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Volume 1, 1995

Discrete & Continuous Dynamical Systems - A

1995 , Volume 1 , Issue 2

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More results on the decay of solutions to nonlinear, dispersive wave equations
Jerry L. Bona and  Laihan Luo
1995, 1(2): 151-193 doi: 10.3934/dcds.1995.1.151 +[Abstract](171) +[PDF](353.0KB)
Abstract:
The asymptotic behaviour of solutions to the generalized regularized long-wave-Burgers equation

$u_{t}+u_x+u^pu_{x}-\nu u_{x x}-u_{x xt}=0$ ($*$)

is considered for $\nu > 0$ and $p \geq 1.$ Complementing recent studies which determined sharp decay rates for these kind of nonlinear, dispersive, dissipative wave equations, the present study concentrates on the more detailed aspects of the long-term structure of solutions. Scattering results are obtained which show enhanced decay of the difference between a solution of ($*$) and an associated linear problem. This in turn leads to explicit expressions for the large-time asymptotics of various norms of solutions of these equations for general initial data for $p > 1$ as well as for suitably restricted data for $p \geq 1.$ Higher-order temporal asymptotics of solutions are also obtained. Our techniques may also be applied to the generalized Korteweg-de Vries-Burgers equation

$u_{t}+u_x+u^pu_{x}-\nu u_{x x}+u_{x x x}=0,$ ($**$)

and in this case our results overlap with those of Dix. The decay of solutions in the spatial variable $x$ for both ($*$) and ($**$) is also considered.

The Riemann problem for scalar CJ-combustion model without convexity
Peng Zhang and  Tong Zhang
1995, 1(2): 195-206 doi: 10.3934/dcds.1995.1.195 +[Abstract](28) +[PDF](952.6KB)
Abstract:
The Riemann problem for scalar Chapman-Jouguet combustion model is considered. The entropy conditions are extended from convex case to nonconvex one. The existence and uniqueness of the corresponding entropy solutions are obtained constructively.The solutions consist of the generalized Chapman-Jouguet detonation and deflagration waves and strong detonation waves as well as noncombustion waves.
The index at infinity of some twice degenerate compact vector fields
A.M. Krasnosel'skii and  Jean Mawhin
1995, 1(2): 207-216 doi: 10.3934/dcds.1995.1.207 +[Abstract](40) +[PDF](189.3KB)
Abstract:
The index at infinity of some compact vector fields associated with Nemytski operators is computed in situations where the linear part is degenerate and the nonlinear part does not satisfy the Landesman-Lazer conditions. Applications are given to the existence and multiplicity of solutions of nonlinear equations depending upon a parameter.
Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems
B. Buffoni and  F. Giannoni
1995, 1(2): 217-222 doi: 10.3934/dcds.1995.1.217 +[Abstract](32) +[PDF](190.5KB)
Abstract:
We deal with indefinite Lagrangian systems of the form

$ x''+\partial_{x}V(x,y)=0,\ x\in R^{n};\qquad \ -y''+\partial_{y}V(x,y)=0,\ y\in R^{m}, $

where $V\in C^{1}(R^{n+m},R)$. We are interested in the existence of a brake periodic orbit of prescribed Hamiltonian. This problem may be considered as a generalization of the classical case $m=0$, for which are known many existence results (Seifert theorem and its developments), and also a generalization of the case $n=1$, whose study has been initiated by Hofer and Toland and which is still under investigation. Here we assume at least a quadratic growth on $V$ in order to find a brake orbit via a linking variational principle.

Suboptimal feedback control for a class of nonlinear systems using spline interpolation
V. Rehbock , K.L. Teo and  L.S. Jennings
1995, 1(2): 223-236 doi: 10.3934/dcds.1995.1.223 +[Abstract](35) +[PDF](522.0KB)
Abstract:
We consider a class of nonlinear quadratic regulator problems where the system dynamics are affine in the control. It has been shown recently that an optimal feedback control law for this class of problems can be given in terms of the solution of a state dependent algebraic Ricatti equation (ARE) at each instance of time. However, in most practical problems it is not possible to find an analytic solution to the ARE and hence numerical schemes to calculate suboptimal controls are required. In this paper, we consider one such scheme based on cubic basis spline interpolation. It is shown that if the chosen partitioning of the state space is sufficiently small, the resulting suboptimal controller leads to a stable closed loop system.
Asymptotic behavior of solutions in nonlinear dynamic elasticity
Irena Lasiecka and  W. Heyman
1995, 1(2): 237-252 doi: 10.3934/dcds.1995.1.237 +[Abstract](32) +[PDF](218.7KB)
Abstract:
A Dynamic system of 2-D nonlinear elasticity with nonlinear interior dissipation is considered. It is assumed that the principal part of elastic operator is perturbed by the unstructured lower order linear terms. Asymptotic behavior of solutions when time $t \rightarrow 0$ is analyzed. It is shown that in the case of zero load applied to the plate, the arbitrarily large decay rates can be achieved provided that both the "damping" coefficient and the "traction" coefficient are suitably large. This result generalizes and extends, to the nonlinear and multidimensional context, the earlier results obtained only for the one-dimensional linear wave equation. In the case of a loaded plate the existence of compact global attractor attracting all solutions is established.
Partially hyperbolic sets from a co-dimension one bifurcation
Todd Young
1995, 1(2): 253-275 doi: 10.3934/dcds.1995.1.253 +[Abstract](29) +[PDF](282.9KB)
Abstract:
We study the saddle-node bifurcation of a partially hyperbolic fixed point in a Lipschitz family of $C^{k}$ diffeomorphisms on a Banach manifold (possibly infinite dimensional) in the case that the fixed point is a saddle along hyperbolic directions and has multiple curves of homoclinic orbits. We show that this bifurcation results in an invariant set which consists of a countable collection of closed invariant curves and an uncountable collection of nonclosed invariant curves which are the topological limits of the closed curves. In addition, it is shown that these curves are $C^k$-smooth and that this invariant set is uniformly partially hyperbolic.
Periodic solutions to unbounded Hamiltonian system
V. Barbu
1995, 1(2): 277-283 doi: 10.3934/dcds.1995.1.277 +[Abstract](56) +[PDF](148.9KB)
Abstract:
The existence of periodic solutions for unbounded Hamiltonian systems in Hilbert spaces is studied.
On a class of nonlinear time optimal control problems
Piermarco Cannarsa and  Carlo Sinestrari
1995, 1(2): 285-300 doi: 10.3934/dcds.1995.1.285 +[Abstract](35) +[PDF](225.6KB)
Abstract:
We consider the minimum time optimal control problem for systems of the form

$ y'(t)=f(y(t),u(t))\,\quad y(t) \in \mathbb{R}^n,\ u(t)\in U \subset \mathbb{R}^d. $

We assume $f(x,U)$ to be a convex set with $C^1$ boundary for all $x\in\mathbb{R}^n$ and the target $\kappa$ to satisfy an interior sphere condition. For such problems we prove necessary and sufficient optimality conditions using the properties of the minimum time function $T(x)$. Moreover, we give a local description of the singular set of $T$.

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