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Mathematical way of attaining a desired output from a dynamic system
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective
Optimal_control
Variables Control
In optimal control theory, a control is a variable chosen by the controller or agent to manipulate state variables, similar to an actual control valve
Control (optimal control theory)
Control_(optimal_control_theory)
Hypothesis in neuroscience
problem – to movement trajectories. Active inference is related to optimal control by replacing value or cost-to-go functions with prior beliefs about
Free_energy_principle
Binary feedback controller
In optimal control problems, it is sometimes the case that a control is restricted to be between a lower and an upper bound. If the optimal control switches
Bang–bang_control
Mathematics concept
unscented optimal control combines the notion of the unscented transform with deterministic optimal control to address a class of uncertain optimal control problems
Unscented_optimal_control
Advanced method of process control
for this reason MPC is also called receding horizon control. Although this approach is not optimal, in practice it has given very good results. Much academic
Model_predictive_control
Principle in optimal control theory for best way to change state in a dynamical system
Bellman's principle of optimality, a related approach to optimal control problems which states that the optimal trajectory remains optimal at intermediate points
Pontryagin's maximum principle
Pontryagin's_maximum_principle
The PROPT MATLAB Optimal Control Software is a new generation platform for solving applied optimal control (with ODE or DAE formulation) and parameters
PROPT
Python package
optimization and optimal control problems. GEKKO is used for advanced control in the Temperature Control Lab (TCLab) for process control education at 20
Gekko_(optimization_software)
Numerical method for solving optimal control problems
Pseudospectral optimal control is a numerical technique for solving optimal control problems. These problems involve finding the best way to control a dynamic
Pseudospectral optimal control
Pseudospectral_optimal_control
Necessary condition for optimality associated with dynamic programming
the optimal policy in the last time period is specified in advance as a function of the state variable's value at that time, and the resulting optimal value
Bellman_equation
object in question. The search for optimal values for object or system characteristics is carried out by means of optimal change to design, geometrical or
IOSO
Multiple ways for multi-joint objects to realize a movement
the natural outcome of an adaptive optimal control process. Optimal control is a way of understanding motor control and the motor equivalence problem,
Degrees_of_freedom_problem
British electronic engineer (1930–2024)
Journal of Control, 9(5):547--559, 1966. D. Q. Mayne, Differential Dynamic Programming---a Unified Approach to Optimal Control, in Advances in Control Systems
David_Mayne
Techniques to maintain quantum coherence
pulse with a varying frequency in time. Optimal control as applied in coherent control seeks the optimal control field for steering a quantum system to
Coherent_control
Experimental design that is optimal with respect to some statistical criterion
same precision as an optimal design. In practical terms, optimal experiments can reduce the costs of experimentation. The optimality of a design depends
Optimal_experimental_design
Ross and F. Fahroo, the Ross–Fahroo lemma is a fundamental result in optimal control theory. It states that dualization and discretization are, in general
Ross–Fahroo_lemma
Study of mathematical algorithms for optimization problems
a cost function where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error. Typically, A is some subset of the Euclidean
Mathematical_optimization
Probabilistic optimal control
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations
Stochastic_control
Linear optimal control technique
In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems, and it can also be operated
Linear–quadratic–Gaussian control
Linear–quadratic–Gaussian_control
Optimality condition in optimal control theory
sufficient conditions for optimality of a control with respect to a loss function. Its solution is the value function of the optimal control problem which, once
Hamilton–Jacobi–Bellman equation
Hamilton–Jacobi–Bellman_equation
Concept in game theory
control of moving systems. Differential games are related closely to optimal control problems. In an optimal control problem there is single control u
Differential_game
American electrical engineer, academic and researcher
including Optimal Control, Optimal Estimation, Aircraft Control and Simulation, Applied Optimal Control and Estimation, and Robot Manipulator Control. Lewis
Frank_L._Lewis
General-purpose MATLAB software
"GPOPS 2") is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse
GPOPS-II
Field of machine learning
In machine learning and optimal control, reinforcement learning (RL) (also known as approximate dynamic programming) is concerned with how an intelligent
Reinforcement_learning
Process of developing trajectory performance
optimization is a technique for computing an open-loop solution to an optimal control problem. It is often used for systems where computing the full closed-loop
Trajectory_optimization
Function used in optimal control theory
The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of
Hamiltonian_(control_theory)
variables. They are one of the necessary conditions for optimality infinite-horizon optimal control problems without an endpoint constraint on the state
Transversality_condition
Optimization algorithm for artificial neural networks
backpropagation appeared in optimal control theory since 1950s. Yann LeCun et al credits 1950s work by Pontryagin and others in optimal control theory, especially
Backpropagation
Indian-American mathematician
arise in optimal control problems that exhibit multiple optimal solutions. A Sethi-Skiba point is an indifference point in an optimal control problem such
Suresh_P._Sethi
Control loop feedback mechanism
the three control terms of proportional, integral and derivative influence on the controller output to apply accurate and optimal control. The block
PID_controller
Software product for solving general-purpose optimal control problems
DIDO (/ˈdaɪdoʊ/ DY-doh) is a MATLAB optimal control toolbox for solving general-purpose optimal control problems. It is widely used in academia, industry
DIDO_(software)
Branch of engineering and mathematics
and the control strategy chosen. List of the main control techniques Optimal control is a particular control technique in which the control signal optimizes
Control_theory
Problem in mathematical optimisation
