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Optimization for dynamical systems
describes Lyapunov optimization for dynamical systems. It gives an example application to optimal control in queueing networks. Lyapunov optimization refers
Lyapunov_optimization
Function in control theory
In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V ( x ) {\displaystyle V(x)} to systems with control
Control-Lyapunov_function
Concept in the analysis of dynamical systems
theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the
Lyapunov_function
{\displaystyle F} and strictly positive ε {\displaystyle \varepsilon } . Lyapunov optimization Foster, F. G. (1953). "On the Stochastic Matrices Associated with
Foster's_theorem
Algorithm in queueing theory
achieves maximum network throughput, which is established using concepts of Lyapunov drift. Backpressure routing considers the situation where each job can
Backpressure_routing
Numerical optimization process
A sum-of-squares optimization program is an optimization problem with a linear cost function and constraints that certain polynomials constructed from
Sum-of-squares_optimization
Type of control method
tolerant adaptive control. Nonlinear control Intelligent control Lyapunov optimization Annaswamy, Anuradha M. (3 May 2023). "Adaptive Control and Intersections
Adaptive_control
Generalization of finite measure to Banach spaces
Tardella, Fabio (1990). "A new proof of the Lyapunov convexity theorem". SIAM Journal on Control and Optimization. 28 (2): 478–481. doi:10.1137/0328026. MR 1040471
Vector_measure
Robust nonlinear control to achieve exponential stabilization
conventional sliding mode controllers. The control design is underpinned by a Lyapunov stability analysis that utilizes an auxiliary function, often referred
RISE_controllers
Java math library
collection of Java classes for basic numerical analysis, statistics, and optimization. It implements a parallel version of the adaptive strassen's algorithm
SuanShu_numerical_library
Mathematical Theory
remaining time it will be x1(t) = 0. Backpressure routing Lyapunov optimization Convex optimization Linear programming M. J. Neely, "Energy Optimal Control
Drift_plus_penalty
Field of mathematics and science based on non-linear systems and initial conditions
scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: Chaotic electrical circuits, about 1 millisecond;
Chaos_theory
Maximized objective function of an optimization problem
The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters
Value_function
Eduardo D. (1983). "A Lyapunov-like characterization of asymptotic controllability". SIAM Journal on Control and Optimization. 21 (3): 462–471. Isidori
Small_control_property
Probabilistic problem-solving algorithm
issues related to simulation and optimization. The traveling salesman problem is what is called a conventional optimization problem. That is, all the facts
Monte_Carlo_method
{\displaystyle L_{p}} average of the norms of the products in the semigroup. The Lyapunov exponent of the set of matrices characterizes the rate of growth of the
Joint_spectral_radius
Israeli professor of Electrical Engineering
efficient analysis and design methods. Additionally she has introduced the Lyapunov-Krasovskii method for systems with fast-varying delays (without any constraints
Emilia_Fridman
Branch of engineering and mathematics
generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called
Control_theory
Examining complex systems as a whole
shown to exhibit stable behavior given a suitable Lyapunov control function by Aleksandr Lyapunov in 1892. Thermodynamic systems were treated as early
Systems_thinking
Methods of mathematical approximation
Eigenvalue perturbation Homotopy perturbation method Interval finite element Lyapunov stability Method of dominant balance Order of approximation Perturbation
Perturbation_theory
American applied mathematician
theory of stochastic stability (based on the concept of supermartingales as Lyapunov functions), the theory of non-linear filtering (based on the Kushner equation)
Harold_J._Kushner
Property of differential equations describing physical phenomena
in the sense above are termed ill-posed. A simple example is a global optimization problem, because the location of the optima is generally not a continuous
Well-posed_problem
Mathematical biology software
metabolic control analysis, computation of Lyapunov exponent, time scale separation, parameter scans, optimization, and parameter estimation. Importing and
COPASI
Approach to controller design that explicitly deals with uncertainty
performance, and together are sought to be optimized by casting control design as a suitable optimization problem. The ability of feedback to cope with
