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Sachkov Y.L.,Program Systems Institute
ESAIM - Control, Optimisation and Calculus of Variations | Year: 2010

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained. © EDP Sciences, SMAI, 2009.


Andresen B.,Copenhagen University | Hoffmann K.H.,TU Chemnitz | Nulton J.,San Diego State University | Tsirlin A.,Program Systems Institute | Salamon P.,San Diego State University
European Journal of Physics | Year: 2011

We present a solution to the minimum time control problem for a classical harmonic oscillator to reach a target energy ET from a given initial state (qi, pi) by controlling its frequency ω, ωmin ≤ ω ≤ ωmax. A brief synopsis of optimal control theory is included and the solution for the harmonic oscillator problem is used to illustrate the theory. © 2011 IOP Publishing Ltd.


Cai Q.,Nanjing University of Science and Technology | Huang T.,Nanjing University of Science and Technology | Huang T.,Zhejiang Sci-Tech University | Sachkov Y.L.,Program Systems Institute | Yang X.,Nanjing University of Science and Technology
Journal of Dynamical and Control Systems | Year: 2015

Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first study some properties of horizontal curves on E. Second, we prove that time-like normal geodesics are locally maximizers in the Engel group and calculate the explicit expression of non-space-like geodesics. © 2015 Springer Science+Business Media New York


Sachkov Y.L.,Program Systems Institute
ESAIM - Control, Optimisation and Calculus of Variations | Year: 2010

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described. © 2009 EDP Sciences, SMAI.


Butt Y.A.,Mohammad Ali Jinnah University | Sachkov Y.L.,Program Systems Institute | Bhatti A.I.,Mohammad Ali Jinnah University
Journal of Dynamical and Control Systems | Year: 2016

Global optimality analysis in sub-Riemannian problem on the Lie group SH(2) is considered. We cutout open dense domains in the preimage and in the image of the exponential mapping based on the description of Maxwell strata. We then prove that the exponential mapping restricted to these domains is a diffeomorphism. Based on the proof of diffeomorphism, the cut time, i.e., time of loss of global optimality, is computed on SH(2). We also consider the global structure of the exponential mapping and obtain an explicit description of cut locus and optimal synthesis. © 2016 Springer Science+Business Media New York


Sachkov Y.L.,Program Systems Institute
ESAIM - Control, Optimisation and Calculus of Variations | Year: 2011

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed. © 2010 EDP Sciences, SMAI.


Butt Y.A.,Mohammad Ali Jinnah University | Sachkov Y.L.,Program Systems Institute | Bhatti A.I.,Mohammad Ali Jinnah University
Journal of Dynamical and Control Systems | Year: 2016

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. © 2016 Springer Science+Business Media New York


Butt Y.A.,Mohammad Ali Jinnah University | Sachkov Y.L.,Program Systems Institute | Bhatti A.I.,Mohammad Ali Jinnah University
Journal of Dynamical and Control Systems | Year: 2014

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. © 2014 Springer Science+Business Media New York.

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