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].     

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.     

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.     

Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝN with Vanishing Potential

In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem \(\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ &u(x)>0, \;\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} \) ?????(0.1) where a, μ > 0, τ > 0, and f(x, u) = h ₁(x)|u| ^m?2 u + λ h ₂(x)|u| ^r?2 u with the parameter λ ∈ ?, 1 < p < N, 1 < r < m < \(p^{*}=\frac {pN}{N-p}\) , and the functions h ₁ (x), h ₂(x) ∈ C(?^N) satisfy some conditions. The potential V(x) > 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.     

Solutions of singular control systems on lie groups

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.     

Sign-Changing and Multiple Solutions of Impulsive Boundary Value Problems Via Critical Point Methods

In this paper, we study the second-order impulsive boundary value problem $$\left\{\begin{array}{ll} -Lu=f(x, u), \, \, x\in [0, 1]\backslash\{x_{1}, x_{2}, \cdots, x_{l}\}, \\ -{\Delta} (p(x_{i}) u'(x_{i}))=I_{i}(u(x_{i})), \quad i=1, 2, \cdots, l, \\ R_{1}(u)=0, \, \, \, R_{2}(u)=0, \end{array}\right.$$ where Lu = (p(x)u′)′ ? q(x)u is a Sturm-Liouville operator, R ₁(u) = αu′(0) ? βu(0) and R ₂(u) = γu′(1) + σu(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.     

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

One-parameter bifurcations of periodic solutions of differential equations in ?ⁿ with a finite symmetry group Γ are studied. The following three types of periodic solutions x(t) with the symmetry group H $\subseteq $ Γ 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, $$h\left( {x\left( t \right)} \right) \equiv x\left( {t + {\tau }\left( h \right)} \right)\quad \left( {{\tau }\left( h \right) \ne 0\,{if}\,h \ne e} \right)$$ ? FS-cycles: H consists of transformations of both F and S types. In the present paper bifurcations of F-cycles at double real multipliers and all codimension one bifurcations of S-cycles were studied. In the present paper a more complicated case of a double pair of complex multipliers for F-cycles is considered and bifurcations of FS-cycles are shortly discussed.     

Explicit solutions of the {\mathfrak{a}_1} -type lie-Scheffers system and a general Riccati equation

For a general differential system $ \dot{x}(t) = \sum\nolimits_{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.     

Sphere Bundles Transverse to Holomorphic Vector Fields

We prove that for every pair of nonzero complex numbers λ ₁ and λ ₂ with \(\frac {\lambda _{1}}{\lambda _{2}}\not \in \mathbb {R}\) there is an embedding \(S^{2}\times S^{1}\rightarrow \mathbb {C}^{2}\) transverse to the linear holomorphic vector field \(Z(x,y)=\lambda _{1}x\frac {\partial }{\partial x}+\lambda _{2} y\frac {\partial }{\partial y}\) . This extends a previous result by Ito (1989).     

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.     

Periodic Solutions for a Class of Second Order Delay Systems

The main purpose of this paper is to study the existence and multiplicity of periodic solutions for the following non-autonomous second order delay systems $$ -\frac{{{d^2}u}}{{d{t^2}}}+Au(t)=f\left( {t,u\left( {t-r} \right)} \right), $$ where A = (a _ij ) _n×n is a symmetric matrix, f ∈ C(R × R ⁿ , R ⁿ ) is r-periodic in the first variable, and r > 0 is a given constant. Some existence and multiplicity theorems of 2r-periodic solutions of such systems are obtained via the linking theorem of Benci and Rabinowitz.     

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

Monotonic homotopy for trajectories of young systems

Let $ \mathbb{Y} $ be a Young system. Assume that the accessible set $ \mathcal{A} $ ( $ \mathbb{Y} $ ; x) of $ \mathbb{Y} $ starting from x is locally and semi-locally simply connected by trajectories of $ \mathbb{Y} $ . We prove that the covering space Γ( $ \mathbb{Y} $ ; x) of p-monotonically homotopic trajectories is identified to the universal covering space of $ \mathcal{A} $ ( $ \mathbb{Y} $ ; x).     

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      

Remarks on the entropy of sofic dynamical system of Blackwell's type

Consider a sofic dynamical system (X, T, μ), where X =A ^Z is the full symbolic compact set with the product topology, and A = {0, 1, . . . , d}. The shift is $ T:\left\{ {{x_n}} \right\}\to \left\{ {{{{x^{\prime}}}_n}} \right\},{{x^{\prime}}_n}={x_n}+1 $ . The measure μ is a T-invariant sofic probability measure. For all words a ₁ . . . a _n the measure is μ(a ₁ . . . a _n ) = μ({x : x ₁ = a ₁, . . . , x _n = a _n }) = $ l{m_{{{a_1}}}}\ldots {m_{{{a_n}}}}r $ . Matrices {m ₀, . . . , m _d }, d ≥ 1, are nonzero substochastic matrices of order J. The matrix P = m ₀ +. . . + m _d is a stochastic matrix, the row l is a left P-invariant probability row and all entries of the column r are equal to 1. We obtain an explicit formula for the entropy h(T, μ) of sofic dynamical system of Blackwell’s type for which rank(m _a ) = 1, a ≠ 0.     

Multifractal Spectrum for Barycentric Averages

Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any x∈X and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:

$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$

and we will obtain a variational formula for this multifractal spectrum.

     

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.     

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:

Limit Cycles for Multidimensional Vector Fields. The Elliptic Case

We consider polynomial vector fields of the form and their polynomial perturbations of degree n. We present a sufficient condition that the perturbed system has an invariant surface close to the plane z = 0. We study limit cycles which appear on this surface. The linearized condition for limit cycles, bifurcating from the curves y ² – x ³ + 3x = h, leads to a certain 2- dimensional integral (which generalizes the elliptic integrals). We show that this integral has a representation R ₁(h)I ₁ + + R _e(h)I _e, where R _j (h) are rational functions with degrees of numerators and denominators bounded by O(n). In the case of constant and one-dimensional matrix A(x,y) we estimate the number of zeros of the integral by const n.     

Invariance properties of time measurable differential inclusions and dynamic programming

Take a multifunctionF: [0,T]×? ⁿ → ? ⁿ andP: [0,?T] → ? ⁿ .P is said to be weakly invariant (or “viable”) forF if for anyx ₀∈P(0) there exists a solutionx tox∈F(t, x) satisfying the constraintx(t)∈P(t) for allt. If all solutions with initial value inP(0) satisfy the constraint thenP is strongly invariant forF. Weak and strong invariance are important concepts connected with existence of optimal controls and stabilizing controls, and dynamic programming. Weak and strong invariance have previously been proved for multifunctionsF, which are assumed merely to be measurable with respect to the time variable, under a regularity hypothesis onP (“local absolute continuity from the left”). We show how the regularity hypothesis can be reduced when apriori knowledge is available of a closed subsetI?[0,T] such thatF is upper semicontinuous at all points inI×? ⁿ . The invariance theorems are applied to characterize the value function for optimal control problems with endpoint constraints as the unique generalized solution of the Hamilton-Jacobi equation, under hypotheses which are in certain cases less restrictive than those imposed hitherto.