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 $\partial(\text{conv}B) \times [-π/2, π/2]$ generated by the billiard in $\mathbb{R}^2 \setminus B$ and characterize the set of measures $\{ν_B \}$. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge-Kantorovich problems.
The notion of a rough two-dimensional (convex) body is introduced, and to each rough body there is assigned a measure on $\TTT^3$ describing billiard scattering on the body. The main result is characterization of the set of measures generated by rough bodies. This result can be used to solve various problems of least aerodynamical resistance.
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 determine the parametric hull of a given volume which minimizes the total water resistance for a given speed of the ship. The total resistance is the sum of Michell's wave resistance and of the viscous resistance, approximated by assuming a constant viscous drag coefficient. We prove that the optimized hull exists, is unique, symmetric, smooth and that it depends continuously on the speed. Numerical simulations show the efficiency of the approach, and complete the theoretical results.
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.
In this work, two mathematical models for malaria under resistance are presented. More precisely, the first model shows the interaction between humans and mosquitoes inside a patch under infection of malaria when the human population is resistant to antimalarial drug and mosquitoes population is resistant to insecticides. For the second model, human-mosquitoes population movements in two patches is analyzed under the same malaria transmission dynamic established in one patch. For a single patch, existence and stability conditions for the equilibrium solutions in terms of the local basic reprod…
Understanding how to limit biological invasion is critical, especially in the context of accelerating anthropogenic ecological changes. Although biological invasion success could be explained by the lack of natural enemies in new regions, recent studies have revealed that resident herbivores often do have a substantial effect on both native and invasive plants. Very few studies have included consideration of native plant resistance while estimating methods of controlling invasion; hence, it is unclear to what extent the interactive effects of controlling approaches and native plants' resistanc…
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.
Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance
Steklov Mathematical Institute of RAS（俄罗斯科学院斯捷克洛夫数学研究所）；HSE University（高等经济学院）；Center for R&D in Mathematics and Applications, Department of Mathematics, University of Aveiro, Portugal（阿威罗大学数学系数学研发与应用中心）；Institute for Information Transmission Problems, Moscow（莫斯科信息传输问题研究所）
Ensemble optimal control for managing drug resistance in cancer therapies
Technical University of Munich（慕尼黑工业大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Istituti Clinici Scientifici Maugeri IRCCS（莫盖里临床科学研究所IRCCS）；University of Pavia（帕维亚大学）
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…
A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, Plakhov & Gouveia (2007, Nonlinearity, 20), shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement. Giving continuity to this study, new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier. This result, now presented, consi…
This paper revisits the optimal shape problem of a single cooling fin using a one-dimensional heat conduction equation with convection boundary conditions. Firstly, in contrast to previous works, we apply an approach using optimality conditions based on requiring stationarity of the Lagrangian functional of the optimisation problem. This yields an optimality condition basis for the commonly touted constant temperature gradient condition. Secondly, we seek to minimise the root temperature for a prescribed thermal power, rather than maximising the heat transfer rate for a constant root temperatu…
In this paper, we consider the robustness of a basic model of a dynamical distribution network. In the first problem, i.e., optimal weight allocation, we minimize the H-inf- norm of the dynamical distribution network subject to allocation of the weights on the edges. It is shown that this optimization problem can be formulated as a semi-definite program. Next we consider the semi-definiteness of the weighted graph Laplacian matrix with negative weights on the edges. A necessary and sufficient condition, using the effective resistance matrix, is established to guarantee the positive semi-defini…
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 consider a state-constrained optimal control problem of a system of two non-local partial-differential equations, which is an extension of the one introduced in a previous work in mathematical oncology. The aim is to minimize the tumor size through chemotherapy while avoiding the emergence of resistance to the drugs. The numerical approach to solve the problem was the combination of direct methods and continuation on discretization parameters, which happen to be insufficient for the more complicated model, where diffusion is added to account for mutations. In the present paper, we propose a…
The ubiquitous expansion and transformation of the energy supply system involves large-scale power infrastructure construction projects. In the view of investments of more than a million dollars per kilometre, planning authorities aim to minimise the resistances posed by multiple stakeholders. Mathematical optimisation research offers efficient algorithms to compute globally optimal routes based on geographic input data. We propose a framework that utilizes a graph model where vertices represent possible locations of transmission towers, and edges are placed according to the feasible distance …
