# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

2011 , Volume 1 , Issue 2

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2011, 1(2): 211-224 doi: 10.3934/naco.2011.1.211 +[Abstract](2085) +[PDF](249.2KB)
Abstract:
The problem of detecting small parameter variations in linear uncertain systems due to incipient faults, with the possibility of injecting an input signal to enhance detection, is considered. Most studies assume that there is only one fault developing. Recently an active approach for two or more simultaneous faults has been introduced for the discrete time case. In this paper we extend this approach to the continuous time case. A computational method for the construction of an input signal for achieving guaranteed detection with specified precision is presented. The method is an extension of a multi-model approach used for the construction of auxiliary signals for failure detection, however, new technical issues must be addressed.
2011, 1(2): 225-244 doi: 10.3934/naco.2011.1.225 +[Abstract](2666) +[PDF](326.6KB)
Abstract:
We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously differentiable solution and present a method of boundary feedback stabilization. We introduce a Lyapunov function which is a weighted and squared $H^1$-norm of the difference between the non-stationary and the stationary state. We develop boundary feedback conditions which guarantee that the Lyapunov function and the $H^1$-norm of the difference between the non-stationary and the stationary state decay exponentially with time. This allows us also to prove exponential estimates for the $C^0$- and $C^1$-norm.
2011, 1(2): 245-260 doi: 10.3934/naco.2011.1.245 +[Abstract](1785) +[PDF](126.9KB)
Abstract:
We study a turnpike property of approximate solutions of a discrete-time control system with a compact metric space of states. In our recent work we prove this turnpike property and show that it is stable under perturbations of an objective function. In the present paper we improve this turnpike result by showing that it also holds for those solutions defined on a finite interval (domain) which are approximately optimal on all subintervals of the domain that have a fixed length which does not depend on the length of the whole domain.
2011, 1(2): 261-274 doi: 10.3934/naco.2011.1.261 +[Abstract](2079) +[PDF](183.3KB)
Abstract:
The purpose of this paper is to investigate the nonemptiness and boundedness of solution set for a generalized mixed variational inequality problem with strict feasibility in reflexive Banach spaces. We introduce a concept of strict feasibility for the generalized mixed variational inequality problem which includes the existing concepts of strict feasibility introduced for variational inequalities and complementarity problems. By using a degree theory developed in Wang and Huang [28], we prove that the monotone generalized mixed variational inequality has a nonempty bounded solution set if and only if it is strictly feasible. The results presented in this paper generalize and extend some known results in [8, 23].
2011, 1(2): 275-282 doi: 10.3934/naco.2011.1.275 +[Abstract](2248) +[PDF](137.5KB)
Abstract:
This paper addresses the optimal control problem for a linear system with respect to a Bolza-Meyer criterion, where both integral and non-integral terms are of the first degree. The optimal solution is obtained as an impulsive control, whereas the conventional linear feedback control fails to provide a causal solution. The theoretical result is complemented with illustrative examples verifying performance of the designed control algorithm in cases of large and short control horizons.
2011, 1(2): 283-299 doi: 10.3934/naco.2011.1.283 +[Abstract](2478) +[PDF](240.8KB)
Abstract:
Nonlinear rescaling (NR) methods alternate finding an unconstrained minimizer of the Lagrangian for the equivalent problem in the primal space (which is an infinite procedure) with Lagrange multipliers update.
We introduce and study a proximal point nonlinear rescaling (PPNR) method that preserves convergence and retains a linear convergence rate of the original NR method and at the same time does not require an infinite procedure at each step.
The critical component of our analysis is the equivalence of the NR method with dynamic scaling parameter update to the interior quadratic proximal point method for the dual problem in the rescaled from step to step dual space.
By adding the classical quadratic proximal term to the primal objective function the PPNR step can be viewed as a primal-dual proximal point mapping. This allows analyzing a wide variety of non-quadratic augmented Lagrangian methods from unique and general point of view using tools typical for the classical quadratic proximal-point technique.
We proved convergence of the primal-dual PPNR sequence under minimum assumptions on the input data and established a $q$-linear rate of convergence under the standard second-order optimality conditions.
2011, 1(2): 301-316 doi: 10.3934/naco.2011.1.301 +[Abstract](1600) +[PDF](272.3KB)
Abstract:
This paper is concerned with how the QR factors change when a real matrix $A$ suffers from a left or right multiplicative perturbation, where $A$ is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in the QR factors in norm are no bigger than a small constant multiple of the norm of the difference between the perturbation and the identity matrix. One of common cases for a left multiplicative perturbation case naturally arises from the computation of the QR factorization. The newly established bounds can be used to explain the accuracy in the computed QR factors. For a right multiplicative perturbation, the bounds on the relative changes in the QR factors are still dependent upon the condition number of the scaled $R$-factor, however. Some optimized'' bounds are also obtained by taking into account certain invariant properties in the factors.
2011, 1(2): 317-331 doi: 10.3934/naco.2011.1.317 +[Abstract](2158) +[PDF](241.7KB)
Abstract:
In this paper, Robinson's metric regularity of a positive order around/at some point of parametric variational systems is discussed. Under some suitable conditions, the relationships among H$\ddot{o}$lder-likeness, H$\ddot{o}$lder calmness, metric regularity of a positive order and Robinson's metric regularity of a positive order are discussed for the parametric variational systems. Then, some applications to the stabilities of the optimal value map and the solution map are studied for a parametric vector optimization problem, respectively.