Robust_control
Sums of sets of vectors are nearly convex
Tardella, Fabio (1990). "A new proof of the Lyapunov convexity theorem". SIAM Journal on Control and Optimization. 28 (2): 478–481. doi:10.1137/0328026. MR 1040471
Shapley–Folkman_lemma
Simulation of a dynamical system of particles
calculated as an intermediate step. Such characteristics include Lyapunov stability, Lyapunov time, various measurements from ergodic theory, etc. Although
N-body_simulation
Form of artificial neural network
Hopfield network has been widely used for optimization. The idea of using the Hopfield network in optimization problems is straightforward: If a constrained/unconstrained
Hopfield_network
Branch of ordinary differential equations
exponents are Lyapunov exponents. The zero solution is asymptotically stable if all Floquet exponents have negative real part. It is Lyapunov stable if all
Floquet_theory
Existence and uniqueness of solutions to initial value problems
corollary Γ will have a unique fixed point. Finally, we have been able to optimize the interval of the solution by taking α = min{a, b/M}. In the end, this
Picard–Lindelöf_theorem
generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called
Jan_Camiel_Willems
Type of functional equation (mathematics)
identity Sturm–Liouville theory Floquet theory Phase portrait Stability Lyapunov Asymptotic Exponential Series solutions Rate of convergence Asymptotic
Differential_equation
Russian mathematician
Pontryagin. M. I. Zelikin was awarded the Chebyshev Award in 1987 and the Lyapunov Award in 2010. He was elected a corresponding member of the Russian Academy
Mikhail_Zelikin
System where changes of output are not proportional to changes of input
especially in Hamiltonian systems Examination of dissipative quantities (see Lyapunov function) analogous to conserved quantities Linearization via Taylor expansion
Nonlinear_system
Polish Professor of automatics and signal processing
Functional Ecology, vol.3, pp. 589–596, 1989. M.Ziółko: Application of Lyapunov Functionals to Studying Stability of Linear Hyperbolic Systems. IEEE Trans
Mariusz_Ziółko
connected networks or recurrent neural networks) and selecting effective optimization algorithms. The choice of deep BSDE network architecture, the number
Deep backward stochastic differential equation method
Deep_backward_stochastic_differential_equation_method
Application of mathematical methods to other fields
mathematical research in its own right, with mathematicians such as Aleksandr Lyapunov, Norbert Wiener, Lev Pontryagin and Fields Medal recipient Pierre-Louis
Applied_mathematics
Method for controlling a prosthesis
developed called Rapid Exponentially Stabilizing Control Lyapunov Functions(RES-CLF). Control Lyapunov function are used to stabilize a nonlinear system to
Robotic_prosthesis_control
Integrated circuit technology
describes memristive memory evolution, revealing tunneling phenomena and Lyapunov functions. Neuromorphic principles extend to sensors, such as the retinomorphic
Neuromorphic_computing
Sequence of data points over time
Correlation dimension Recurrence plots Recurrence quantification analysis Lyapunov exponents Entropy encoding Time-series metrics or features that can be
Time_series
Branch of engineering
influence N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits Engineering and efficiency Preflight engineering Mass ratio Payload
Aerospace_engineering
ISBN 978-0-86341-170-0. Zinober, Alan S.I., ed. (1994). Variable Structure and Lyapunov Control. Lecture Notes in Control and Information Sciences. Vol. 193. London:
Variable_structure_control
French professor of engineering
Optimization Approach to Supply Chain Management". In Bienstock, Daniel; Nemhauser, George (eds.). Integer Programming and Combinatorial Optimization
Aurelie_Thiele
Nonlinear two-terminal fundamental circuit element
properties in common with a Hopfield network, such as the existence of Lyapunov functions and classical tunnelling phenomena. In the context of memristive
Memristor
Argentine American mathematician
stability (ISS), a stability theory notion for nonlinear systems, and control-Lyapunov functions. Many of the subsequent results were proved in collaboration
Eduardo_D._Sontag
Goldengorin, Boris (ed.). Optimization Problems in Graph Theory: In Honor of Gregory Z. Gutin's 60th Birthday. Springer Optimization and Its Applications 139
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Measure of a civilization's evolution
on the basis of this universal characteristic, which allowed Aleksandr Lyapunov to define life as "a highly stable state of matter, which uses information
Kardashev_scale
