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Numerical Algebra, Control & Optimization

2012 , Volume 2 , Issue 3

A special issue
Dedicated to Professor Helmut Maurer on the occasion of his 65th birthday

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Preface
Volker Rehbock and Ryan Loxton
2012, 2(3): i-i doi: 10.3934/naco.2012.2.3i +[Abstract](194) +[PDF](74.2KB)
Abstract:
This issue of Numerical Algebra, Control and Optimization is dedicated to Professor Helmut Maurer on the occasion of his 65th birthday. Professor Maurer has made many outstanding contributions to the field of optimal control, particularly in terms of sufficient optimality conditions, sensitivity analysis, and computational methods for singular and state-constrained optimal control problems. Professor Maurer also has a keen interest in solving practical optimal control models arising in applications such as robotics, economics, chemistry and biomedical engineering.

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Path planning and collision avoidance for robots
Matthias Gerdts, René Henrion, Dietmar Hömberg and Chantal Landry
2012, 2(3): 437-463 doi: 10.3934/naco.2012.2.437 +[Abstract](243) +[PDF](1317.2KB)
Abstract:
An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented. The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set strategy is added to the resolution technique.
Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems
Georg Vossen and Stefan Volkwein
2012, 2(3): 465-485 doi: 10.3934/naco.2012.2.465 +[Abstract](216) +[PDF](478.7KB)
Abstract:
The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal model reduction (H2), balanced truncation (BT), and proper orthogonal decomposition (POD) are studied. The proposed approach is based on semi-discretization of the underlying dynamics for the state and the adjoint equations as a large scale linear time-invariant (LTI) system. This system is reduced to a lower-dimensional one using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The size of the reduced-order system is iteratively increased until the error in the optimal control, computed with the a-posteriori error estimator, satisfies a given accuracy. The method is illustrated with numerical tests.
A direct method for the solution of an optimal control problem arising from image registration
Marcus Wagner
2012, 2(3): 487-510 doi: 10.3934/naco.2012.2.487 +[Abstract](241) +[PDF](2468.4KB)
Abstract:
In the present paper, the the elastic/hyperelastic image registration problem is treated as a multidimensional control problem of Dieudonné-Rashevsky type. For its numerical solution, we describe a direct method based on discretization methods and large-scale optimization techniques. Selected numerical results will be presented and discussed. The quality of the results obtained with the optimal control method competes well with those generated from a standard variational method.
Quadratic order conditions for bang-singular extremals
M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk and Pablo A. Lotito
2012, 2(3): 511-546 doi: 10.3934/naco.2012.2.511 +[Abstract](209) +[PDF](454.0KB)
Abstract:
This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case.
Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions
Walter Alt, Robert Baier, Matthias Gerdts and Frank Lempio
2012, 2(3): 547-570 doi: 10.3934/naco.2012.2.547 +[Abstract](339) +[PDF](298.8KB)
Abstract:
We analyze the Euler discretization to a class of linear-quadratic optimal control problems. First we show convergence of order $h$ for the optimal values of the objective function, where $h$ is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the continuous controls coincide except on a set of measure $O(\sqrt{h})$. Under a slightly stronger assumption on the smoothness of the coefficients of the system equation we obtain an error estimate of order $O(h)$.
Control parameterization for optimal control problems with continuous inequality constraints: New convergence results
Ryan Loxton, Qun Lin, Volker Rehbock and Kok Lay Teo
2012, 2(3): 571-599 doi: 10.3934/naco.2012.2.571 +[Abstract](278) +[PDF](325.0KB)
Abstract:
Control parameterization is a powerful numerical technique for solving optimal control problems with general nonlinear constraints. The main idea of control parameterization is to discretize the control space by approximating the control by a piecewise-constant or piecewise-linear function, thereby yielding an approximate nonlinear programming problem. This approximate problem can then be solved using standard gradient-based optimization techniques. In this paper, we consider the control parameterization method for a class of optimal control problems in which the admissible controls are functions of bounded variation and the state and control are subject to continuous inequality constraints. We show that control parameterization generates a sequence of suboptimal controls whose costs converge to the true optimal cost. This result has previously only been proved for the case when the admissible controls are restricted to piecewise continuous functions.
Optimal control strategies for tuberculosis treatment: A case study in Angola
Cristiana J. Silva and Delfim F. M. Torres
2012, 2(3): 601-617 doi: 10.3934/naco.2012.2.601 +[Abstract](265) +[PDF](342.2KB)
Abstract:
We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions. Numerical simulations are given using data from Angola.
Noether's symmetry Theorem for variational and optimal control problems with time delay
Gastão S. F. Frederico and Delfim F. M. Torres
2012, 2(3): 619-630 doi: 10.3934/naco.2012.2.619 +[Abstract](281) +[PDF](199.0KB)
Abstract:
We extend the DuBois--Reymond necessary optimality condition and Noether's symmetry theorem to the time delay variational setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations and optimal control with delays.
Perturbation feedback control: A geometric interpretation
Heinz Schättler and Urszula Ledzewicz
2012, 2(3): 631-654 doi: 10.3934/naco.2012.2.631 +[Abstract](202) +[PDF](303.9KB)
Abstract:
Perturbation feedback control is a classical procedure in control engineering that is based on linearizing a nonlinear system around some locally optimal nominal trajectory. In the presence of terminal constraints defined by a $k$-dimensional embedded submanifold, the corresponding flow of extremals for the underlying system gives rise to a canonical foliation in the $(t,x)$-space consisting of $(n-k+1)$-dimensional leaves and $k$-dimensional cross sections. In this paper, the connections between the formal computations in the engineering literature and the geometric meaning underlying these constructions are described.

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