Extremal Trajectories Research Articles

Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms

We consider differential systems in RN driven by a nonlinear nonhomogeneous second order differential operator, a maximal monotone term and a multivalued perturbation F(t,u,u′). For periodic systems we prove the existence of extremal trajectories, that is solutions of the system in which F(t,u,u′) is replaced by extF(t,u,u′) (= the extreme points of F(t,u,u′)). For Dirichlet systems we show that the extremal trajectories approximate the solutions of the “convex” problem in the C1(T,RN)-norm (strong relaxation).

Strong Local Optimality for a Bang-Bang-Singular Extremal: The Fixed-Free Case

In this paper we give sufficient conditions for a Pontryagin extremal trajectory, consisting of two bang arcs followed by a partially or totally singular one, to be a strong local minimizer for a Mayer problem. The problem is defined on $\mathbb{R}^n$ and the end-points constraints are of fixed-free type. We use a Hamiltonian approach and its connection with the second order conditions in the form of a linear quadratic accessory problem. An example is proposed. All the results are coordinate free so they also hold on a manifold.

Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) long-time average among solution trajectories, as well as the extremal trajectories themselves. Upper bounds on time averages can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization problem. We prove that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on time averages. Moreover, any nearly minimal auxiliary function provides phase space volumes in which all nearly maximal trajectories are guaranteed to lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming, which we illustrate using the Lorenz system.

Periodic solutions for time-dependent subdifferential evolution inclusions

We consider evolution inclusions driven by a time dependent subdifferential plus a multivalued perturbation. We look for periodic solutions. We prove existence results for the convex problem (convex valued perturbation), for the nonconvex problem (nonconvex valued perturbation) and for extremal trajectories (solutions passing from the extreme points of the multivalued perturbation). We also prove a strong relaxation theorem showing that each solution of the convex problem can be approximated in the supremum norm by extremal solutions. Finally we present some examples illustrating these results.

The brachistochronic motion of a vertical disk rolling on a horizontal plane without slip

This paper deals with the brachistochronic motion of a thin uniform disk rolling on a horizontal plane without slip. The problem is formulated and solved within the frame of the optimal control theory. The brachistochronic motion of the disk is controlled by three torques. The possibility of the realization of the brachistochronic motion found in presence of Coulomb dry friction forces is inspected. Also, the influence of values of the coefficient of dry friction on the structure of the extremal trajectory is analyzed. Two illustrative numerical examples are provided.

Strong-field atomic ionization in an elliptically polarized laser field and a constant magnetic field

Within the framework of the quasistationary quasienergy state (QQES) formalism, the tunneling and multiphoton ionization of atoms and ions subjected to a perturbation by a high intense laser radiation field of an arbitrary polarization and a constant magnetic field are considered. On the basis of the exact solution of the Schr\odinger equation and the Green's function for the electron moving in an arbitrary laser field and crossed constant electric and magnetic fields, the integral equation for the complex quasienergy and the energy spectrum of the ejected electron are derived. Using the ``imaginary-time'' method, the extremal subbarrier trajectory of the photoelectron moving in a nonstationary laser field and a constant magnetic field are considered. Within the framework of the QQES formalism and the quasiclassical perturbation theory, ionization rates when the Coulomb interaction of the photoelectron with the parent ion is taken into account at arbitrary values of the Keldysh parameter are derived. The high accuracy of rates is confirmed by comparison with the results of numerical calculations. Simple analytical expressions for the ionization rate with the Coulomb correction in the tunneling and multiphoton regimes in the case of an elliptically polarized laser beam propagating at an arbitrary angle to the constant magnetic field are derived and discussed. The limits of small and large magnetic fields and low and high frequency of a laser field are considered in details. It is shown that in the presence of a nonstationary laser field perturbation, the constant magnetic field may either decrease or increase the ionization rate. The analytical consideration and numerical calculations also showed that the difference between the ionization rates for an $s$ electron in the case of right- and left-elliptically polarized laser fields is especially significant in the multiphoton regime for not-too-high magnetic fields and decreases as the magnetic field increases. The paper generalizes the results obtained earlier [V. M. Rylyuk, Phys. Rev. A 86, 013402 (2012)].

Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2)

We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group SH(2). We obtain the complete description of the Maxwell points corresponding to the discrete symmetries of the vertical subsystem of the Hamiltonian system. An effective upper bound on the cut time is obtained in terms of the first Maxwell times. We study the local optimality of extremal trajectories and prove the lower and upper bounds on the first conjugate times. We also obtain the generic time interval for the n-th conjugate time which is important in the study of sub-Riemannian wavefront. Based on our results of n-th conjugate time and n-th Maxwell time, we prove a generalization of Rolle's theorem that between any two consecutive Maxwell points, there is exactly one conjugate point along any geodesic.

A Characterization of Switched Linear Control Systems With Finite $L_{2}$-Gain

Motivated by an open problem posed by J. P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the $L_{2}$ -gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one.

Conditions for the absence of jumps of the solution to the adjoint system of the maximum principle for optimal control problems with state constraints

Properties of Lagrange multipliers from the Pontryagin maximum principle for problems with state constraints are investigated. Sufficient conditions for the continuity of the solution of the adjoint equation depending on how the extremal trajectory approaches the state constraint boundary are obtained. The proof uses the notion of closure with respect to measure of a Lebesgue measurable function and the Caratheodory theorem.

