Showing papers in "Journal of Dynamical and Control Systems in 2012"
TL;DR: In this paper, a complete classification of left-invariant sub-Riemannian structures on three-dimensional Lie groups in terms of the basic differential invariants is given.
Abstract: We give a complete classification of left-invariant sub-Riemannian structures on three-dimensional Lie groups in terms of the basic differential invariants. As a consequence, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and A +( $ \mathbb{R} $ )?×?S 1, where A +( $ \mathbb{R} $ ) denotes the group of orientation preserving affine maps on the real line.
117 citations
TL;DR: In this paper, the authors studied Gevrey asymptotic properties of singularly perturbed singular nonlinear partial differential equations of irregular type in the complex domain and constructed actual holomorphic solutions of these problems with the help of the Borel-Laplace transforms.
Abstract: We study Gevrey asymptotic properties of solutions of singularly perturbed singular nonlinear partial differential equations of irregular type in the complex domain. We construct actual holomorphic solutions of these problems with the help of the Borel---Laplace transforms. Using the Malgrange---Sibuya theorem, we show that these holomorphic solutions have a common formal power series asymptotic expansion of Gevrey order 1 in the perturbation parameter.
36 citations
TL;DR: In this paper, an intrinsic formulation of the kinematic problem of two n-dimensional manifolds rolling one on another without twisting or slipping is presented, where the conditions of no twisting and no slipping are encoded by means of a distribution of rank n. The configuration space of the system is an n(n?+?3)/2-dimensional manifold.
Abstract: We present an intrinsic formulation of the kinematic problem of two n-dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an n(n?+?3)/2-dimensional manifold. The conditions of no-twisting and no-slipping are encoded by means of a distribution of rank n. We compare the intrinsic point of view versus the extrinsic one. We also show that the kinematic system of rolling the n-dimensional sphere over $ {\mathbb{R}^n} $ is controllable. In contrast with this, we show that in the case of SE(3) rolling over $ \mathfrak{s}\mathfrak{e}(3) $ the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.
26 citations
TL;DR: In this article, the authors provided a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation, where the boundary conditions were considered to be Dirichlet or Neumann type related to the degeneracy rate of a(?).
Abstract: The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation $ {u_t} - {\left( {a(x){u_x}} \right)_x} - \frac{\lambda }{{{x^\beta }}}u = 0 $ , (t, x) ? (0, T) × (0, 1), where the diffusion coefficient a(?) is degenerate at x = 0. Also the boundary conditions are considered to be Dirichlet or Neumann type related to the degeneracy rate of a(?). Under some conditions on the function a(?) and parameters β, ?, we prove global Carleman estimates. The proof is based on an improved Hardy-type inequality.
26 citations
TL;DR: In this paper, the authors considered the Cauchy problem for a general inhomogeneous linear partial differential equation with constant coefficients in two complex variables and obtained necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of some functions connected with the inhomogeneity.
Abstract: We consider the Cauchy problem for a general inhomogeneous linear partial differential equation with constant coefficients in two complex variables. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of some functions connected with the inhomogeneity. The results are presented in the general framework of 1/p-fractional equations.
17 citations
TL;DR: In this paper, the concept of dynamic Morse decomposition for an action of a semigroup of homeomorphisms was introduced, and it was shown that the concept is equivalent to Morse decompositions for control systems on manifolds.
Abstract: In this paper, we introduce the concept of dynamic Morse decomposition for an action of a semigroup of homeomorphisms. Conley has shown in [5, Sec. 7] that the concepts of Morse decomposition and dynamic Morse decompositions are equivalent for flows in metric spaces. Here, we show that a Morse decomposition for an action of a semigroup of homeomorphisms of a compact topological space is a dynamic Morse decomposition. We also define Morse decompositions and dynamic Morse decompositions for control systems on manifolds. Under certain condition, we show that the concept of dynamic Morse decomposition for control system is equivalent to the concept of Morse decomposition.
16 citations
TL;DR: In this article, the stability of a linearized inverted pendulum system with two delayed position feedbacks was investigated using the semigroup approach and it was shown that the spectrum of the system is located in the left complex halfplane.
Abstract: In this paper, we examine the stability of a linearized inverted pendulum system with two delayed position feedbacks. The semigroup approach is adopted in investigation for the well-posedness of the closed loop system. We prove that the spectrum of the system is located in the left complex half-plane and its real part tends to???? when the feedback gains satisfy some additional conditions. The asymptotic eigenvalues of the system is presented. By estimating the norm of the Riesz spectrum projection of the system operator that does not have the uniformly upper bound, we show that the eigenfunctions of the system do not form a basis in the state Hilbert space. Furthermore, the spectrum determined growth condition of the system is concluded and the exponential stability of the system is then established. Finally, numerical simulation is presented by applying the MATLAB software.
