Applications of variational methods to some three-point boundary value problems with instantaneous and noninstantaneous impulses

In this paper, we study the multiple solutions for some second-order p-Laplace differential equations with three-point boundary conditions and instantaneous and noninstantaneous impulses. By applying the variational method and critical point theory the multiple solutions are obtained in a Sobolev space. Compared with other local boundary value problems, the three-point boundary value problem is less studied by variational method due to its variational structure. Finally, two examples are given to illustrate the results of multiplicity

Optimal control in ink-jet printing via instantaneous control

This paper concerns the optimal control of a free surface flow with moving contact line, inspired by an application in ink-jet printing. Surface tension, contact angle and wall friction are taken into account by means of the generalized Navier boundary condition. The time-dependent differential system is discretized by an arbitrary Lagrangian-Eulerian finite element method, and a control problem is addressed by an instantaneous control approach, based on the time discretization of the flow equations. The resulting control procedure is computationally highly efficient and its assessment by numerical tests show its effectiveness in deadening the natural oscillations that occur inside the nozzle and reducing significantly the duration of the transient preceding the attainment of the equilibrium configuration

Three solutions for a three-point boundary value problem with instantaneous and non-instantaneous impulses

In this paper, we consider the multiplicity of solutions for the following three-point boundary value problem of second-order $p$-Laplacian differential equations with instantaneous and non-instantaneous impulses: \begin{equation*} \left\{ {\begin{array}{l} -(\rho(t)\Phi_{p} (u'(t)))'+g(t)\Phi_{p}(u(t))=\lambda f_{j}(t,u(t)),\quad t\in(s_{j},t_{j+1}],\; j=0,1,...,m,\\ \Delta (\rho (t_{j})\Phi_{p}(u'(t_{j})))=\mu I_{j}(u(t_{j})), \quad j=1,2,...,m,\\ \rho (t)\Phi_{p} (u'(t))=\rho(t_{j}^{+}) \Phi_{p} (u'(t_{j}^{+})),\quad t\in(t_{j},s_{j}],\; j=1,2,...,m,\\ \rho(s_{j}^{+})\Phi_{p} (u'(s_{j}^{+}))=\rho(s_{j}^{-})\Phi_{p} (u'(s_{j}^{-})),\quad j=1,2,...,m,\\ u(0)=0, \quad u(1)=\zeta u(\eta), \end{array}} \right. \end{equation*} where \Phi_{p}(u): = |u|^{p-2}u, \; p > 1, \; 0 = s_{0} < t_{1} < s_{1} < t_{2} < ... < s_{m_{1}} < t_{m_{1}+1} = \eta < ... < s_{m} < t_{m+1} = 1, \; \zeta > 0, \; 0 < \eta < 1 , $\Delta (\rho (t_{j})\Phi_{p}(u'(t_{j}))) = \rho (t_{j}^{+})\Phi_{p}(u'(t_{j}^{+}))-\rho (t_{j}^{-})\Phi_{p}(u'(t_{j}^{-}))$ for $u'(t_{j}^{\pm}) = \lim\limits_{t\to t_{j}^{\pm}}u'(t)$, $j = 1, 2, ..., m$, and $f_{j}\in C((s_{j}, t_{j+1}]\times\mathbb{R}, \mathbb{R})$, $I_{j}\in C(\mathbb{R}, \mathbb{R})$. $\lambda\in (0, +\infty)$, $\mu\in\mathbb{R}$ are two parameters. $\rho(t)\geq 1$, $1\leq g(t)\leq c$ for $t\in (s_{j}, t_{j+1}]$, $\rho(t), \; g(t)\in L^{p}[0, 1]$, and $c$ is a positive constant. By using variational methods and the critical points theorems of Bonanno-Marano and Ricceri, the existence of at least three classical solutions is obtained. In addition, several examples are presented to illustrate our main results

Stochastic Variational Integrators

This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevin-type equations for constrained mechanical systems and implements a stochastic analog of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discretized stochastic variational principle. The paper shows that the discrete flow of an SVI is a.s. symplectic and in the presence of symmetry a.s. momentum-map preserving. A first-order mean-square convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigid-bodies interacting via a potential.Comment: 21 pages, 8 figure

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equation (FSIDE) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of an approximate solution, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators and fixed point approach. Furthermore, we examine the convergence of Faedo-Galerkin(F-G) approximate solution to the mild solution of our given problem. Finally, a concrete example involving partial differential equation is provided to validate the main abstract results

Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays

In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance

Variational approach to second-order impulsive dynamic equations on time scales

The aim of this paper is to employ variational techniques and critical point theory to prove some conditions for the existence of solutions to nonlinear impulsive dynamic equation with homogeneous Dirichlet boundary conditions. Also we will be interested in the solutions of the impulsive nonlinear problem with linear derivative dependence satisfying an impulsive condition.Comment: 17 page

Variational approach to p-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses

In this paper, we examine the existence of solutions of p-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses. New criteria guaranteeing the existence of infinitely many solutions are established for the considered problem. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of an energy functional. The main result of the present work is established by using a variational approach and a mountain pass lemma. Finally, an example is given to illustrate our main result