Infinite time optimal control and periodicity

For smooth nonlinear systems

Controllability properties of nonlinear behaviors

This paper proposes a topological framework for the analysis of the time shift on behaviors. It is shown that controllability is not a property of the time shift, while chain controllability is. This also leads to a global decomposition of behaviors.

     

A General Deterministic Treatment of Derivatives in Particle Methods

A unified approach to approximating spatial derivatives in particle methods using integral operators is presented. The approach is an extension of particle strength exchange, originally developed for treating the Laplacian in advection–diffusion problems. Kernels of high order of accuracy are constructed that can be used to approximate derivatives of any degree. A new treatment for computing derivatives near the edge of particle coverage is introduced, using “one-sided” integrals that only look for information where it is available. The use of these integral approximations in wave propagation applications is considered and their error is analyzed in this context using Fourier methods. Finally, simple tests are performed to demonstrate the characteristics of the treatment, including an assessment of the effects of particle dispersion, and their results are discussed.     

Covering space for monotonic homotopy of trajectories of control systems

This paper considers monotonic (or causal) homotopy between trajectories of control systems. The main result is the construction of an analogue of the simply connected covering space. The constructed covering Γ(Σ,x) has the structure of a manifold and satisfies the property that two trajectories are monotonic homotopic if and only if the end points of their liftings coincide.     

A Model for Kidney Tissue Damage under High Speed Loading

In a medical procedure to comminute kidney stones the patient is subjected to hypersonic waves focused at the stone. Unfortunately such shock waves also damage the surrounding kidney tissue. We present here a model for the mechanical response of the soft tissue to such a high speed loading regime. The material model combines shear induced plasticity with irreversible volumetric expansion as induced, e.g., by cavitating bubbles. The theory is based on a multiplicative decomposition of the deformation gradient and on an internal variable formulation of continuum thermodynamics. By the use of logarithmic and exponential mappings the stress update algorithms are extended from small‐strain to the finite deformation range. In that way the time‐discretized version of the porous‐viscoplastic constitutive updates is described in a fully variational manner. By numerical experiments we study the shock‐wave propagation into the tissue and analyze the resulting stress states. A first finite element simulation shows localized damage in the human kidney. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Output least squares stability in elliptic systems

In this paper the stability of the solutions of parameter estimation problems in their output least squares formulation is analyzed. The concepts of output least squares stability (OLS stability) is defined and sufficient conditions for this property are proved for abstract elliptic equations. These results are applied to the estimation of the diffusion, convection, and friction coefficient in second-order elliptic equations in ⁿ ,n=2, 3. Results on Tikhonov regularization in a nonlinear setting are also given.Part of this research was carried out while both authors were visitors at the Division of Applied Mathematics at Brown University. The research of F. Colonius was supported in part by the Deutsche Forschungsgemeinschaft. Both authors also acknowledge support from the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under Grant No. S3206.     

Normal Forms for Control Systems at Singular Points

A normal form for open loop control systems is provided, based on their interpretation as skew product flows and on normal forms for nonautonomous differential equations.     

Numerical simulation of shock propagation in a polydisperse bubbly liquid

The effect of distributed bubble nuclei sizes on shock propagation in a bubbly liquid is numerically investigated. An ensemble-averaged technique is employed to derive the statistically averaged conservation laws for polydisperse bubbly flows. A finite-volume method is developed to solve the continuum bubbly flow equations coupled to a single-bubble-dynamic equation that incorporates the effects of heat transfer, liquid viscosity and compressibility. The one-dimensional shock computations reveal that the distribution of equilibrium bubble sizes leads to an apparent damping of the averaged shock dynamics due to phase cancellations in oscillations of the different-sized bubbles. If the distribution is sufficiently broad, the phase cancellation effect can dominate over the single-bubble-dynamic dissipation and the averaged shock profile is smoothed out.