16 citations
TL;DR: In this paper, for a general differential system, the explicit solution in terms of iterated integrals of products of u d's is obtained by infinite quadratures, and the solution of a general Riccati equation is obtained as a byproduct.
Abstract: For a general differential system $ \dot{x}(t) = \sum
olimits_{d = 1}^3 {u_d } (t){X_d} $ , where X d generates the simple Lie algebra of type $ {\mathfrak{a}_1} $ , we compute the explicit solution in terms of iterated integrals of products of u d 's. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures.
16 citations
TL;DR: In this paper, the authors extend the notion of regional observability of the gradient for linear systems to a class of semilinear parabolic systems and propose a method to reconstruct the gradient in the subregion of the system evolution domain.
Abstract: In this paper, we extend the notion of regional observability of the gradient for linear systems to a class of semilinear parabolic systems. To reconstruct the gradient in the subregion of the system evolution domain, we begin with the first approach which combines the extension of the HUM method and the fixed point techniques. The analytical case is then tackled using sectorial property of the considered dynamic operator and converted to a fixed point problem. The two approaches lead to algorithms which are successfully implemented numerically and illustrated with examples and simulations.
12 citations
TL;DR: In this article, the authors studied phase portraits at singular points of vector fields of the special type where all components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space.
Abstract: We study phase portraits at singular points of vector fields of the special type where all components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. Also we assume some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three applications in differential geometry: singularities of geodesic flows in various singular metrics on two-dimensional manifolds.
12 citations
TL;DR: In this article, a three-dimensional autonomous chaotic system is presented, where three control parameters and three different nonlinear terms in the governed equations are analyzed via theoretical analysis and numerical simulation, and an adaptive control law is derived to make the states of two identical chaotic systems asymptotically synchronized.
Abstract: In this paper, a new three-dimensional autonomous chaotic system is presented. There are three control parameters and three different nonlinear terms in the governed equations. The new chaotic system has six equilibrium points. Basic dynamic properties of the new system are investigated via theoretical analysis and numerical simulation. The nonlinear characteristic of the new chaotic system are demonstrated in terms of equilibria, Jacobian matrices, Lyapunov exponents, a dissipative system, Poincare maps and bifurcations. Then, an adaptive control law is derived to make the states of two identical chaotic systems asymptotically synchronized based on the Lyapunov stability theory. Finally, a numerical simulation is presented to verify the effectiveness of the proposed synchronization scheme.
TL;DR: In this article, a system of differential equations for geodesics is derived from Pontryagin maximum principle and by using Hamiltonian structure for a left-invariant sub-Riemannian metric on a 3D solvable Lie group.
Abstract: In this paper we study geodesics of a left-invariant sub-Riemannian metric on a three-dimensional solvable Lie group. A system of differential equations for geodesics is derived from Pontryagin maximum principle and by using Hamiltonian structure. In a generic case the normal geodesics are described by elliptic functions, and their qualitative behavior is quite complicated.
TL;DR: In this article, the equivalence of the maximum principle in strong pointwise and integral forms is investigated in an optimum control problem in which dynamics is described by a functional-differential equation of pointwise type (a differential equation with deviating argument).
Abstract: The problem of equivalence of the maximum principle in strong pointwise and integral forms is investigated in an optimum control problem in which dynamics is described by a functional-differential equation of pointwise type (a differential equation with deviating argument). The maximum principle in the strong pointwise form is formulated in the aspect of a two-parametrical set of finite-dimensional extremal problems [1,2]. Like it is in ordinary systems, the first parameter is time t and the second parameter is a length of the words which are composed by the generators of finitely generated group Q of homeomorphisms of the real line and which are generated by the functions of a deviation of the argument (for the ordinary systems the corresponding group Q is trivial). The maximum principle in the strong pointwise form follows from the maximum principle in the integral form. The basic obstacle for the equivalence of two forms is a combinatorial condition on the group Q. The presence of noted combinatorial property involves the existence of metric invariants for the group Q that allows to describe its structure in more detail.
TL;DR: In this article, the first-order differential (P C ) inclusions in the gradient form on a square domain were studied and necessary and sufficient conditions for the problems (P A and P C ) were derived in the Euler-Lagrange form.
