Author

# Vlastimil Pták

Bio: Vlastimil Pták is an academic researcher from Czechoslovak Academy of Sciences. The author has contributed to research in topic(s): Matrix (mathematics) & Matrix analysis. The author has an hindex of 20, co-authored 81 publication(s) receiving 1656 citation(s).

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##### Papers

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202 citations

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TL;DR: The method of nondiscrete mathematical induction is applied to the Newton process and yields a very simple proof of the convergence and sharp apriori estimates.

Abstract: The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].

89 citations

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01 Jan 1994

TL;DR: In this paper, the authors present a brief history of LMIs in control theory and discuss some of the standard problems involved in LMIs, such as linear matrix inequalities, linear differential inequalities, and matrix problems with analytic solutions.

Abstract: Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

10,744 citations

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TL;DR: The role of problems of the form w and z satisfying w = q + Mz, w = or 0, z = or0, zw = 0 play a fundamental role in mathematical programming.

Abstract: : Problems of the form: Find w and z satisfying w = q + Mz, w = or 0, z = or 0, zw = 0 play a fundamental role in mathematical programming. This paper describes the role of such problems in linear programming, quadratic programming and bimatrix game theory and reviews the computational procedures of Lemke and Howson, Lemke, and Dantzig and Cottle.

728 citations

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TL;DR: In this paper, the authors present new conditions ensuring existence, uniqueness, and global asymptotic stability of the equilibrium point for a large class of neural networks, which are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous non-reasing neuron activation functions.

Abstract: In this paper, we present new conditions ensuring existence, uniqueness, and Global Asymptotic Stability (GAS) of the equilibrium point for a large class of neural networks. The results are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous nondecreasing neuron activation functions. Such functions may be unbounded (but not necessarily surjective), may have infinite intervals with zero slope as in a piece-wise-linear model, or both. The conditions on GAS rely on the concept of Lyapunov Diagonally Stable (or Lyapunov Diagonally Semi-Stable) matrices and are proved by employing a class of Lyapunov functions of the generalized Lur'e-Postnikov type. Several classes of interconnection matrices of applicative interest are shown to satisfy our conditions for GAS. In particular, the results are applied to analyze GAS for the class of neural circuits introduced for solving linear and quadratic programming problems. In this application, the principal result here obtained is that these networks are GAS also when the constraint amplifiers are dynamical, as it happens in any practical implementation. >

713 citations

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682 citations

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TL;DR: A survey of computational methods in linear algebra can be found in this article, where the authors discuss the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, and more traditional questions such as algebraic eigenvalue problems and systems with a square matrix.

Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

667 citations