Showing papers in "Journal of Dynamical and Control Systems in 2009"
TL;DR: In this paper, the authors investigated the practical stability problem of nonlinear time-varying cascade systems and gave sufficient conditions that guarantee global uniform asymptotic stability and practical global uniform exponential stability.
Abstract: In this paper, we investigate the practical stability problem of nonlinear time-varying cascade systems. We give some sufficient conditions that guarantee practical global uniform asymptotic stability and practical global uniform exponential stability of such dynamical systems.
64 citations
TL;DR: In this paper, a discrete variational optimal control problem for a rigid body is studied, where the cost is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition.
Abstract: In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange---d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra $\mathfrak{s}\mathfrak{o}(3)$ . We use the Lagrange method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.
45 citations
TL;DR: In this article, the authors studied the approximate controllability of semilinear neutral functional-differential systems and impulsive functional differential systems with finite delay, and derived the fraction power theory and the?-norm.
Abstract: In this paper, we study the approximate controllability of semilinear neutral functional-differential systems and impulsive functional-differential systems with finite delay. Since the considered equations admit nonlinear terms involving spatial derivatives, the fraction power theory and ?-norm is used to discuss the problem so that the established results can be applied to them. An example is provided to illustrate the application of the obtained results.
34 citations
TL;DR: In this paper, the authors give sufficient conditions under which there exist actual sectorial holomorphic solutions which are Gevrey asymptotic to the given formal series solutions for given 1-summable formal series initial conditions.
Abstract: We construct formal power series solutions of nonlinear partial integro-differential equations with Fuchsian and irregular singularities at the origin of $$ \mathbb{C}^2 $$ for given initial conditions being formal power series. We give sufficient conditions under which there exist actual sectorial holomorphic solutions which are Gevrey asymptotic to the given formal series solutions for given 1-summable formal series initial conditions. A phenomenon of small divisors is observed for the appearance of singularities of the Borel transform of the constructed formal series due to the presence of the Fuchsian singularity. This property has an effect on the Gevrey asymptotic order for the constructed holomorphic solutions which becomes larger than the Gevrey order of the initial conditions.
30 citations
TL;DR: The existence of multiple positive solutions for the singular Dirichlet boundary value problem was shown in this article, where the fixed point index was used to measure the existence of positive solutions.
Abstract: The existence of multiple positive solutions for the singular Dirichlet boundary-value problem $$\begin{array}{*{20}c} {{x^{{\prime \prime }} + \Phi {\left( t \right)}f{\left( {t,x{\left( t \right)},x^{\prime } {\left( t \right)}} \right)} = 0,\,\,\,\,0 < t < 1,}} \\ {{x{\left( 0 \right)} = 0,x{\left( 1 \right)} = 0,}} \\ \end{array} $$ is presented by using the fixed point index; here f may be singular at x?=?0.
16 citations
TL;DR: In this article, the time-optimal control problem for (2 × 2)-co-operative hyperbolic systems with variable coefficients and involving Laplace operator is considered, where the objective is to steer the initial state ((y(0, y?(0)) so that the observation z(t) hitting a given target set in minimum time.
Abstract: In this paper, the time-optimal control problem for (2 × 2)-co-operative hyperbolic system with variable coefficients and involving Laplace operator is considered. This problem is, steering the initial state ((y(0), y?(0)) so that the observation z(t) hitting a given target set in minimum time. For different cases of the observation, the time-optimal controls are characterized in terms of the adjoint, this characterization (in particular case of the space of controls) is used to derive specific properties of the optimal control (bang-bangness, uniqueness, etc.). These results are extended to the case of (n × n)-co-operative hyperbolic system.
15 citations
TL;DR: In this paper, the existence of global holomorphic solutions of linear q-difference-differential equations with meromorphic coefficients having complex singularities along q-spirals was investigated.
Abstract: We investigate the existence of global holomorphic solutions of linear q-difference-differential equations with meromorphic coefficients having complex singularities along q-spirals. We also study the rate of growth of the constructed solutions near these singular points.
