Showing papers in "Journal of Dynamical and Control Systems in 2014"
TL;DR: In this article, the authors considered boundary stabilization for a one-dimensional Euler-Bernoulli equation with boundary moment control and disturbance, and the active disturbance rejection control (ADRC) and sliding mode control (SMC) approaches were adopted in investigation.
Abstract: We consider boundary stabilization for a one-dimensional Euler-Bernoulli equation with boundary moment control and disturbance. The active disturbance rejection control (ADRC) and sliding mode control (SMC) approaches are adopted in investigation. By the ADRC approach, a state feedback disturbance estimator with time-varying gain is designed to estimate the disturbance. It is shown that the closed-loop system is asymptotically stable by canceling the disturbance in the feedback loop with its online estimation. In the second part, the SMC is applied to reject the disturbance. The well-posedness of the closed-loop system via SMC is proven, and the monotonicity of the "reaching condition" is presented without differentiating the sliding mode function which may not always exist for the weak solution. The numerical experiments are presented to illustrate the convergence and the peaking value reduction caused by the constant high gain. In addition, the control energies are compared numerically for two approaches.
45 citations
TL;DR: In this article, the Euler problem on optimal configurations of elastic rods in the plane with fixed endpoints and tangents at the endpoints is considered, and the global structure of the exponential mapping that parameterises extremal trajectories is described.
Abstract: 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.
30 citations
TL;DR: In this article, the authors considered the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2), which comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional.
Abstract: 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.
25 citations
TL;DR: In this paper, left-invariant control affine systems, evolving on three-dimensional matrix Lie groups, are investigated for equivalence and controllability, and a representative is identified for each equivalence class.
Abstract: We consider left-invariant control affine systems, evolving on three-dimensional matrix Lie groups. Equivalence and controllability are investigated. All full-rank systems are classified, under detached feedback equivalence. A representative is identified for each equivalence class. The controllability nature of these representatives is determined.
14 citations
TL;DR: In this paper, the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves, was studied and a positive answer was given.
Abstract: The famous conjecture of V. Ya. Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper, we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: (1) triangular orbits; (2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to k-reflective real analytic pseudo-billiards with odd k, the real piecewise-algebraic Ivrii's conjecture and its analogue in the invisibility theory: Plakhov's invisibility conjecture.
13 citations
TL;DR: In this paper, the authors introduced the concepts of topological entropy and invariance entropy for linear control systems on infinite-dimensional state spaces with finite-dimensional unstable subspaces.
Abstract: For linear (control) systems on infinite-dimensional state spaces with finite-dimensional unstable subspace, this paper introduces the concepts of topological entropy and invariance entropy. For linear dynamical systems on Banach spaces, described by a strongly continuous semigroup, the topological entropy is given by the sum of the real parts of the unstable eigenvalues of the infinitesimal generator. An application is provided by computing the topological entropy of delay equations and of a parabolic partial differential equation. Furthermore, the invariance entropy for infinite-dimensional linear control systems is equal to the topological entropy of the homogeneous equation and so it is also described by the eigenvalues of the infinitesimal generator.
12 citations
TL;DR: In this article, the controllability, observability, and stability of the solution of time-varying Volterra integro-dynamic systems on time scales were investigated.
Abstract: This paper deals with the controllability, observability, and stability of the solution of time-varying Volterra integro-dynamic system on time scales. We obtain new results about controllability and observability and generalize to a time scale some known properties about stability from the continuous case.
10 citations
TL;DR: In this article, the authors discuss the semicontinuities of orbital and limit set maps in an impulsive semidynamical system and investigate their relationships with the stabilities of orbits.
Abstract: In this paper, we discuss the semicontinuities of orbital and limit set maps in an impulsive semidynamical system and investigate their relationships with the stabilities of orbits. Actually, we only deal with infinite impulsive trajectories under the hypotheses that each prolongational set is compact in the phase space. We prove that if the limit set is stable (eventually stable or eventually weakly stable), then the corresponding limit set map is upper semicontinuous or lower semicontinuous, respectively. And if the limit set map is upper semicontinuous (lower semicontinuous), then the corresponding limit set is stable (eventually stable or eventually weakly stable, respectively). Furthermore, we give several sufficient conditions to guarantee that limit sets are minimal.
10 citations
TL;DR: In this article, the authors investigate the structure of reachable sets from a given point q 0 for a class of analytic control affine systems characterized by possessing two singular trajectories initiating at q 0.
Abstract: In this paper, we investigate the structure of reachable sets from a given point q 0 for a class of analytic control affine systems characterized, among other things, by possessing two singular trajectories initiating at q 0. The aim of the paper is to establish the connection between the minimal number of analytic functions needed for describing reachable sets and the number of geometrically optimal singular trajectories. The paper is written in a language of the sub-Lorentzian geometry. Also, the sub-Lorentzian geometry methods are used to prove theorems.
