Topological insulators are solid bodies characterized by a broad energy gap that is stable under small deformations. This motivates the use of topological methods in their study. A key role in solid-state theory is played by symmetry groups. Kitaev pointed out a relation between the symmetries of some classes of solids, including topological insulators, and Clifford algebras. According to this observation, the quantization of topological insulators should reduce to the theory of irreducible representations of Clifford algebras. The next important step was made by Kennedy and Zirnbauer, who introduced the notion of pseudosymmetries. While the algebra of observables of a topological insulator is generated by Hamiltonians satisfying commutation relations with symmetry operators, the quantum observables are described by complex structures on the Nambu space that satisfy anticommutation relations with pseudosymmetries. This correspondence determines the quantization of topological insulators.
The article addresses problems concerning bodies of zero resistance and invisible bodies within a model where a solid body interacts with a medium of non-interacting particles that reflect from the body in a perfectly elastic manner. Known results are reviewed, some statements are proved, and several conjectures and open problems are formulated.
We consider an optimal control problem with Fuller-type symmetry in which the control takes values in a two-dimensional disk and the optimal synthesis contains a second-order singular trajectory. This problem was first investigated in the monograph by M. I. Zelikin and V. F. Borisov, where self-similar trajectories of the optimal synthesis were constructed. We derive effectively computable lower and upper bounds on the Bellman function of this problem. To this end we bound the set of admissible controls from above and below by rectangles. For these control sets the Bellman functions of the modified problems can be computed analytically and provide the aforementioned bounds on the Bellman function in the original problem.
We consider two almost Lorentzian geometries in the Grushin α -plane as optimal control problems. Using methods of geometric control theory, we study extremal trajectories, compute the reachable set, investigate the existence of optimal trajectories, and describe the length maximizers and the distance in the Lorentzian metric.
We study the Hamiltonian system of the perturbed two-dimensional Fuller problem with control in a square in a neighborhood of a second-order singular point. We prove the existence of a two-dimensional surface filled with chattering extremals that reach the origin with countably many control switches in finite time. The proof is based on the resolution of singularities technique for the Poincaré map. Applying the invariant manifold theorem, we obtain an asymptotic expansion for the switching curve of the family of chattering extremals.
We address a nonconvex optimal control problem (OCP) governed by a state-linear control system, with a Bolza cost functional whose terminal term and integrand are representable as differences of convex functions (DC functions), and with additional equality-type constraints whose left-hand sides are also given by Bolza functionals with DC functions. By employing exact penalty theory, we reduce the original constrained OCP to a penalized unconstrained OCP whose objective functional can be represented as the difference of two state-convex Bolza functionals. Exploiting this DC representation of the objective functional of the penalized problem, we establish new conditions for global (approximate) ε -optimality. These conditions provide the basis for efficient local and global search procedures that are not only capable of escaping “local and stationary traps” but also lead to globally ε -optimal controls. We also present several low-dimensional OCPs to demonstrate the effectiveness of the proposed ε -optimality conditions.
The homogeneity-based approach is well recognized in both optimal control and robust finite-time stabilization. M. I. Zelikin was the first to investigate weighted homogeneity in optimal synthesis. This paper is a tribute to Professor Zelikin and his contribution to geometric optimal control. After reviewing the background of homogeneous control applications, we focus on the problem of finite-horizon linear quadratic regulator design. The problem is studied under certain assumptions on the dilation symmetry (homogeneity) of the system. We show that the corresponding optimal solution can be obtained by solving an algebraic Riccati equation, similarly to the infinite-horizon case. The theoretical results are supported by numerical simulations.
The antipolar of a set Ω⊂ℝ^n is the set Ω^♢ = {p∈ℝ^n*| p(x)≥ 1, x∈Ω} . It is known that if the antipolar defined by the Euclidean inner product ℬ on ℝ^n is regarded as a subset of ℝ^n , namely, Ω^♢_ℬ = {y∈ℝ^n|ℬ(y,x)≥ 1, x∈Ω} , then for n>1 there exist infinitely many self-dual sets Ω=Ω^♢_ℬ ; moreover, for n>2 , a complete classification of such sets is unknown. We prove that if ℬ is a nondegenerate bilinear form of signature (1,-1,…,-1) , then for every n≥ 1 there are exactly two self-dual sets, which are the connected components of the hyperboloid {ℬ(x,x)≥ 1} .
We study the motion of a finite rod whose center of mass coincides with its geometric center in a planar medium consisting of noninteracting point particles. Collisions between the particles and the rod are perfectly elastic, and after colliding with the rod the particles no longer interact with it. We prove that the rod motion converges either to a steady state or to a state of motion with nonzero velocity along its axis. Moreover, we show that in the latter case the angular velocity of the rod decays asymptotically as 1/(t^2ln t) when the mass is distributed uniformly along the rod, and as 1/t^2b with some b>1 or as 1/t^2 when the rod’s moment of inertia is, respectively, less than or greater than that corresponding to a uniform mass distribution.