Abstract: This paper is devoted to optimization of so-called first-order differential (P C ) inclusions in the gradient form on a square domain. As a supplementary problem, discrete-approximation problem (P A ) is considered. In the Euler---Lagrange form, necessary and sufficient conditions are derived for the problems (P A ) and partial differential inclusions (P C ), respectively. The results obtained are based on a new concept of locally adjoint mappings. The duality theorems are proved and duality relation is established.
TL;DR: In this paper, the authors give criteria for a given set K to be approximately weakly invariant with respect to approximate or exact solutions of the differential inclusion in Banach spaces, using the concept of a tangent set to another set.
Abstract: We give here criteria for a given set K to be approximately weakly invariant with respect to approximate or exact solutions of the differential inclusion x?(t) ? F(x(t)) in Banach spaces, using the concept of a tangent set to another set. We give an application regarding the Lipschitz dependence of ?-solutions upon the initial states.
TL;DR: In this article, the reachable sets of the Heisenberg group were studied by the time-like future directed curves and a complete description of the Hamiltonian geodesics was given.
Abstract: Let $ {\mathbb{H}^n} $ be the Heisenberg group in $ {\mathbb{R}^{2n + 1 }} $ and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first study the reachable sets by the time-like future directed curves. Second, we give a complete description of the Hamiltonian geodesics. Third, we compute the time-like conjugate locus of the origin.
TL;DR: In this article, the authors analyze some basic subsets characterized by the tangential singularities of the graph of a real polynomial in two variables and obtain an upper bound of the number of Gaussian cusps that holds even if the parabolic curve is non-convex.
Abstract: Given a surface defined as the graph of a real polynomial in two variables, we analyze some basic subsets characterized by its tangential singularities. If the parabolic curve is compact we provide certain criteria to determine when the unbounded component of its complement is hyperbolic. Moreover, we obtain an upper bound of the number of Gaussian cusps that holds even if the parabolic curve is non compact.
TL;DR: In this article, an age-cycle structured population endowed with a general biological rule is analyzed and the asymptotic behavior of the generated semigroup in the uniform topology is described.
Abstract: In this work, we complete [4] in which we have mathematically analyzed an age-cycle structured population endowed with a general biological rule. We describe the asymptotic behavior of the generated semigroup in the uniform topology.
TL;DR: In this article, the multiplicity of solutions for a class of non-cooperative p(x)-Laplacian operator elliptic systems was studied and the authors obtained a sequence of radially symmetric solutions by using the limit index theory due to Li.
Abstract: In this paper, we study the multiplicity of solutions for a class of noncooperative p(x)-Laplacian operator elliptic system. Under suitable assumptions, we obtain a sequence of radially symmetric solutions by using the Limit Index Theory due to Li [4].
TL;DR: In this article, the authors show a process to determine the solution of a singular control system on a connected Lie group with Lie algebra through every state x?=?yv; where v is any admissible initial condition on V.
Abstract: Let G be a connected Lie group with Lie algebra $$ \mathfrak{g} $$ . A singular control system $$ {\mathcal{S}_G} $$ on G is defined by a pair (E, D) of $$ \mathfrak{g} $$ -derivations. Through a fiber bundle decomposition of TG in [1] ; the authors decompose $$ {\mathcal{S}_G} $$ in two subsystems $$ {\mathcal{S}_{G/V}} $$ and $$ {\mathcal{S}_V} $$ ; as in the linear case on Euclidean spaces, see for instance [9] : Here, $$ V \subset G $$ is the Lie subgroup with Lie algebra $$ \mathfrak{v} $$ ; the generalized 0-eigenspace of E: On the other hand, D defines the drift vector field of the system. We assume that the subspace $$ \mathfrak{v} $$ is invariant under D. With this hypothesis we show a process to determine the solution of $$ {\mathcal{S}_G} $$ through every state x?=?yv; where v is any admissible initial condition on V. From this information, we are able to build the global solution. Finally, in order to illustrate our processes we develop some examples on nilpotent simply connected Lie groups.
TL;DR: In this article, the existence of traveling wave fronts connecting the zero equilibrium and the positive equilibrium is established by using an iterative technique and a nonstandard ordering for the set of profiles of the corresponding wave system.
Abstract: In this paper, we consider a modified disease model with distributed delay. The existence of traveling wave fronts connecting the zero equilibrium and the positive equilibrium is established by using an iterative technique and a nonstandard ordering for the set of profiles of the corresponding wave system. We also study the critical wave speed and give a detailed analysis on its location and asymptotic behavior with respect to the time delay. Our work extends some previous results.