10 citations
TL;DR: In this paper, the authors restrict their attention to sub-Riemannian manifolds where the associated distribution is an Engel distribution, which is a regular and bracket-generating distribution of codimension 2 in a four-dimensional manifold.
Abstract: A sub-Riemannian manifold is a smooth manifold which carries a metric defined only on a smooth distribution $\mbox{$\cal D$}$
. In this paper, we will restrict our attention to sub-Riemannian manifolds where the associated distribution is an Engel distribution which means that $\mbox{$\cal D$}$
is a regular and bracket-generating distribution of codimension 2 in a four-dimensional manifold. We obtain a parallelism on a sub-Riemannian structure of Engel type, and then, we classify all simply connected four-dimensional sub-Riemannian manifolds which are homogeneous spaces by using a canonical linearization of the structure.
10 citations
TL;DR: In this paper, the Coulomb potential of point charges placed at the vertices of a polygonal linkage is considered and it is proved that any convex configuration of a quadrilateral linkage is the point of global minimum of Coulomb Potential for appropriate values of charges of vertices.
Abstract: Equilibria of polygonal linkage with respect to Coulomb potential of point charges placed at the vertices of linkage are considered It is proved that any convex configuration of a quadrilateral linkage is the point of global minimum of Coulomb potential for appropriate values of charges of vertices Similar problems are treated for the equilateral pentagonal linkage Some corollaries and applications in the spirit of control theory are also presented
TL;DR: In this article, the existence of positive solutions to the p?Kirchhoff elliptic problem was studied and the nontrivial solution was obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.
Abstract: In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem
a + μ ? ? N ( | ? u | p + V ( x ) | u | p ) dx ? ? Δ p u + V ( x ) | u | p ? 2 u = f ( x , u ) , in ? N , u ( x ) > 0 , in ? N , u ? D 1 , p ( ? N ) , $$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|
abla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ u\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} $$ ?????(0.1)
where a, μ > 0, ? > 0, and f(x, u) = h 1(x)|u| m?2 u + ? h 2(x)|u| r?2 u with the parameter ? ? ?, 1 0 is continuous in ? N and V(x)?0 as |x|?+?. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.
TL;DR: In this article, the authors considered a controlled stochastic delay partial differential equation withNeumann boundary conditions in which the derivative of the unknown is equal to the sum of the control and of a white noise in time.
Abstract: We consider a controlled stochastic delay partial differential equation withNeumann boundary conditions in which the derivative of the unknown is equal to the sum of the control and of a white noise in time We study the optimal control problem by means of the associated backward stochastic differential equations
TL;DR: In this article, a predator-prey model with nonlinear diffusion and time delay is considered, and the stability is investigated and Hopf bifurcation is demonstrated, based on the normal form theory and the center manifold argument.
Abstract: In this paper, a class of predator---prey model with nonlinear diffusion and time delay is considered. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, main conclusions are included.
TL;DR: In this article, the authors derived a simple formula for the Puiseux characteristic of the curve corresponding to a Goursat germ with given small growth vector, and showed that such germs correspond to finite jets of Legendrian curve germs, and that the RVT coding corresponds to the classical invariant in the singularity theory of planar curves.
Abstract: Germsof Goursat distributions can be classified according to a geometric coding called an RVT code. Jean (ESAIM Control Optim Calc Var. 1:241---266, 1996) and Mormul (Cent Eur J Math 2:859---883, 2004) have shown that this coding carries precisely the same data as the small growth vector. Montgomery and Zhitomirskii (Mem Amer Math Soc 203(956):x+137, 2010) have shown that such germs correspond to finite jets of Legendrian curve germs, and that the RVT coding corresponds to the classical invariant in the singularity theory of planar curves: the Puiseux characteristic. Here, we derive a simple formula, Theorem 2, for the Puiseux characteristic of the curve corresponding to a Goursat germ with given small growth vector. The simplicity of our theorem (compared with the more complex algorithms previously known) suggests a deeper connection between singularity theory and the theory of nonholonomic distributions.
TL;DR: This paper sketches some of the algebraic methods of studying the integrability of dynamical systems, and presents a constructive algorithm issued from the Ziglin’s approach.
Abstract: In this paper, we continue the description of the possibilities to use numerical simulations for mathematically rigorous computer-assisted analysis of integrability of dynamical systems. We sketch some of the algebraic methods of studying the integrability and present a constructive algorithm issued from the Ziglin's approach. We provide some examples of successful applications of the constructed algorithm to physical systems.
TL;DR: In this article, the existence and non-existence of positive solutions to a 3 × 3 competition interaction system with non-negative cross-diffusion under Dirichlet boundary conditions was investigated.