Null controllable region of delta operator systems subject to actuator saturation

ABSTRACTIn this paper, we give exact description of null controllable regions for delta operator systems subject to actuator saturation. The null controllable region is in terms of a set of extremal trajectories of anti-stable subsystems. For the delta operator system with real eigenvalues or complex eigenvalues, the description is simplified to an explicit formula which is used to characterise the boundary of a null controllable region. The relations of null controllable regions are shown separately for continuous-time systems, discrete-time systems and delta operator systems. Two numerical examples are given to illustrate the effectiveness of the proposed techniques on null controllable regions.

Optimization problems for WSNs: trade-off between synchronization errors and energy consumption

We discuss a class of optimization problems related to stochastic models of wireless sensor networks (WSNs). We consider a sensor network that consists of a single server node and m groups of identical client nodes. The goal is to minimize the cost functional which accumulates synchronization errors and energy consumption over a given time interval. The control function u(t) = (u1(t),...,um(t)) corresponds to the power of the server node transmitting synchronization signals to the groups of clients. We find the structure of extremal trajectories. We show that optimal solutions for such models can contain singular arcs.

Cut time in sub-riemannian problem on engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.

Extremal Trajectories and Maxwell Strata in Sub-Riemannian Problem on Group of Motions of Pseudo-Euclidean Plane

We consider the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin maximum principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates, the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the xy-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata.

Exponential Mapping in Euler’s Elastic Problem

The Euler problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories is described. It is proved that open domains cut out by the Maxwell strata in the preimage and image of the exponential mapping are mapped diffeomorphically. As a consequence, computation of globally optimal elasticae with given boundary conditions is reduced to solving systems of algebraic equations having unique solutions in the open domains. For certain special boundary conditions, optimal elasticae are presented.

Optimality of affine control system of several species in competition on a sequential batch reactor

In this paper, we analyse the optimality of affine control system of several species in competition for a single substrate on a sequential batch reactor, with the objective being to reach a given (low) level of the substrate. We allow controls to be bounded measurable functions of time plus possible impulses. A suitable modification of the dynamics leads to a slightly different optimal control problem, without impulsive controls, for which we apply different optimality conditions derived from Pontryagin principle and the Hamilton–Jacobi–Bellman equation. We thus characterise the singular trajectories of our problem as the extremal trajectories keeping the substrate at a constant level. We also establish conditions for which an immediate one impulse (IOI) strategy is optimal. Some numerical experiences are then included in order to illustrate our study and show that those conditions are also necessary to ensure the optimality of the IOI strategy.

The optimal rolling of a sphere, with twisting but without slipping

The problem of a sphere rolling on the plane, with twisting but without slipping, is considered. It is required to roll the sphere from one configuration to another in such a way that the minimum of the action is attained. We obtain a complete parametrization of the extremal trajectories and analyse the natural symmetries of the Hamiltonian system of the Pontryagin maximum principle (rotations and reflections) and their fixed points. Based on the estimates obtained for the fixed points we prove upper estimates for the cut time, that is, the moment of time when an extremal trajectory loses optimality. We consider the problem of re-orienting the sphere in more detail; in particular, we find diffeomorphic domains in the pre-image and image of the exponential map which are used to construct the optimal synthesis.Bibliography: 15 titles.

Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints

We consider the optimal control problem of minimizing an objective function that is quadratic in the control over a fixed interval for a multi-input bilinear dynamical system in the presence of control constraints. Such models are motivated by and applied to mathematical models for cancer chemotherapy over an a priori specified fixed therapy horizon. The necessary conditions for optimality of the Pontryagin maximum principle are easily evaluated and give a functional description of optimal controls as continuous functions of states and multipliers. However, there is no a priori guarantee that a numerically computed extremal controlled trajectory is locally optimal. In this paper, we formulate sufficient conditions for strong local optimality that are based on the existence of a bounded solution to a matrix Riccati differential equation. The theory is applied to a $3$-compartment model for multi-drug cancer chemotherapy with cytotoxic and cytostatic agents. The numerical results are compared with those for a corresponding optimal control problem when the objective is taken linear in the controls.

Tools for improving feeding strategies in a SBR with several species

This paper analyzes feeding strategies in a sequential batch reactor (SBR) with the objective of reaching a given (low) substrate level as quickly as possible for a given volume of water. Inside the SBR, several species compete for a single substrate, which leads to a minimal time control problem in which the control variable is the feeding rate. Following Gajardo et al. (2008) SIAM J Control Optim 47(6):2827-2856, we allow the control variable to be a bounded measurable function of time combined with possible impulses associated with instantaneous dilutions. For this problem, the extremal trajectories of the singular arc type are characterized as the strategies used to maintain the substrate at a constant level. Since this optimization problem is difficult to solve, this characterization provides a valuable tool for investigating the optimality of various feeding strategies. Our aim is thus to illustrate the use of this tool by proposing potential optimal feeding strategies, which may then be compared with other more intuitive strategies. This aim was accomplished via several numerical experiments in which two specific strategies are compared.

Conjugate points in nilpotent sub-Riemannian problem on the Engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. This problem is very important as nilpotent approximation of nonholonomic systems in four-dimensional space with two-dimensional control, for instance of a system which describes motion of mobile robot with a trailer. We study local optimality of extremal trajectories and estimate conjugate time in this article.

Time-Open Orbital Transfer in a Transformed Hamiltonian Setting

This paper develops a new technique for the determination of the solutions of the time-open optimal orbital transfer for the unlimited thrust case. The problem is transformed in a setup where it is possible to derive the Hamiltonian. In this setup four initial costates are sufficient to identify any extremal trajectory starting from the initial orbit, with respect to the five necessary with the standard Lawden primer theory. The extremal flow has three invariant manifolds on which it is possible to obtain global results. The method has been used to present some global and local results, and to develop a two-point boundary-problem solution.