TL;DR: In this paper, it was shown that the non-wandering set of those consists of k fixed points on a closed n-manifold M n (n???4) and that the closure of any separatrix of f t? MS flow (M n, 3) is a locally flat $ \frac{n}{2} $ -sphere while there is f? MS diff (M 4, 3) such that the closed separatrix is a wildly embedded 2-sphere.
Abstract: Let MS flow (M n , k) and MS diff (M n , k) be Morse-Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold M n (n???4). We prove that the closure of any separatrix of f t ? MS flow (M n , 3) is a locally flat $ \frac{n}{2} $ -sphere while there is f t ? MS flow (M n , 4) the closure of separatrix of those is a wildly embedded codimension two sphere. For n???6, one proves that the closure of any separatrix of f ? MS diff (M n , 3) is a locally flat $ \frac{n}{2} $ -sphere while there is f ? MS diff (M 4, 3) such that the closure of any separatrix is a wildly embedded 2-sphere.
TL;DR: In this article, the authors consider a control problem where the state variable is defined as the solution of a variational inequality and obtain a system of optimality to the exact problem via Lagrangian.
Abstract: We consider a control problem where the state variable is defined as the solution of a variational inequality. This system describes the vertical displacement of points of a thin plate with the presence of crack inside [7]. As the control we define the force that originates the deection of the plate. In order to get the system of optimality for the control problem we use a penalized problem [1] and its reformation as a Lagrangian problem. We prove the existence of a Lagrange multiplier to obtain a system of optimality to the exact problem via Lagrangian. Applying the method of bounded increments [19] we get the final result that characterizes the optimal state and control.
TL;DR: In this paper, the dispersion of the infectious agents is assumed to follow a nonlocal diffusion law modelled by a convolution operator with symmetric or asymmetric kernel, and the authors proved the existence of travelling wave fronts of this model.
Abstract: This paper is concerned with a nonlocal version of the man-environment-man epidemic model in which the dispersion of the infectious agents is assumed to follow a nonlocal diffusion law modelled by a convolution operator with symmetric or asymmetric kernel. By constructing appropriate upper and lower solutions, we prove the existence of travelling wave fronts of this model. Moreover, we show that the minimal wave speed exists in this model with symmetric or asymmetric dispersion kernel, and the temporal delay in epidemic model can reduce the speed of epidemic spread.
TL;DR: In this article, an isomorphism stability property for Cartesian products of either flows with joining primeness property or flows which are?-weakly mixing is shown. But this property is not applicable to flows with weak mixing.
Abstract: We show an isomorphism stability property for Cartesian products of either flows with joining primeness property or flows which are ?-weakly mixing.
TL;DR: In this paper, the authors investigate the periodic solutions for a class of non-autonomous Hamiltonian systems and give new sufficient conditions for the existence of multiple periodic solutions, using a decomposition technique of space and variational approaches.
Abstract: In this paper, we investigate the periodic solutions for a class of non-autonomous Hamiltonian systems. By using a decomposition technique of space and variational approaches we give new sufficient conditions for the existence of multiple periodic solutions.
TL;DR: In this article, the modified Michaelis-Menten equation in biochemistry was considered and the integrability of the first integrals of the system was studied. But it was not shown that the system is analytically integrable.
Abstract: In this work we consider the modified Michaelis-Menten equation in biochemistry $$ \dot{x} $$ ?=??a(E???y)x?+?by; $$ \dot{y} $$ ?=?a(E???y)x???(b?+?r)y; $$ \dot{z} $$ ?=?ry: It models the enzyme kinetics. We contribute to the description of the topological structure of its orbits by studying the integrability problem. We prove that a?=?0, or r?=?0, or E?=?0 are the unique values of the parameters for which the system is analytically integrable, and in this case we provide an explicit expression for its first integrals.
TL;DR: In this paper, an n-dimensional nonlinear control system is considered, whose nominal vector field has a double-zero eigenvalue and no other eigen value on the imaginary axis, and the idea is to find under which conditions there exists a scalar control law such that it is possible to establish a priori, that the closed loop system undergoes the controllable Takens-Bogdanov bifurcation.
Abstract: An n-dimensional nonlinear control system is considered, whose nominal vector field has a double-zero eigenvalue and no other eigenvalue on the imaginary axis, and then the idea is to find under which conditions there exists a scalar control law such that it is possible to establish a priori, that the closed loop system undergoes the controllable Takens-Bogdanov bifurcation. One application of this result is discussed.
TL;DR: In this article, it was shown that the monodromy group of a Liouvillian function is almost solvable and the first proof of this theorem was given by A Khovanskii.
Abstract: We present a proof of the fact that the monodromy group of a Liouvillian function is almost solvable The first proof of this theorem was given by A Khovanskii