Analytic Solutions of Systems

In this paper, functional series solutions of the nonlinear analytic system for the unknown state variable x(t), and functional series solutions of the analytic infinite-dimension

General energy decay rates for a weakly damped Timoshenko system

In this paper, we consider the following Timoshenko-type system:

Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

Nonoscillation and Exponential Stability of Delay Differential Equations with Oscillating Coefficients

We prove that under some additional conditions, the nonoscillation of the scalar delay differential equation

Optimal Control on Nilpotent Lie Groups

Let G be a nilpotent Lie group and let = {X ₁,X ₂} be a bracket generating left invariant distribution on G. In this paper we study the left invariant optimal control problem on G defined by the differential equation

The Deligne–Simpson Problem and the Connectedness of some Varieties

The Deligne–Simpson problem is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes or so that there exist irreducible (p+1)-tuples of matrices M _j ∈ C _j or A _j ∈ c _j satisfying the equality M ₁ ⋯ M _p+1 = I or A ₁ + ⋯ + A _p+1 = 0. The matrices M _j and A _j are interpreted as monodromy operators of regular linear systems and as matrices-residues of Fuchsian ones on the Riemann sphere. We prove that in the so-called simple case the subset or of the variety or consisting of all irreducible (p+1)-tuples (if nonempty) is connected. “Simple” means that the greatest common divisor of all quantities of Jordan blocks of a given size, of a given matrix M _j or A _j , and with a given eigenvalue is 1. To the memory of my mother     

Algorithms for Formal Reduction of Vector Field Singularities

We present several algorithms transforming a Pfaffian form of the following type:

An Estimate for the Entropy of Hamiltonian Flows

In this paper, we present a generalization to Hamiltonian flows on symplectic manifolds of the estimate proved by Ballmann and Wojtkovski in [4] for the dynamical entropy of the geodesic flow on a compact Riemannian manifold of nonpositive sectional curvature. Given such a Riemannian manifold M, Ballmann and Wojtkovski proved that the dynamical entropy h _μ of the geodesic flow on M satisfies the inequalitywhere v is a unit vector in T _p M if p is a point in M, SM is the unit tangent bundle on M, K(v) is defined as , where is the Riemannian curvature of M, and μ is the normalized Liouville measure on SM. We consider a symplectic manifold M of dimension 2n, and a compact submanifold N of M, given by the regular level set of a Hamiltonian function on M; moreover, we consider a smooth Lagrangian distribution on N, and we assume that the reduced curvature of the Hamiltonian vector field with respect to the distribution is non-positive. Then we prove that under these assumptions, the dynamical entropy h _μ of the Hamiltonian flow with respect to the normalized Liouville measure on N satisfies      

Dynamical Entropy for Markov Operators

Let (X,,µ) be a probability space. An operator P : L ¹(X,,µ) L ¹(X,,µ) is called a Markov operator if it satisfies the following conditions:(1)

Bifurcations of Cycles in Systems of Differential Equations with a Finite Symmetry Group – I

One-parameter bifurcations of periodic solutions of differential equations in ⁿ with a finite symmetry group are studied. The following three types of periodic solutions x(t) with the symmetry group H are considered separately. F-cycles: H consists of transformations that do not change the periodic solution, h(x(t)) x(t); S-cycles: H consists of transformations that shift the phase of the solution,

Optimal Control on the Heisenberg Group

Let H denote either the Heisenberg group , or the Cartesian product of n copies of the three-dimensional Heisenberg group . Let {X ₁, Y ₁, ...;, X _n, Y _n} be an independent set of left-invariant vector fields on H. In this paper, we study the left-invariant optimal control problem on H with the dynamics the cost functional with arbitrary positive parameters ₁, ...;, _n , and admissible controls taken from the set of measurable functions The above control system is encoded either in the kernel of a contact 1-form (for ), or in the kernel of a Pfaffian system (for ). In both cases, the action of the semi-direct product of the torus T ⁿ with H describe the symmetries of the problem.The Pontryagin maximum principle provides optimal controls; extremal trajectories are solutions to the Hamiltonian system associated with the problem. Abnormal extremals (which do not depend on the cost functional) yield solutions that are geometrically irrelevant.An explicit integration of the extremal equations provides a tool for studying some aspects of the sub-Riemannian structure defined on H by means of the above optimal control problem.     

Limit Cycles for a Class of Three-Dimensional Polynomial Differential Systems

Perturbing the system inside the family of polynomial differential systems of degree n in , we obtain at most n ² limit cycles using the first-order averaging theory. Moreover, there exist such perturbed systems having at least n ² limit cycles.      

