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Solvability regions of affinely parameterized quadratic equations [article]

Krishnamurthy Dvijotham, Hung Nguyen, Konstantin Turitsyn
2017 arXiv   pre-print
Quadratic systems of equations appear in several applications.  ...  The results in this paper are motivated by quadratic systems of equations that describe equilibrium behavior of physical infrastructure networks like the power and gas grids.  ...  CONCLUSIONS We have developed a general framework for computing inner approximations of the solvability regions of affinely parameterized quadratic systems of equations.  ... 
arXiv:1703.08881v1 fatcat:bccngbypufhtnlfkb7u226icau

Planar C2 cubic spline interpolation under geometric boundary conditions

A Ginnis
2002 Computer Aided Geometric Design  
The solvability of the resulting non-linear problem, which is equivalent to a quadratic system with respect to the lengths of the boundary tangent vectors, is exhaustively studied, leading to necessary  ...  A robust scheme for the numerical solution of the quadratic system is presented, and the use of the new boundary conditions is illustrated in the context of three examples.  ...  Special thanks are due to the referee whose suggestions, on adopting a more elegant and compact approach for handling the case of non-parallel unit tangent vectors and non-zero curvatures (Section 3.1)  ... 
doi:10.1016/s0167-8396(02)00091-2 fatcat:sntghwttxfflljddsesziotvsi

Explicit Solutions for Safety Problems Using Control Barrier Functions [article]

Han Wang, Kostas Margellos, Antonis Papachristodoulou
2022 arXiv   pre-print
In this paper we aim at explicitly synthesizing a safe control law as a function of the state for nonlinear control-affine systems with limited control ability.  ...  We address the infeasible cases by solving a parameterized adaptive control Barrier function-based quadratic programming problem.  ...  The proposed approach was based on parameterized control Barrier functions-based quadratic programming.  ... 
arXiv:2204.09380v2 fatcat:linl4ohq5rgete35jfstv77aua

A symbolic-numerical envelope algorithm using quadratic MOS patches

Bohumír Bastl, Jiří Kosinka, Miroslav Lávička
2009 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling on - SPM '09  
Recently, it has been proved that quadratic triangular Bézier patches in R 3,1 belong to the class of MOS surfaces (i.e., surfaces providing rational envelopes of the associated twoparameter family of  ...  Moreover, since quadratic MOS patches are capable of producing C 1 approximations of MSTs, this algorithm offers a good basis for consequent methods, e.g. computing rational approximations of envelopes  ...  For patches of class (v), the exceptional lines can be computed by solving the polynomial equation P (u, v) = 0 of total degree 6 in u, v, which is not generally solvable in radicals.  ... 
doi:10.1145/1629255.1629278 dblp:conf/sma/BastlKL09 fatcat:idakxy3bcvcydcf3cxe6bhgtsq

A Constructive Procedure based on Dynamic Scaling for Adaptive Control of Nonlinearly Parameterized Nonlinear Systems

Xiangbin Liu, Romeo Ortega, Hongye Su, Jian Chu
2011 IFAC Proceedings Volumes  
In this paper this result is extended providing some answers to the questions of characterization of "monotonizable" systems and solvability of the PDE.  ...  a partial differential equation (PDE).  ...  Acknowledgment The authors express their gratitude to Daizhan Cheng for his help in the proof of Proposition 5.  ... 
doi:10.3182/20110828-6-it-1002.01708 fatcat:2lkojtwzyvdznh3pdkgytmlk74

Linearly Solvable Optimal Control [chapter]

K. Dvijotham, E. Todorov
2013 Reinforcement Learning and Approximate Dynamic Programming for Feedback Control  
An example of an optimal control method relying on iterative linearizations of the dynamics (and quadratizations of the cost) is the iterative LQG method [34] .  ...  Linearly Solvable Optimal Control. 1 2 LINEARLY SOLVABLE OPTIMAL CONTROL INTRODUCTION Optimal control is of interest in many fields of science and engineering [4, 21] , and is arguably at the core of robust-yet-efficient  ...  In other words, the controller shifts probability mass from one region of the state space to another.  ... 
doi:10.1002/9781118453988.ch6 fatcat:kb2sfuqgw5fa7fmneim5ij33f4

Steady affine motions and morphs

Jarek Rossignac, Álvar Vinacua
2011 ACM Transactions on Graphics  
We propose to measure the quality of an affine motion by its steadiness, which we formulate as the inverse of its Average Relative Acceleration (ARA).  ...  To facilitate the design of pleasing in-betweening motions that interpolate between an initial and a final pose (affine transformation), B and C, we propose the Steady Affine Morph A SAM is affine-invariant  ...  Solvability is not affected by changing the angle c. Therefore, 579 we only examine the solvability for (a, s) pairs, where unsolvable 580 regions are painted red.  ... 
doi:10.1145/2019627.2019635 fatcat:w23ninwsbveondt2pkpq2lzgmm