15 citations
TL;DR: In this paper, the authors considered the open-loop system of a weakly coupled linear wave-plate equation with Dirichlet boundary control and collocated observation and showed that the system is wellposed in the sense of D. Salamon and regular in the senses of G. Weiss.
Abstract: We consider the open-loop system of a weakly coupled linear wave-plate equation with Dirichlet boundary control and collocated observation. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. With the multiplier method, the feedthrough operator is explicitly represented.
15 citations
TL;DR: In this paper, it was shown that the degree of singularity of a control-affine optimal control problem is strongly related to the order of the singularity in the problem.
Abstract: An open question contributed by Y. Orlov to a recently published volume Unsolved Problems in Mathematical Systems and Control Theory (V. D. Blondel and A. Megretski, eds.), Princeton Univ. Press (2004), concerns regularization of optimal control-affine problems. These noncoercive problems admit, in general, "cheap (generalized) controls" as minimizers; it has been questioned whether and under what conditions infima of the regularized problems converge to the infimum of the original problem. Starting from a study of this question, we show by simple functional-theoretic reasoning that it admits, in general, a positive answer. This answer does not depend on commutativity/noncommutativity of controlled vector fields. Instead, it depends on the presence or absence of a Lavrentiev gap.
We set an alternative question of measuring "singularity" of minimizing sequences for control-affine optimal control problems by socalled degree of singularity. It is shown that, in the particular case of singular linear-quadratic problems, this degree is tightly related to the "order of singularity" of the problem. We formulate a similar question for nonlinear control-affine problem and establish partial results. Some conjectures and open questions are formulated.
13 citations
TL;DR: In this paper, the authors describe the Zermelo navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method of evaluating its control curvature.
Abstract: The aim of this paper is to describe the Zermelo navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method of evaluating its control curvature. We will show that, up to the change of the Riemannian metric on the manifold, the control curvature of the Zermelo problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss---Bonnet formula in the form of an inequality. This Gauss---Bonnet inequality allows one to generalize the Zermelo problems and obtain a theorem of E. Hopf that establishes the flatness of Riemannian tori without conjugate points.
13 citations
TL;DR: In this paper, the authors design differential inclusions that generate, in one generic format, two distinct phases of system dynamics: the first ensures feasibility in finite time; the second maintains that property forever after.
Abstract: It is common to tolerate that a system's performance be unsustainable during an interim period. To live long however, its state must eventually satisfy various constraints. In this regard we design here differential inclusions that generate, in one generic format, two distinct phases of system dynamics. The first ensures feasibility in finite time; the second maintains that property forever after.
TL;DR: In this paper, the authors reduced the Cauchy problem for discrete inclusions to problem with geometric constraints in Hilbert space and derived necessary and sufficient condition for optimality for both convex and non-convex partial differential inclusions.
Abstract: Optimization of Cauchy problem for discrete inclusions is reduced to problem with geometric constraints in Hilbert space ?2 and necessary and sufficient condition for optimality is derived. Both for convex and non-convex partial differential inclusions the Cauchy type optimization is stated and on the basis of apparatus of locally conjugate mappings sufficient conditions are formulated. The obtained results are generalized to the multidimensional case.
TL;DR: In this article, it was shown that under some additional conditions, the nonoscillation of the scalar delay differential equation implies the exponential stability of the equation, and the same result holds for non-oscillation conditions for equations with positive and negative coefficients and for equations of arbitrary signs.
Abstract: We prove that under some additional conditions, the nonoscillation of the scalar delay differential equation $$ \dot x\left( t \right) + \sum\limits_{k = 1}^m {a_k \left( t \right)x\left( {h_k \left( t \right)} \right) = 0} $$ implies the exponential stability. New nonoscillation conditions are obtained for equations with positive and negative coefficients and for equations of arbitrary signs. As an example, we present an exponentially stable equation with two delays and two oscillating coefficients.
TL;DR: In this article, the Taylor series of a function constructed via elements of the sequence has nonnegative coefficients, and it turns out that this problem is closely connected with properties of core Lie subalgebras of affine control systems which are responsible for a homogeneous approximation.