We discuss integrability questions for some ODE systems. We study the structure of the universal covering Lie group G of the full connected isometry Lie group of the Heisenberg group endowed with a left-invariant sub-Riemannian metric and analyze left-invariant sub-Riemannian metrics on G . Special attention is paid to the sub-Riemannian metric on G for which geodesics can be expressed in elementary functions. We prove that the geodesic equations for every left-invariant sub-Riemannian metric on G with a two- or three-dimensional vector subspace V generating the Lie algebra and, in the case V=3 , containing the center of the Lie algebra are integrable in quadratures (but not in elementary functions when V=2 ).
We consider the Cauchy problem for a path-dependent Hamilton–Jacobi equation with coinvariant derivatives subject to a right-end boundary condition. Assuming that the boundary functional is lower semicontinuous and locally bounded from above, we introduce a notion of a minimax (generalized) solution, which is in general discontinuous. We prove existence and uniqueness theorems for such solutions. In addition, we show that the minimax solution is lower semicontinuous and locally bounded from above.
We consider a control problem for a mobile robot with two trailers moving in the plane without obstacles. The trailers are sequentially attached to the robot at coupling points. The control is two-dimensional and unbounded; it corresponds to the linear and angular velocities of the robot. A nonholonomic system arises from the no-slip wheel condition. We study the controllability of this system from an arbitrary initial position of the robot with trailers to an arbitrary final position. We determine the conditions (depending on the locations of the coupling points relative to the robot and trailers) under which the system is completely controllable. In the remaining cases, we establish uncontrollability and describe the corresponding invariant sets.
A well-known result in optimal control of linear time-invariant systems with scalar control states that if the spectrum of the system matrix is real, then the number of switching points of the optimal control is uniformly bounded, irrespective of the length of the time interval. We study the complementary case where the governing matrix has complex eigenvalues and show that, generically, the number of switching points is bounded from below by a linear function of the length of the time interval. Our method is based on establishing a linear lower bound for the L_1(0,T) norm of the switching function. An alternative approach, which we develop in a separate paper, relates the zeros of the switching function to the classical mean motion problem studied by Hermann Weyl.
We consider an N -person noncooperative game and prove the existence of a Nash equilibrium under the assumption that the players’ loss functions are compositions of monotone and continuous functions. As a corollary, we establish sufficient conditions for the existence of a saddle point.
We consider the control system ẋ=u f(x)+v g(x) on the plane, where the smooth vector fields f and g are linearly independent at every point and the controls u and v are nonnegative. We provide a complete description of the reachable set from a fixed point of the plane. In particular, we show that its boundary consists of orbits of the vector fields and may be disconnected. In addition, several equivalent sufficient conditions for the connectedness of the boundary are given.
We study continuously sub-Riemannian differentiable mappings of Carnot groups and construct their compositions according to the graph-mapping principle. For the resulting mappings, we derive analogs of differential properties and an area formula for the image surface.
We provide examples of Finsler and sub-Finsler manifolds whose geodesics exhibit chattering, that is, infinitely many switches occurring within an arbitrarily small time interval. We also present an explicit left-invariant structure on a Carnot group whose geodesics exhibit chattering. This provides a negative answer to a question posed by Le Donne. In addition, we present a sufficient condition for normal extremals of the Pontryagin maximum principle in (sub-)Finsler problems to exhibit chattering.
For a class of closed nonconvex sets in the two-dimensional Euclidean space, we propose an approach to evaluating the Chebyshev layer based on two well-known concepts that generalize the definition of a convex set. We consider a family of planar sets with finitely many pseudovertices and select three sets of pseudovertices for analysis which differ in the order of smoothness of their pseudovertices. Within each of these three cases (the case of a piecewise smooth boundary of the set, the case of a discontinuity in the curvature of the boundary of the set, and the classical case, where the curvature of the boundary is continuous), we find a formula for the limit value of the radii of Efimov–Stechkin support balls. Specifically, we consider balls with centers on a branch of the bisector (a one-dimensional manifold of the nonuniqueness set) corresponding to the chosen pseudovertex. Using these formulas, we evaluate analytically the Chebyshev layers for nonconvex sets, including those with boundaries of variable smoothness. We also give an illustrative example and its interpretation from the viewpoint of optimal control theory.
In the author’s previous paper on adjacency operators of locally finite graphs, it was proven that, for any infinite locally finite graph and any field of characteristic zero, only algebraic over the prime subfield of the field elements (in particular, only algebraic numbers when the field is ℂ ) may not be eigenvalues of the adjacency operator of the graph over the field. There were also given examples of infinite locally finite connected graphs for which certain algebraic numbers are not eigenvalues of their adjacency operators over ℂ . In the present paper, we give examples of infinite locally finite connected graphs for each of which infinitely many algebraic numbers are not eigenvalues of its adjacency operator over ℂ . More precisely, for every prime integer p , we construct an infinite locally finite connected graph such that no positive integer multiple of p is an eigenvalue of the adjacency operator over ℂ of the graph. In addition, in this paper, a necessary condition (based on the results of the abovementioned previous paper) is given for an algebraic number not to be an eigenvalue of the adjacency operator over ℂ of at least one infinite locally finite connected graph.
This paper deals with a terminal problem of optimal guaranteed control for a linear continuous system with disturbances whose output signals are measured with a bounded error. We formulate a problem of constructing an optimal multi-closed control strategy, and define an optimal closed-loop measurement feedback on its basis. Algorithms for calculating optimal strategies and for implementing optimal closed-loop feedback in real time are proposed.