Biharmonic Volumetric Mapping Using Fundamental Solutions

Huanhuan Xu, Wuyi Yu, Shiyuan Gu, Xin Li
2013 IEEE Transactions on Visualization and Computer Graphics  
We demonstrate the efficacy of our mapping framework on various geometric models with complex geometry (which are decomposed into subparts with simpler and solvable geometry) or heterogeneous interior  ...  This new computational model aims to facilitate the mapping of solid models with complicated geometry or heterogeneous inner structures.  ...  If the affine transformation is degenerated, e.g., a planar local region is transformed into another planar region, the rank of the coefficient matrix of the system (3) reduces to 3 and the linear system  ... 
doi:10.1109/tvcg.2012.173 pmid:23492380 fatcat:5qw6vv2yyfa4haxsx2ahexhknq

Solving asymmetric variational inequalities via convex optimization

Michele Aghassi, Dimitris Bertsimas, Georgia Perakis
2006 Operations Research Letters  
We thereby identify a class of VIs that includes monotone affine VIs over polyhedra, which may be solved by commercial optimization solvers.  ...  Using duality, we reformulate the asymmetric variational inequality (VI) problem over a conic region as an optimization problem. We give sufficient conditions for the convexity of this reformulation.  ...  Acknowledgments We would like to thank Rob Freund for enligtening discussions of this material. We are also grateful to the reviewers of this paper for their insightful comments.  ... 
doi:10.1016/j.orl.2005.09.006 fatcat:ejo2uq2vnrerrl7rugkq5baauq

A Survey of the S-Lemma

Imre Pólik, Tamás Terlaky
2007 SIAM Review  
The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well.  ...  In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure.  ...  The authors would like to thank Etienne de Klerk, Jean-Baptiste Hiriart-Urruty, Dima Pasechnik, and Gábor Pataki for their comments and suggestions during the development of this paper.  ... 
doi:10.1137/s003614450444614x fatcat:5wnzytza7zhfdfgpqrmi7ccu34

A Lie-Group Approach to Rigid Image Registration [article]

Martin Schröter, Uwe Helmke, Otto Sauer
2010 arXiv   pre-print
locally quadratically convergent algorithms.  ...  In this paper we assume that the correspondence is given either by an Euclidean, or by an affine volume-preserving transformation.  ...  (The region of quadratic convergence is a subset of the region in which the cost function is convex.)  ... 
arXiv:1007.5160v1 fatcat:p4qfsdjp35am3c7rvemnkxtsqm

Polynomial integration on regions defined by a triangle and a conic

David Sevilla, Daniel Wachsmuth
2010 Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10  
We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type for quadratic polynomials f, φ1, φ2 on a plane triangle  ...  The naive approach would involve consideration of the many possible shapes of T ∩ {f ≥ 0} (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables  ...  As an prototype, we consider the minimization of a convex and quadratic functional subject to a linear elliptic partial differential equation and inequality constraints on the control.  ... 
doi:10.1145/1837934.1837968 dblp:conf/issac/SevillaW10 fatcat:tahxcresrzdkziatotcsb36qr4

Neurodynamic Programming and Zero-Sum Games for Constrained Control Systems

M. Abu-Khalaf, F.L. Lewis, Jie Huang
2008 IEEE Transactions on Neural Networks  
control, of nonlinear systems affine in input with the control policy having saturation constraints.  ...  The result is a closed-form representation, on a prescribed compact set chosen a priori, of the feedback strategies and the value function that solves the associated Hamilton-Jacobi-Isaacs (HJI) equation  ...  region of the state space.  ... 
doi:10.1109/tnn.2008.2000204 fatcat:llywvngzozcx7otknn2eliehi4

An Efficient Method to Estimate the Suboptimality of Affine Controllers

M. J. Hadjiyiannis, P. J. Goulart, D. Kuhn
2011 IEEE Transactions on Automatic Control  
We investigate the loss of optimality due to the use of such affine policies.  ...  For such systems, we consider the problem of designing robust causal controllers that minimize the expected value of a convex quadratic cost function, subject to mixed linear state and input constraints  ...  An attractive feature of such affine parameterizations is that they can be shown to be equivalent (in the state feedback case) to parameterizations of control policies as affine functions of prior states  ... 
doi:10.1109/tac.2011.2139390 fatcat:3y4ajhyzuvhrlpgo6y6mu3seq4

Contrast Invariant and Affine sub-pixel Optical Flow

Neus Sabater, Sebastien Leprince, Jean-Philippe Avouac
2012 2012 19th IEEE International Conference on Image Processing  
In addition, the proposed model considers local affine displacements instead of simpler translations.  ...  It is robust to drastic changes in the images' content thanks to an adaptive weighting of the neighboring pixels.  ...  At each scale, when the linear equation is not solvable or when the estimation is unreliable, a bilateral filter [11] extrapolates the disparity field.  ... 
doi:10.1109/icip.2012.6466793 dblp:conf/icip/SabaterLA12 fatcat:lyws4a7a4feozgek7qng66koku
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