Abstract: In this paper, the following problem is considered: for a given nondecreasing finite sequence of positive integers, determine if there exists an affine control system whose growth vector coincides with this sequence. The answer is "yes" if and only if the Taylor series of a certain function constructed via elements of the sequence has nonnegative coefficients. It turns out that this problem is closely connected with properties of "core Lie subalgebras" of affine control systems which are responsible for a homogeneous approximation. We give a representation of core Lie subalgebras as free Lie algebras and describe sets of all possible core Lie subalgebras for systems with a fixed growth vector.
TL;DR: In this paper, the authors investigated the existence of mild solutions of second-order initial-values problems for a class of semilinear differential inclusions with nonlocal conditions, using suitable fixed-point theorems for multi-valued maps.
Abstract: In this paper, we investigate the existence of mild solutions of second-order initial-values problems for a class of semilinear differential inclusions with nonlocal conditions. By using suitable fixed-point theorems for multi-valued maps, we study the case where the multi-valued map F has convex or nonconvex values.
TL;DR: In this article, it is shown that no observable systems exist on semisimple groups, and necessary conditions for the existence of such a system on a general Lie group are given.
Abstract: A vector field on a Lie group is linear if its flow is a oneparameter group of automorphisms. A linear system is obtained by adding left invariant controlled vector fields. The observability of such a system, whenever the output function is a Lie group morphism, was studied by Ayala and Hacibekiroglu.
Within this framework, it is shown that no observable systems exist on semisimple groups, and necessary conditions for the existence of such a system on a general Lie group are given.
The case where the output morphism is replaced by the projection on a homogeneous space is briefly discussed.
TL;DR: In this paper, the authors studied the generation problem of Riesz basis for a general network of strings with joint damping at each vertex and showed that the spectrum of the system operator is distributed in a strip parallel to the imaginary axis.
Abstract: In this paper, we study the generation problem of Riesz basis for a general network of strings with joint damping at each vertex. First, we give a basic spectral property of the system operator $$ \mathcal{A} $$ . Under certain conditions, we prove that the spectrum of $$ \mathcal{A} $$ is distributed in a strip parallel to the imaginary axis. By the discussion of the completeness of generalized eigenvectors of the operator $$ \mathcal{A} $$ , we prove further that there exists a sequence of generalized eigenvectors of $$ \mathcal{A} $$ that forms a Riesz basis with parentheses in the Hilbert state space.
TL;DR: For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, a region near the point where the exponential map is a local diffeomorphism is described in this paper.
Abstract: For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. This is proved by taking the Taylor series of the exponential map and calculating the first nonzero term, which has order $ 2{\left( {{{\mathcal{D}}}_{{{\mathcal{H}}}} - n} \right)} $ , where n is the topological dimension and $ {{\mathcal{D}}}_{{{\mathcal{H}}}} $ is the Hausdorff dimension of the metric space associated to the sub-Riemannian manifold.
TL;DR: In this article, it was shown that the local topological linearization implies the local smooth linearization, at generic points, at arbitrary points, and that, except at "strongly" singular points, this homeomorphism can be chosen to be a smooth mapping.
Abstract: We consider the problem of topological linearization of smooth (C ? or C ?) control systems, i.e., of their local equivalence to a linear controllable system via point-wise transformations on the state and the control (static feedback transformations) that are topological but not necessarily differentiable. We prove that the local topological linearization implies the local smooth linearization, at generic points. At arbitrary points, it implies the local conjugation to a linear system via a homeomorphism that induces a smooth diffeomorphism on the state variables, and, except at "strongly" singular points, this homeomorphism can be chosen to be a smooth mapping (the inverse map needs not be smooth). Deciding whether the same is true at "strongly" singular points is tantamount to solve an intriguing open question in differential topology.
TL;DR: In this paper, a non-cooperative hierarchical game approach was applied to a three-dimensional model with five bounded controls and the best optimal strategy for each player was found analytically with the use of the Pontryagin Maximum Principle.
