In a typical adaptive update law, the rate of adaptation is generally a function of the state feedback error. Ideally, the adaptive update law would also include some feedback of the parameter estimation error. The desire to include some measurable form of the parameter estimation error in the adaptation law resulted in the development of composite adaptive update laws that are functions of a p...

We construct large families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. A Riemann-Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudo-differential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler-...

This paper investigates virtual holonomic constraints for Euler-Lagrange systems with n degreesof-freedom and n− 1 controls. In our framework, a virtual holonomic constraint is a relation specifying n − 1 configuration variables in terms of a single angular configuration variable. The enforcement by feedback of such a constraint induces a desired repetitive behavior in the system. We give condi...

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where aD α t x(t)) and 0 < α < 1, such that the following is the corresponding Euler-Lagrange tD α b ( c aD α t )x(t) + b(t, x(t))( c aD α t x(t)) + f(t, x(t)) = 0. (1) At last, exact solutions for some Euler-Lagrange equations ar...

This paper addresses the classical and discrete Euler-Lagrange equations for systems of n particles interacting quadratically in R. By highlighting the role played by the center of mass of the particles, we solve the previous systems via the classical quadratic eigenvalue problem (QEP) and its discrete transcendental generalization. The roots of classical and discrete QEP being given, we state ...

An effective action is obtained for the area and mass aspect of a thin shell of radiating selfgravitating matter. On following a mini-superspace approach, the geometry of the embedding space-time is not dynamical but fixed to be either Minkowski or Schwarzschild inside the shell and Vaidya in the external space filled with radiation. The Euler-Lagrange equations of motion are discussed and show...

This paper obtains feedback stabilization of an inverted pendulum on a rotor arm by the “method of controlled Lagrangians”. This approach involves modifying the Lagrangian for the uncontrolled system so that the Euler-Lagrange equations derived from the modified or “controlled” Lagrangian describe the closed-loop system. For the closed-loop equations to be consistent with available control inpu...

We investigate essential spectrum of the Euler equation linearized about an arbitrary smooth steady flow in dimension 3. It is proved that for every Lyapunov-Oseledets exponent μ of the associated bicharacteristic-amplitude system, the circle of radius e has a common point with the spectrum. If, in addition, μ is attained on an aperiodic point, then the spectrum contains the entire circle.

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss...

A coupled Euler–Lagrange solution approach is used to model blast loading on a buried structure. The coupling algorithm is discussed along with a benchmark calculation.