Optimal Synthesis of the Zermelo–Markov–Dubins Problem in a Constant Drift Field

We consider the optimal synthesis of the Zermelo–Markov–Dubins problem, that is, the problem of steering a vehicle with the kinematics of the Isaacs–Dubins car in minimum time in the presence of a drift field. By using standard optimal control tools, we characterize the family of control sequences that are sufficient for complete controllability and necessary for optimality for the special case of a constant field. Furthermore, we present a semianalytic scheme for the characterization of an optimal synthesis of the minimum-time problem. Finally, we establish a direct correspondence between the optimal syntheses of the Markov–Dubins and the Zermelo–Markov–Dubins problems by means of a discontinuous mapping.     

Refinements of the Dubins–Savage Inequality

This paper sharpens the Dubins–Savage inequality for certain supermartingales whose conditional moment-generating functions are suitably bounded. In particular, sharper inequalities are derived for generalized gaussian, sub-normal and sub-Poisson sequences. A related inequality due to Khan is also refined.     

Markov–Dubins path via optimal control theory

Markov–Dubins path is the shortest planar curve joining two points with prescribed tangents, with a specified bound on its curvature. Its structure, as proved by Dubins in 1957, nearly 70 years after Markov posed the problem of finding it, is elegantly simple: a selection of at most three arcs are concatenated, each of which is either a circular arc of maximum (prescribed) curvature or a straight line. The Markov–Dubins problem and its variants have since been extensively studied in practical and theoretical settings. A reformulation of the Markov–Dubins problem as an optimal control problem was subsequently studied by various researchers using the Pontryagin maximum principle and additional techniques, to reproduce Dubins’ result. In the present paper, we study the same reformulation, and apply the maximum principle, with new insights, to derive Dubins’ result again. We prove that abnormal control solutions do exist. We characterize these solutions, which were not studied adequately in the literature previously, as a concatenation of at most two circular arcs and show that they are also solutions of the normal problem. Moreover, we prove that any feasible path of the types mentioned in Dubins’ result is a stationary solution, i.e., that it satisfies the Pontryagin maximum principle. We propose a numerical method for computing Markov–Dubins path. We illustrate the theory and the numerical approach by three qualitatively different examples.     

Moments of the Mean of Dubins–Freedman Random Probability Distributions

Explicit formulas are given to recursively generate the moments of the mean M for Dubins–Freedman random distribution functions with arbitrary base measure . Using a standard inversion formula for moments of a distribution on the unit interval, the distribution of M is approximated for several natural choices of . The support of the mean is also considered. It is shown that the support of M is connected whenever is concentrated on the vertical bisector of the unit square S, but may have arbitrarily many gaps otherwise.     

Bojanov–Naidenov Problem for Functions with Asymmetric Restrictions for the Higher Derivative

Ukrainian Mathematical Journal - For given r ∈ N, p, ??, β,μ > 0, we solve the extreme problems$$ underset{a}{overset{b}{int }}{x}_{pm}^q(t) dtto sup,...     

Optimal control of the convective Cahn–Hilliard equation

This article studies the problem for optimal control of the convective Cahn–Hilliard equation in one-space dimension. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.     

Optimal control of the viscous Camassa–Holm equation

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.     

Optimal control of the viscous Dullin–Gottwalld–Holm equation

This paper studies the problem for optimal control of the viscous DGH equation. The existence and uniqueness of weak solution to the equation are proved in a short interval. The optimal control of the viscous DGH equation under boundary condition is given and the existence of optimal solution to the equation is proved.     

Optimal stopping of Markov switching Lévy processes

We consider a finite time horizon optimal stopping of a regime-switching Lévy process. We prove that the value function of the optimal stopping problem can be characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequalities.     

L p -Version of the Dubins–Savage Inequality and Some Exponential Inequalities

The Dubins–Savage inequality is generalized by using the pth (1<p≤2) conditional moment of the martingale differences. This inequality is further extended under suitable conditions when p>2. Another martingale inequality due to Freedman is also generalized when 0<p≤2. Implications of these inequalities for strong convergence are discussed. Some general exponential inequalities are also given for martingales (supermartingales) under suitable conditions.      

Proof of Stability in the Brower–Paul Problem

Doklady Mathematics - We study the stability of equilibrium in the problem known as “a ball on a rotating saddle,” which was first considered by the famous Dutch mathematician Brauer in...     

Optimal Control Problem for Cancer Invasion Reaction–Diffusion System

Abstract

An optimal control problem constrained by a reaction–diffusion mathematical model which incorporates the cancer invasion and its treatment is considered. The state equations consisting of three unknown variables namely tumor cell density, normal cell density, and drug concentration. The main goal of the considered optimal control problem is to minimize the density of cancer cells and decreasing the side effects of treatment. Moreover, existence of a weak solution of brain tumor reaction–diffusion system and the corresponding adjoint system of optimal control problem is also investigated. Further, existence of minimizer for the optimal control problem is established and also the first-order optimality conditions are derived.     

An Optimal Control Problem for a Singular System of Solid–Liquid Phase Transition

In this article, we deal with a control problem for a singular system regarding a phase-field model which describes a solid–liquid transition by the Ginzburg–Landau theory. The purpose is to control the system by the means of the heat supply r able to guide it into a certain state with a solid (or liquid) part in a prescribed subset Ω₀ of the space domain Ω, and maintain it in this state during a period of time. The transition is described by a nonlinear differential system of two equations for the phase field and temperature. The control problem is set for some expressions of the cost functional which might reveal cases of physical interest. An approximating control problem is introduced and the existence of at least an optimal pair is proved. The first-order optimality conditions for the approximating problem are determined and a convergence result is given.     

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.     

Optimal large-time decay of the relativistic Landau–Maxwell system

The Cauchy problem of the relativistic Landau–Maxwell system in R³${\mathbb{R}}^3$ is investigated. For perturbative initial data with suitable regularity and integrability, we obtain the optimal large-time decay rates of the relativistic Landau–Maxwell system. For the proof, a new interactive instant energy functional is introduced to capture the macroscopic dissipation and the very weak electromagnetic dissipation of the linearized system. The iterative method is applied to handle the time-decay rates of the full instant energy functional because of the regularity-loss property of the electromagnetic field.     

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.     

Optimal control of the freezing time in the Hegselmann–Krause dynamics

We study the optimal control problem of minimizing the freezing time in the discrete Hegselmann–Krause (HK) model of opinion dynamics. The underlying model is extended with a set of strategic agents that can freely place their opinion at every time step. Indeed, if suitably coordinated, the strategic agents can significantly lower the freezing time of an instance of the HK model. We give several lower and upper worst-case bounds for the freezing time of a HK system with a given number of strategic agents, while still leaving some gaps for future research.     

On the use of the Borel–Cantelli lemma in Markov chains

In the present paper, we propose technical generalizations of the Borel–Cantelli lemma. These generalizations can be further used to derive strong limit results for Markov chains. In our work, we obtain some strong limit results.     

Asymptotics of Regular Solutions to the Camassa–Holm Problem

Computational Mathematics and Mathematical Physics - A periodic boundary value problem is considered for a modified Camassa–Holm equation, which differs from the well-known classical equation...