Abstract: In this paper, we report on our investigation of the existence and non-existence of positive solutions to a 3 × 3 competition interaction system with non-negative cross-diffusion under Dirichlet boundary conditions. First, it is shown that the system with constant diffusions can have a positive solution under suitable assumptions even though each of three 2 × 2 sub-systems coupled from the three equations of the system does not have a positive solution. Second, we show that the emergence of cross-diffusion in one equation of the system may generate a positive solution in case that the corresponding competition interaction system without cross-diffusion does not have a positive solution.
TL;DR: In this paper, the structure of analytic continuation of solutions of an even rank system of linear ordinary differential equations of Okubo normal form (ONF) was studied and an adjustment of the method for evaluating the connection formulas of the Gauss hypergeometric function by means of the Euler integral to the system of ONF was developed.
Abstract: We study the structure of analytic continuation of solutions of an even rank system of linear ordinary differential equations of Okubo normal form (ONF). We develop an adjustment of the method for evaluating the connection formulas of the Gauss hypergeometric function by means of the Euler integral to the system of ONF. We obtain recursive relations between connection coefficients for the system of ONF and ones for the underlying system of ONF of half rank.
TL;DR: In this paper, the authors apply the nonlinear superposition principle to nonholonomic systems, in particular, those in chained and power forms, which are used to represent the kinematic equations of various non-holonomic wheeled vehicles.
Abstract: The aim of this paper is to apply the nonlinear superposition principle to some non-holonomic systems, in particular, those in chained and power forms, which are used to represent the kinematic equations of various non-holonomic wheeled vehicles. The existence of nonlinear superposition formulas is studied on the basis of Lie algebraic analysis. First, it is shown that nonlinear superposition formulas can be constructed using the knowledge of n + 1 particular solutions, using the affine structure of a system in chained form and the fact that a system in power form is diffeomorphic to a system in chained form. Secondly, using the notion of first integral, it is shown that only one particular solution is sufficient.
TL;DR: In this article, the authors investigated the finite time blow-up of solutions with supercritical boundary/interior sources and nonlinear boundary and interior damping and proved that the solution grows as an exponential function.
Abstract: The goal of this paper is to investigate the finite time blow-up of solutions with supercritical boundary/interior sources and nonlinear boundary/interior damping. First, we prove that if the interior and boundary sources dominate their corresponding damping terms, then every weak solution blows up in finite time with positive initial energy. Second, without any restriction on the boundary source, we prove the finite time blow-up of solutions, provided that the interior sources dominate both interior and boundary damping and the initial energy is nonnegative. A similar result has been shown when the boundary source is absent. Moreover, in the absence of the interior sources, we prove that the solution grows as an exponential function.
TL;DR: In this article, it was proved that for an open class Γ = {A, ± B} and a generic pair (A, B) in L × L, if S contains a subgroup isomorphic to SL(2,?), associated to an arbitrary root, then S is the whole G. In control theory, this case is specially important since the control system, a? = (A + uB)g, where u??, is controllable on G if and only if S = G.
Abstract: In this paper, we consider a subsemigroup S of a real connected simple Lie group G generated by {exp tX : X ? Γ, t ? 0} for some subset Γ of L, the Lie algebra of G. It is proved that for an open class Γ = {A, ± B} and a generic pair (A, B) in L × L, if S contains a subgroup isomorphic to SL(2, ?), associated to an arbitrary root, then S is the whole G. In a series of previous papers, analogous results have been obtained for the maximal root only. Recently, a similar result for complex connected simple Lie groups was proved. The proof uses special root properties that characterize some particular subalgebras of L. In control theory, this case Γ = {A, ± B} is specially important since the control system, a? = (A + uB)g, where u ? ?, is controllable on G if and only if S = G.
TL;DR: In this article, the stabilization problem of the nonlinear vibrating Timoshenko systems of Kirchhoff-type with boundary control conditions is considered, and the explicit energy decay rates for solutions of the system are established, depending on boundary control feedback.
Abstract: In this work, the stabilization problem of the nonlinear vibrating Timoshenko systems of Kirchhoff-type with boundary control conditions is considered. By virtue of the multiplier method, the explicit energy decay rates for solutions of the system are established, depending on boundary control feedback. In the view of control, the result of this work implies that, by choosing suited feedback boundary controls, the Kirchhoff-type Timoshenko system can be achieved by various decay rates, not only exponential and polynomial.
TL;DR: In this paper, the optimisation of Cauchy problem for partial differential inclusions of parabolic type is considered, and sufficient condition for optimality is derived, where the apparatus of locally conjugate mappings is used.
Abstract: Optimization of Cauchy problem for partial differential inclusions of parabolic type is considered, and sufficient condition for optimality is derived. For derivation of sufficient conditions both for convex and nonconvex partial differential inclusions, the apparatus of locally conjugate mappings is used. Besides, some special limiting conditions such as the equality of the coordinate-wise limits of the conjugate variables and their derivatives are formulated. The obtained results are generalized to the multidimensional cases. The linear problem illustrates the fact that some of the conjugate variables in concrete problems can be eliminated.
