Problems For Second-order Systems Research Articles

Finite Time Prescribed Performance Control for Uncertain Second-Order Nonlinear Systems

In this article, we discuss the finite time stability problem for second-order systems with an uncertain nonlinear function. A finite time performance function with the sinusoidal function is constructed, and the constrained problem of the original system is transformed into the stability problem of the equivalent system. Combining prescribed performance control and fuzzy logic systems, an effective control method is proposed. The simulation results also prove that the method we adopted is effective.

Bifurcation problems for second order systems

In this paper, we consider the following linear system of second order differential equations (0.1)u′′+A(t)u=0,0≤t<∞where, for each t, A(t) is an n×n matrix with real components, and positive with respect to the usual cone K in Rn. Conditions are provided in order that the first conjugate point T of t=0, i.e. the smallest T such that the above equation has a nontrivial solution u:[0,T]→K satisfying the boundary conditions u(0)=0=u(T)will be a bifurcation point for higher order perturbations of the equation. The paper is mainly motivated by results in Ahmad and Lazer (1997, 1980); Ahmad and Salazar (1981) and Schmitt (1975); Schmitt and Smith (1978). Some additional new consequences are discussed.

Optimal actuation in vibration control

The paper addresses the problem of finding the optimal location of actuators and their relative gain so that the control effort in an actively controlled vibrating system is minimized. In technical terms the problem is finding the optimal input vector of unit norm that minimizes the norm of the control gain vector. This problem is addressed in the context of the active natural frequency modification problem associated with resonance avoidance in undamped systems, and in the context of the single-input-multi-output pole assignment problem for second order systems.

Initial-boundary value problems for second order systems of partial differential equations

We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to ...

The Robust Pole Assignment Problem for Second-Order Systems

Pole assignment problems are special algebraic inverse eigenvalue problems. In this paper, we research numerical methods of the robust pole assignment problem for second-order systems. The problem is formulated as an optimization problem. Depending upon whether the prescribed eigenvalues are real or complex, we separate the discussion into two cases and propose two algorithms for solving this problem. Numerical examples show that the problem of the robust eigenvalue assignment for the quadratic pencil can be solved effectively.

The Tricomi problem for second order systems of mixed type with parabolic degeneracy

The present article deals with the Tricomi problem for systems of second order equations of mixed type with parabolic degeneracy. First the formulation of the problem for the systems is given, next the representations and estimates of solutions for the above problem are obtained, finally the existence of solutions for the problem is proved by the successive iteration and the method of parameter extension.

Second order initial boundary-value problems of variational type

We consider linear hyperbolic boundary-value problems for second order systems, which can be written in the variational form δ L = 0 , with L [ u ] : = ∫ ∫ ( | ∂ t u | 2 − W ( x ; ∇ x u ) ) d x d t , F ↦ W ( x ; F ) being a quadratic form over M d × n ( R ) . The domain of L is the homogeneous Sobolev space H ˙ 1 ( Ω × R t ) n , with Ω either a bounded domain or a half-space of R d . The boundary condition inherent to this problem is of Neumann type. Such problems arise for instance in linearized elasticity. When Ω is a half-space and W depends only on F, we show that the strong well-posedness occurs if, and only if, the stored energy ∫ Ω W ( ∇ x u ) d x is convex and coercive over H ˙ 1 ( Ω ) n . Here, the energy density W does not need to be convex but only strictly rank-one convex. The “only if” part is the new result. A remarkable fact is that the classical characterization of well-posedness by the Lopatinskiĭ condition needs only to be satisfied at real frequency pairs ( τ , η ) with τ ⩾ 0 , instead of pairs with R τ ⩾ 0 . Even stronger is the fact that we need only to examine pairs ( τ = 0 , η ) , and prove that some Hermitian matrix H ( η ) is positive definite. Another significant result is that every such well-posed problem admits a pair of surface waves at every frequency η ≠ 0 . These waves often have finite energy, like the Rayleigh waves in elasticity. When we vary the density W so as to reach non-convex stored energies, this pair bifurcates to yield a Hadamard instability. This instability may occur for some energy densities that are quasi-convex, contrary to the case of the pure Cauchy problem, as shown in several examples. At the bifurcation, the corresponding stationary boundary-value problem enters the class of ill-posed problems in the sense of Agmon, Douglis and Nirenberg. For bounded domains and variable coefficients, we show that the strong well-posedness is equivalent to a Korn-like inequality for the stored energy.

The Brockett Problem for Linear Discrete Control Systems

The Brockett problem for linear discrete control systems is studied. A method for designing time-varying stabilizing feedback is developed. The Brockett problem for second-order systems with scalar feedback and nondegenerate transfer function is shown to have a positive solution.

Systems of Difference Equations Associated with Boundary Value Problems for Second Order Systems of Ordinary Differential Equations

We establish existence results for solutions to boundary value problems for systems of second order difference equations associated with systems of second order ordinary differential equations subject to nonlinear boundary conditions.

On the Numerical Solution of Singular Boundary Value Problems of Second Order by a Difference Method

The standard three-point discretization applied to the numerical solution of nonlinear boundary value problems for second-order systems with a singularity of the first kind is investigated. The results are extended to the boundary value problems arising in practical problems from mechanics and chemistry. A number of numerical examples illustrating the theoretical results is presented.

A difference method for a singular boundary value problem of second order

The standard three-point discretization applied to the numerical solution of linear boundary value problems for second order systems with a singularity at the origin is investigated. A number of numerical examples illustrating the theoretical results are presented.

A Nonlinear Singular Perturbation Problem for Second Order Systems

The existence and asymptotic behavior as $\varepsilon \to 0^ + $ of solutions of nonlinear boundary value problems for second order systems are studied using differential inequality techniques. Conditions are given under which two point problems for $\varepsilon x'' = f(t,x,x',\varepsilon )$ have unique solutions which converge uniformly as $\varepsilon \to 0^ + $ outside boundary layers at each endpoint of width $\sqrt \varepsilon $ to a solution of the reduced equation $0 = f(t,x,x',0)$.

A geometric method of studying two point boundary value problems for second order systems

A geometric method of studying two point boundary value problems for second order systems