Function that measures dissimilarity between two probability distributions
are the Lyapunov functions of the Kolmogorov forward equations. The converse statement is also true: If H ( P ) {\displaystyle H(P)} is a Lyapunov function
F-divergence
Mathematical models of strategic interactions
Clempner, Julio (2006). "Modeling shortest path games with Petri nets: a Lyapunov based theory". International Journal of Applied Mathematics and Computer
Game_theory
American control theorist
and nonlinear damping adaptive CLFs and ISS-CLFs general frameworks for Lyapunov and ISS-based adaptive nonlinear control output-feedback nonlinear and
Miroslav_Krstić
applied mathematician in France, studies magnetic resonance imaging and Lyapunov equations Na Li, Chinese-American electrical engineer, applied mathematician
List_of_women_in_mathematics
ISBN 978-0-86341-170-0. Zinober, Alan S.I., ed. (1994). Variable Structure and Lyapunov Control. Lecture Notes in Control and Information Sciences. Vol. 193. London:
Variable_structure_system
stability problem in nonlinear differential equations in terms of the Lyapunov function. JPL · 10690 10691 Sans 1981 EJ19 Juan Diego Sans (1922–2005)
Meanings of minor-planet names: 10001–11000
Meanings_of_minor-planet_names:_10001–11000
Method in numerical analysis
above) the Lyapunov-Schmidt decomposition may be used to produce two systems to which the Implicit Function Theorem applies. The Lyapunov-Schmidt decomposition
Numerical_continuation
theory and related areas: control design, estimation, identification, optimization, pattern recognition, signal processing, image processing, etc. Physical
Cybernetical_physics
Barrier certificates play the analogical role for safety to the role of Lyapunov functions for stability. For every ordinary differential equation that
Barrier_certificate
"Robust stability for polytopic systems via polynomially parameter-dependent Lyapunov functions". Proceedings of the 42nd IEEE Conference on Decision and Control
Polynomial_SOS
Theory of stochastic partial differential equations
the model but also on their "superpartners", whose evolution determines Lyapunov exponents. This structure enables an extended form of memory that includes
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Type of artificial neural network
of time series with nearly identical initial conditions is known as the Lyapunov exponent. We assume the output of the logistic map can be manipulated through
Radial_basis_function_network
Mathematical transform that expresses a function of time as a function of frequency
Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well,
Fourier_transform
regulator Matrix differential equation PDE-constrained optimization Riccati equation Shape optimization Clohessy–Wiltshire equations Planar reentry equations
List of named differential equations
List_of_named_differential_equations
Russian physicist and mathematician (1930–2023)
term tropical mathematics, in which the operations of the conditional optimization problem were considered. In the early 1970s, he met Lê Vũ Anh, the daughter
Viktor_Maslov_(mathematician)
Interdisciplinary study of systems
self-reproducing systems, again with only pencil and paper. Aleksandr Lyapunov and Jules Henri Poincaré worked on the foundations of chaos theory without
Systems_theory
Overview of and topical guide to electrical engineering
System properties: Exponential stability Marginal stability BIBO stability Lyapunov stability (i.e., asymptotic stability) Input-to-state (ISS) stability Controllability
Outline of electrical engineering
Outline_of_electrical_engineering
Speed of convergence of a mathematical sequence
definitions that asymptotic rates of convergence have. Among formal techniques, Lyapunov theory is one of the most powerful and widely applied frameworks for characterizing
Rate_of_convergence
Type of boundary condition in mathematics
Carroll, M. S. (2013). "Quantum computer aided design simulation and optimization of semiconductor quantum dots". Journal of Applied Physics. 114 (16)
Robin_boundary_condition
Property of functions which is weaker than continuity
measures is compact. More generally, many functionals of interest—such as Lyapunov exponents, dimension spectra, or return time statistics—are semicontinuous
Semi-continuity
Long term dynamical interactions that disrupt the Solar System
longer time scales, in such a way that the whole Solar System possesses a Lyapunov time in the range of 2~230 million years. In all cases, this means that
Stability_of_the_Solar_System
Method in nonlinear control theory
conditions come entirely from this space, the Lyapunov function candidate V ( σ ) {\displaystyle V(\sigma )} is a Lyapunov function and x {\displaystyle \mathbf
