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The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula

Chein-Shan Liu, Botong Li (2024)

Applications of Mathematics

The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an n -dimensional matrix eigenvalue problem is derived with a special matrix 𝐀 : = [ a i j ] , that is, a i j = 0 if i + j is odd.Based on the product formula, an integration method with a fictitious time, namely...

The equivalence of controlled lagrangian and controlled hamiltonian systems

Dong Eui Chang, Anthony M. Bloch, Naomi E. Leonard, Jerrold E. Marsden, Craig A. Woolsey (2002)

ESAIM: Control, Optimisation and Calculus of Variations

The purpose of this paper is to show that the method of controlled lagrangians and its hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity)...

The Equivalence of Controlled Lagrangian and Controlled Hamiltonian Systems

Dong Eui Chang, Anthony M. Bloch, Naomi E. Leonard, Jerrold E. Marsden, Craig A. Woolsey (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity)...

The existence of limit cycle for perturbed bilinear systems

Hanen Damak, Mohamed Ali Hammami, Yeong-Jeu Sun (2012)

Kybernetika

In this paper, the feedback control for a class of bilinear control systems with a small parameter is proposed to guarantee the existence of limit cycle. We use the perturbation method of seeking in approximate solution as a finite Taylor expansion of the exact solution. This perturbation method is to exploit the “smallness” of the perturbation parameter ε to construct an approximate periodic solution. Furthermore, some simulation results are given to illustrate the existence of a limit cycle for...

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