Abstract: In this work, a microeconomic model describing interactions between a manufacturer, retailer and bank is created and investigated. The manufacturer produces a single product, the retailer buys the good in order to resell it to the third party for a profit, and the bank lends the money. For the first time, a non-cooperative hierarchical game approach will be applied to a three-dimensional model with five bounded controls. The best optimal strategy for each player will be found analytically with the use of the Pontryagin Maximum Principle. A simulation software package is developed to demonstrate the performance of our proposed optimal algorithms.
TL;DR: In this paper, the authors introduced a definition of the index for binary differential equations that coincides with the classical Hopf's definition for positive binary differential equation, where a, b, and c are real analytic functions.
Abstract: In this paper, we study binary differential equations a(x, y)dy 2 + 2b(x, y) dx dy + c(x, y)dx 2 = 0, where a, b, and c are real analytic functions. Following the geometric approach of Bruce and Tari in their work on multiplicity of implicit differential equations, we introduce a definition of the index for this class of equations that coincides with the classical Hopf’s definition for positive binary differential equations. Our results also apply to implicit differential equations F(x, y, p) = 0, where F is an analytic function, p = dy/dx, F p = 0, and F pp ≠ 0 at the singular point. For these equations, we relate the index of the equation at the singular point with the index of the gradient of F and index of the 1-form ω = dy − pdx defined on the singular surface F = 0.
TL;DR: In this article, it was shown that there exists an α-rigid transformation with α ≤ 1/2 whose spectrum has a Lebesgue component and introduced a new criterion to identify a large class of αrigid transformations with singular spectrum.
Abstract: We show that there exists an α-rigid transformation with α ≤ 1/2 whose spectrum has a Lebesgue component. This answers the question stated by Klemes and Reinhold in [30]. Moreover, we introduce a new criterion to identify a large class of α-rigid transformations with singular spectrum.
TL;DR: In this paper, a contractible trajectory of a monotonically homotopy trajectory for a driftless control system Σ coincides with the simply connected universal covering manifold of a given initial state.
Abstract: Covering space for monotonic homotopy of trajectories of control systems without drift vector field was considered in [2]. More precisely, we initially proved that for a given initial state x ? M, the covering space Γ(Σ, x) by equivalent classes of monotonically homotopic trajectories of a driftless control system Σ coincides with the simply connected universal covering manifold of M. In this brief note, we improve upon some aspects of the paper [2] by providing an explicit construction of a contractible trajectory of Σ that was needed therein.
TL;DR: In this article, it was shown that for any α e [0, 1/2], there exists an α-rigid transformation whose spectrum has a Lebesgue component.
Abstract: It is shown that for any α e [0, 1/2], there exists an α-rigid transformation whose spectrum has a Lebesgue component. This answers the question posed by Klemes and Reinhold in [7]. We apply a certain correspondence between weak limits of powers of a transformation and their skew products.
TL;DR: In this article, the authors established the wellposedness of solutions and regularity properties for the boundary control systems with nonlocal conditions with the aid of an intermediate property and the contraction mapping principle.
Abstract: In this paper, we establish the well-posedness of solutions and regularity properties for the boundary control systems with nonlocal conditions with the aid of an intermediate property and the contraction mapping principle. A numerical example illustrating our equivalence results is given.
TL;DR: In this article, the authors considered the manifold construction of the irreducible root system of Coxeter type and proved the existence of Bethe and Dunkl manifolds in the Calogero model.
Abstract: Let R be a root system, for example, the root system associated to a semisimple Lie algebra. In [1], V. A. Golubeva and V. P. Lexin constructed two algebraic manifolds (Bethe and Dunkl manifolds) using the "universal" Dunkl operators. These manifolds were defined as subsets of the complex space ? N of dimension equal to the number of roots of the root system under consideration.
The first manifold (Bethe manifold) is characterized by the following property: the Laplace operator constructed by means of Dunkl operators coincides with the "universal" Hamiltonian of the Calogero model. The second one (Dunkl manifold) is characterized by the property: the "universal" Dunkl operators commute.
In this paper, the manifolds associated with the irreducible root system of Coxeter type are considered. We give their construction supposing that these manifolds are embedded in ? N/2. A theoremon the coincidence of Bethe and Dunkl manifolds is proved.