Sliding_mode_control
Infinitely detailed mathematical structure
the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations
Fractal
Idealised system for theoretical analysis
measure zero) of its possible trajectories are ergodic and it has a positive Lyapunov exponent. Sinai's great achievement with this model was to show that the
Dynamical_billiards
Serbian electrical engineer (1933–2025)
uncertain interconnections, based on graph-theoretic methods and vector Lyapunov Functions. He applied the theory to the decentralized control of the Large
Dragoslav_D._Šiljak
Method for representing and evaluating partial differential equations
"Chapter 3, Section 3.1". Compact heat exchangers : analysis, design and optimization using FEM and CFD approach. Seetharamu, K. N. Hoboken, NJ. ISBN 978-1-119-42435-2
Finite_volume_method
Marginal value theorem (biology, optimization) Artstein's theorem (control theory) Krener's theorem (control theory) Lyapunov–Malkin theorem (stability theory)
List_of_theorems
Fractal analysis technique
always known ahead of time, so box counting algorithms attempt to find an optimized way of cutting a pattern up that will reveal the scaling factor. The fundamental
Box_counting
the foundation for automatic on-off aircraft control in jets. Alexander Lyapunov 1892 His paper Sur le problème général de la stabilité du mouvement (in
List of people in systems and control
List_of_people_in_systems_and_control
Problem in physics and celestial mechanics
learning, kernel density estimation, and kernel machines. Alternative optimizations to reduce the O(n2) time complexity to O(n) have been developed, such
N-body_problem
List of definitions of terms and concepts used in electrical engineering and electronics
control The branch of control theory studying optimization of a control system to fit some optimization criterion. oscillation A periodic cyclical motion
Glossary of electrical and electronics engineering
Glossary_of_electrical_and_electronics_engineering
Measure theory theorems
particular, in the problem of cake-cutting, it was studied by Marco Dall'Aglio. Lyapunov vector-measure theorem Weller's theorem Dubins, Lester Eli; Spanier, Edwin
Dubins–Spanier_theorems
Polish mathematician (1931–2015)
Mathematici. 51: 7–13. doi:10.4064/ap-51-1-7-13. Olech, Czesraw (1990). "The Lyapunov Theorem: Its extensions and applications". Methods of Nonconvex Analysis
Czesław_Olech
Method to analyze non-binary inputs
Rafael; Haber, Rodolfo; Haber-Guerra, Rodolfo E.; Reyes, Fernando (1999). "Lyapunov Stable Control of Robot Manipulators: A Fuzzy Self-Tuning Procedure". Intelligent
Fuzzy_control_system
Transfer manoeuvre between two orbits
maneuver. Transfer orbits using electrical propulsion or low-thrust engines optimize the transfer time to reach the final orbit and not the delta-v as in the
Hohmann_transfer_orbit
Numerical method for solving physical or engineering problems
lemma Computer experiment Direct stiffness method Discontinuity layout optimization Discrete element method Finite difference method Finite element machine
Finite_element_method
central limit theorem Central limit theorem for directional statistics Lyapunov's central limit theorem Martingale central limit theorem Central moment
List_of_statistics_articles
Programming language to describe game rules
Clempner, Julio (2006). "Modeling shortest path games with Petri nets: a Lyapunov based theory". International Journal of Applied Mathematics and Computer
Game_Description_Language
Type of Monte Carlo algorithms for signal processing and statistical inference
authors list (link) Del Moral, Pierre (2003). "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups"
Particle_filter
minor planets at Brorfelde Observatory JPL · 5323 5324 Lyapunov 1987 SL Aleksandr Mikhailovich Lyapunov (1857–1918), Russian mathematician, engineer and physicist
Meanings of minor-planet names: 5001–6000
Meanings_of_minor-planet_names:_5001–6000
the direct method of Lyapunov" (with R.B. Warden and N.R. Amundson). Chem. Eng. Sci. 19, 173–190 (1964). "Studies in optimization–VI: The application of
Rutherford_Aris_bibliography
Probabilistic problem-solving algorithms
1016/S0304-4149(99)00094-0. Del Moral, Pierre (2003). "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups"
Mean-field_particle_methods
Danish-Canadian academic (1929–2007)
systems, a continuation of his Ph.D. work. He simplified the solutions to Lyapunov's stability theorems so that they are now useful to the electric power engineers
Gustav_S._Christensen
LYAPUNOV OPTIMIZATION
LYAPUNOV OPTIMIZATION
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LYAPUNOV OPTIMIZATION
LYAPUNOV OPTIMIZATION
LYAPUNOV OPTIMIZATION