence rate. Our proposed algorithm is practically…
An Inexact Proximal Newton Method for Nonconvex Composite Minimization
本文提出一种求解非凸复合问题的不精确近端牛顿型方法，在KKT残差范数下实现O(k^{-1/2})全局收敛与局部超线性收敛，无需线搜索且数值实验验证其性能媲美现有最优方法。
An inexact semismooth Newton-Krylov method for semilinear elliptic optimal control problem
针对半线性椭圆最优控制问题，提出结合GMRES（Krylov子空间方法）与非单调线搜索的不精确半光滑Newton方法，经理论分析得超线性收敛性，数值实验验证其准确性与效率。
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