TL;DR: In this paper, the authors investigate local and metric geometry of weighted Carnot-Caratheodory spaces, which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations, etc.
Abstract: We investigate local and metric geometry of weighted Carnot---Caratheodory spaces which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations, etc. For such spaces, the intrinsic Carnot---Caratheodory metric might not exist, and some other new effects take place. We describe the local algebraic structure of such a space, endowed with a natural quasimetric (first introduced by A. Nagel, E. M. Stein, and S. Wainger) and compare local geometries of the initial Carnot---Caratheodory (CC) space and its tangent cone at some fixed (possibly nonregular) point. The main results of the present paper, in particular, the theorem on divergence of integral lines and other estimates obtained for the quasimetrics, are new even for the case of sub-Riemannian manifolds.
TL;DR: In this article, phase portraits of a first-order implicit differential equation in a neighborhood of its pleated singular point that is a nondegenerate singular point of the lifted field were studied.
Abstract: We study phase portraits of a first-order implicit differential equation in a neighborhood of its pleated singular point that is a nondegenerate singular point of the lifted field. Although there is no visible local classification of implicit differential equations at pleated singular points (even in the topological category), we show that there exist only six essentially different phase portraits, which are presented.
TL;DR: In this paper, it was shown that the topological entropy of lower dimensional subspaces in the fibers is determined by the Morse spectrum over chain recurrent components of the induced flows on Grassmann bundles.
Abstract: For linear flows on vector bundles, it is shown that the topological entropy of lower dimensional subspaces in the fibers is determined by the Morse spectrum over chain recurrent components of the induced flows on Grassmann bundles.
TL;DR: In this paper, the Dirichlet problem on a coupled system for degenerate parabolic equations is investigated, and the existence, uniqueness of maximum solution, the continuity on the coupled functions, and time-dependent estimates are obtained.
Abstract: The Dirichlet problem on a coupled system for degenerate parabolic equations is investigated in this paper. The aim is to show the existence, uniqueness of maximum solution, the continuity on the coupled functions, and time-dependent estimates.
TL;DR: In this paper, the second-order impulsive boundary value problem is studied and the existence of sign-changing and multiple solutions is obtained based on minimax methods and invariant sets of descending flow.
Abstract: In this paper, we study the second-order impulsive boundary value problem ? Lu = f ( x , u ) , x ? [ 0 , 1 ] ? { x 1 , x 2 , ? , x l } , ? Δ ( p ( x i ) u ? ( x i ) ) = I i ( u ( x i ) ) , i = 1 , 2 , ? , l , R 1 ( u ) = 0 , R 2 ( u ) = 0 , $$\left\{\begin{array}{ll} -Lu=f(x, u), \, \, x\in [0, 1]\backslash\{x_{1}, x_{2}, \cdots, x_{l}\}, \\ -{\Delta} (p(x_{i}) u'(x_{i}))=I_{i}(u(x_{i})), \quad i=1, 2, \cdots, l, \\ R_{1}(u)=0, \, \, \, R_{2}(u)=0, \end{array}\right.$$ where Lu = (p(x)u?)? ? q(x)u is a Sturm-Liouville operator, R 1(u) = ?u?(0) ? βu(0) and R 2(u) = ?u?(1) + ?u(1). The existence of sign-changing and multiple solutions is obtained. The technical approach is based on minimax methods and invariant sets of descending flow.
TL;DR: In this article, an extension of the abstract Hopf bifurcation theorem stated in Jacimovic (Non Anal TMA 73(8):2426---2432, 2010) to abstract integral equations (AIE) and retarded functional differential equations (RFDE) is presented.
Abstract: In this paper, we apply an extension of the abstract Hopf bifurcation theorem stated in Jacimovic (Non Anal TMA 73(8):2426---2432, 2010) to abstract integral equations (AIE) and retarded functional differential equations (RFDE). This yields sufficient conditions of what we refer to as extended Hopf bifurcation for AIE and RFDE, in which we have a relaxation of the non-resonance condition on the eigenvalues of the generator of corresponding semigroup. We illustrate our results with an explicit example of a system of two delay differential equations (DDE), undergoing extended Hopf bifurcation at the resonant eigenvalue.
TL;DR: In this article, qualitative properties of real analytic bounded maps are studied and the Sundman-Poincare method in the Newtonian three-body problem is revisited and applications to collision detection problem are considered.
Abstract: In this work, we study qualitative properties of real analytic bounded maps. The main tool is approximation of real-valued functions analytic in rectangular domains of the complex plane by continued g-fractions of Wall (1948). As an application, the Sundman–Poincare method in the Newtonian three-body problem is revisited and applications to collision detection problem are considered.