Singularities of Second-Order Implicit Differential Equations: A Geometrical Approach

A second-order implicit differential equation

First- and Second-Order Integral Functionals of the Calculus of Variations which Exhibit the Lavrentiev Phenomenon

We study the possible mechanisms of occurrence of the Lavrentiev phenomenon for the basic problem of the calculus of variations $$\mathcal{J}(x) = \int\limits_0^1 {L(t,x(t),\dot x(t))dt \to \inf , x(0) = x_0 } , x(1) = x_1$$ ,when the infimum of the problem in the class of absolutely continuous functionsW _1,1[0, 1] is strictly less than the infimum of the same problem in the class of Lipshitzian functionsW _1,∞[0, 1]. We suggest an approach to constructing new classes of integrands which exhibit the Lavrentiev phenomenon (Theorem 2.1). A similar method is used to construct (Theorem 3.1) a class of autonomousC ¹-differentiable integrandsL(x, .x, ..x) of the calculus of variations which are regular, i.e., convex, coercive w.r.t. ..x, and exhibit theW _2,1–W _2,∞ Lavrentiev gap, i.e., for some choice of boundary conditions of the variational problem $$\begin{array}{*{20}c} {\mathcal{J}(x( \cdot )) = \int\limits_0^1 {L(x(t),\dot x(t),\ddot x(t)) dt \to \inf ,} } \\ {x(0) = x_0 , \dot x(0) = \upsilon _0 , x(1) = x_1 , \dot x(1) = \upsilon _1 } \\ \end{array}$$ ,the infimum of this problem over the spaceW _{2, 1}[0, 1] is strictly less than its infimum over the spaceW _2,∞[0, 1]. This provides a negative answer to the question of whether functionals with regular autonomous second-order integrands should only have minimizers with essentially bounded second derivative.     

Controllability of the Semilinear Beam Equation

In this paper, we prove the approximate controllability of the following semilinear beam equation: $$ \left\{ \begin{array}{lll} \displaystyle{\partial^{2} y(t,x) \over \partial t^{2}} & = & 2\beta\Delta\displaystyle\frac{\partial y(t,x)}{\partial t}- \Delta^{2}y(t,x)+ u(t,x) + f(t,y,y_{t},u),\; \mbox{in}\; (0,\tau)\times\Omega, \\ y(t,x) & = & \Delta y(t,x)= 0 , \ \ \mbox{on}\; (0,\tau)\times\partial\Omega, \\ y(0,x) & = & y_{0}(x), \ \ y_{t}(x)=v_{0}(x), x \in \Omega, \end{array} \right. $$ in the states space $Z_{1}=D(\Delta)\times L^{2}(\Omega)$ with the graph norm, where β?>?1, Ω is a sufficiently regular bounded domain in IR ^N , the distributed control u belongs to L ²([0,τ];U) (U?=?L ²(Ω)), and the nonlinear function $f:[0,\tau]\times I\!\!R\times I\!\!R\times I\!\!R\longrightarrow I\!\!R$ is smooth enough and there are a,c?∈?IR such that $a<\lambda_{1}^{2}$ and $$ \displaystyle\sup\limits_{(t,y,v,u)\in Q_{\tau}}\mid f(t,y,v,u) - ay -cu\mid<\infty, $$ where Q _τ ?=?[0,τ]×IR×IR×IR. We prove that for all τ?>?0, this system is approximately controllable on [0,τ].     

Confluence of Several Regular Singular Points into an Irregular Singular One

Consider the systems and , where s and t are variables, is a parameter, and A and B = diag(₁,..., _n ) are n by n matrices. (1) has only regular singular points, and (2) has an irregular singular point at t = . Several kinds of special solutions having particular behavior near singular points were selected in previous papers. In the present paper, the author shows how (2) results from (1) in a process of confluence as . It is analyzed how the special solutions of (1) converge to those of (2) in that process. As a consequence new proofs of earlier results about connection problems are obtained.     

Special representations of ℤ n -actions

In this paper we generalize the following statement (Alpern's theorem). Given a relatively prime set $$\{ h_i \} \subset \mathbb{N},i = 1,...,N \leqslant \omega ,$$ , and a probability distribution {α _i }, for any antiperiodicT there is a representation $X = \coprod\nolimits_{i = 1}^N {\left( {\coprod\nolimits_{j = 0}^{h_i - 1} {T^j B_i } } \right)}$ , where μ(B _i )=α _i /h _i . Our main result is the similar statement for free ? ⁿ -actions. Both theorems are generalizations of the well-known Rokhlin-Halmos lemma.     

Controllability of Systems on Solvable Lie Groups: The Generic Case

A Lie group G with Lie algebra is called SID-controllable if there exist such that the (Single Input with Drift) control system , is controllable. This is equivalent to saying that the semigroup generated by is all of G. This definition is due to Sachkov who also classified SID-controllable solvable Lie algebras, cf. [7]-[9], [11]. It turns out that SID-controllability is actually a property of the Lie algebra (rather than of a control system): if a solvable is SID-controllable, then a generic SID-system will be controllable. In this paper we generalize this result to systems with multiple inputs and drifts: G is I _n D _m-controllable if there exist inputs and drifts such that

Null controllability of degenerate/singular parabolic equations

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, λ, we prove global Carleman estimates. The proof is based on an improved Hardy-type inequality.     

Nonlinear evolution inclusions with one-sided perron right-hand side

In a Banach space X with uniformly convex dual, we study the evolution inclusion of the form $ {x}^{\prime}(t)\in Ax(t)+F\left( {x(t)} \right) $ , where A is an m-dissipative operator and F is an upper hemicontinuous multifunction with nonempty convex and weakly compact values. If X^* is uniformly convex and F is one-sided Perron with sublinear growth, then, we prove a variant of the well known Filippov-Pli? theorem. Afterward, sufficient conditions for near viability and (strong) invariance of a set $ K\subseteq \overline{D(A)} $ are established. As applications, we derive ε - δ lower semicontinuity of the solution map and, consequently, the propagation of continuity of the minimum time function associated with the null controllability problem.