Sylvester's Law Of Inertia Research Articles

Spectral stability of multi-solitons for generalized Hamiltonian system I: The Caudrey–Dodd–Gibbon–Sawada–Kotera equation

We consider a generalized Hamiltonian system called the Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, which is a natural extension of the Korteweg–de Vries (KdV) equation. Our main results are two-fold. Firstly, spectral stability of solitons and multi-solitons are presented by employing some inverse scattering transform techniques. The main difficulty arises from the third order eigenvalue problem of the equation. The second result is a Hamilton–Krein index identity, which verifies a stability criterion established by Maddocks and Sachs (1993) in showing the Lyapunov stability of KdV N-solitons. However, due to generalized Hamiltonian system nature of the CDGSK equation, it does not possess the L2 conservation law. The linearized operators around the multi-solitons have no spectra gap any more. The main ingredient of the proof is new operator identities for higher order linearized Hamiltonians motivated by the Sylvester Law of Inertia. Such operator identities are shown by employing the recursion operators of the CDGSK equation.

Matrix-Binary Codes based Genetic Algorithm for path planning of mobile robot

The paper presents the new variant of genetic algorithm using the binary codes through matrix for mobile robot navigation (MRN) in static and dynamic environment. The path planning strategy is established using the trace theory as the optimum controller, Sylvester Law of Inertia (SLI) and matrix simulation. The Matrix-Binary Codes based Genetic Algorithm (MGA) transforms the environment from chaos to array. Importantly, the paper proves the SLI by the real and experimental results for the statement “any robotics randomized environment transforms into the array.” Hence the model is presented as pre-optimized robotics by SLI and post-optimized robotics by trace. The simulation result is presented by using MATLAB software and the result obtained by proposed controller is optimal with respect to path and time when compared to other intelligent navigational controller.

Orbital Stability of Double Solitons for the Benjamin-Ono Equation

In this paper we prove the orbital stability of double solitons for the Benjamin-Ono equation. In the case of the KdV equation, this stability has been proved in [17]. Parts of the proof given there rely on the fact that the Euler-Lagrange equations for the conserved quantities of the KdV equation are ordinary differential equations. Since this is not the case for the Benjamin-Ono equation, new methods are required. Our approach consists in using a new invariant for multi-solitons, and certain new identities motivated by the Sylvester Law of Inertia.

Corrections to "Bisection algorithm for computing the frequency response gain of sampled-data systems - infinite-dimensional congruent transformation approach"

This paper derives a bisection algorithm for computing the frequency response gain of sampled-data systems with their intersample behavior taken into account. The properties of the infinite-dimensional congruent transformation (i.e., the Schur complement arguments and the Sylvester law of inertia) play a key role in the derivation. Specifically, it is highlighted that counting up the numbers of the negative eigenvalues of self-adjoint operators is quite important for the computation of the frequency response gain. This contrasts with the well-known arguments on the related issue of the sampled-data H/sub /spl infin// problem, where the key role is played by the positivity of operators and the loop-shifting technique. The effectiveness of the derived algorithm is demonstrated through a numerical example.

Bisection algorithm for computing the frequency response gain of sampled-data systems - infinite-dimensional congruent transformation approach

This paper derives a bisection algorithm for computing the frequency response gain of sampled-data systems with their intersample behavior taken into account. The properties of the infinite-dimensional congruent transformation (i.e., the Schur complement arguments and the Sylvester law of inertia) play a key role in the derivation. Specifically, it is highlighted that counting up the numbers of the negative eigenvalues of self-adjoint operators is quite important for the computation of the frequency response gain. This contrasts with the well-known arguments on the related issue of the sampled-data H/sub /spl infin// problem, where the key role is played by the positivity of operators and the loop-shifting technique. The effectiveness of the derived algorithm is demonstrated through a numerical example.