navigation problem, proposed in 1931 by Ernst Zermelo, is a classic optimal control problem that deals with a boat navigating on a body of water, originating
Zermelo's_navigation_problem
Director of Control and Optimization at the Naval Postgraduate School in Monterey, CA. He has published a highly-regarded textbook on optimal control theory
I._Michael_Ross
Linear optimal control technique
The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set
Linear–quadratic_regulator
Maximized objective function of an optimization problem
on the parameters of the problem. In a controlled dynamical system, the value function represents the optimal payoff of the system over the interval [
Value_function
Artificial-intelligence researcher (born 1971)
and director of the Movement Control Laboratory at the University of Washington. He introduced the use of optimal control as a formal explanatory framework
Emanuel_Todorov
and open source symbolic framework for automatic differentiation and optimal control. Automatic differentiation JModelica.org "Optimization in Engineering
CasADi
Topics referred to by the same term
regulations on trade Internal control, a process to help achieve specific goals typically related to managing risk Control (optimal control theory), a variable
Control
In forestry, the optimal rotation age is the growth period required to derive maximum value from a stand of timber. The calculation of this period is
Optimal_rotation_age
Branch of applied mathematics
systems. The problem of finding optimal functions for such changes is studied in variational calculus and in optimal control theory. Before the Second World
Mathematical_economics
method for optimal control based on Bellman's principle of optimality. It is part of the larger theory of pseudospectral optimal control, a term coined
Bellman_pseudospectral_method
Local boundary-limited electrical grid
Tertiary control is the last (and the slowest) control level, which considers economical concerns in the optimal operation of the microgrid (sampling time
Microgrid
for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control, a
Chebyshev pseudospectral method
Chebyshev_pseudospectral_method
Gauss, is a direct transcription method for discretizing a continuous optimal control problem into a nonlinear program (NLP). The Gauss pseudospectral method
Gauss_pseudospectral_method
Nonlinear equation which arises on linear optimal control problems
nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time. A typical algebraic Riccati
Algebraic_Riccati_equation
American mathematician
Along with I. M. Ross, she has published papers in pseudospectral optimal control theory. The Ross–Fahroo lemma and the Ross–Fahroo pseudospectral method
Fariba_Fahroo
Use of various control systems for operating equipment
(1961), navigation (1960), optimal control and estimation theory (1962), nonlinear control theory (1969), digital control and filtering theory (1974)
Automation
Probabilistic problem-solving algorithm
"Estimation and nonlinear optimal control: Particle resolution in filtering and estimation". Studies on: Filtering, optimal control, and maximum likelihood
Monte_Carlo_method
Magnetic resonance simulation software
polarization (DNP), magic angle spinning (MAS), spin chemistry, and quantum optimal control. The package was introduced in 2011 in the Journal of Magnetic Resonance
Spinach_(software)
In optimal control, problems of singular control are problems that are difficult to solve because a straightforward application of Pontryagin's minimum
Singular_control
Engineering discipline that deals with control systems
included developments in optimal control in the 1950s and 1960s followed by progress in stochastic, robust, adaptive, nonlinear control methods in the 1970s
Control_engineering
Problem optimization method
solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure
Dynamic_programming
Dynamic system property
problems, obtaining optimal control strategies, or, simply prescribing an input that has a desired effect on the state. Controllability and observability
Controllability
Lacitignola, Deborah; Tilioua, Mouhcine (2024). "Optimal social distancing through cross-diffusion control for a disease outbreak PDE model". Communications
PDE-constrained_optimization
Ross, is a result in computational optimal control. Based on generating Carathéodory-π solutions for feedback control, Ross' π-lemma states that there is
Ross'_π_lemma
1957 technique for modelling problems of decision making under uncertainty
s t ) {\displaystyle f_{t}(s_{t})} represent the optimal cost/reward obtained by following an optimal policy over stages t , t + 1 , … , n {\displaystyle
Stochastic dynamic programming
Stochastic_dynamic_programming
Control paradigm in which errors are measured before they can affect a system
Engineering, 1987. Alberts, T.E., Sangveraphunsiri, V. and Book, Wayne J., Optimal Control of a Flexible Manipulator Arm: Volume I, Dynamic Modeling, MHRC Technical
Feed_forward_(control)
Concept in control theory
and optimal control are two closely related frameworks used to address sequential decision making problems; optimal control originates from control theory
Sequential_decision_making
Method to solve constrained optimization problems
the optimal profit to a player is calculated subject to a constrained space of actions, where a Lagrange multiplier is the change in the optimal value
Lagrange_multiplier
pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined
Legendre pseudospectral method
Legendre_pseudospectral_method
Academic journal
that specializes in articles on control theory, optimization, optimal control and calculus of variations. "ESAIM: Control, Optimisation and Calculus of
ESAIM: Control, Optimisation and Calculus of Variations
ESAIM:_Control,_Optimisation_and_Calculus_of_Variations
available thrust is limited, the transfer is occasionally posed as an optimal control problem subjected to the required objective and constraints. Relative
Low thrust relative orbital transfer
Low_thrust_relative_orbital_transfer
Georgian and Soviet mathematician (1927–2025)
of Optimal Processes. Translated to English by Trirogoff. Interscience Publishers. MR 0166037 R.V. Gamkrelidze (1978). Principles of Optimal Control Theory
Revaz_Gamkrelidze
arise in optimal control problems that exhibit multiple optimal solutions. A Sethi-Skiba point is an indifference point in an optimal control problem such
Sethi-Skiba_point
Fastest curve descent without friction
problem can be solved using tools from the calculus of variations and optimal control. The curve is independent of both the mass of the test body and the
Brachistochrone_curve
Mathematical concept
f(x^{*})} ) is called Pareto optimal if there does not exist another solution that dominates it. The set of Pareto optimal outcomes, denoted X ∗ {\displaystyle
Multi-objective_optimization
linear. Witsenhausen obtained the following results: An optimum exists (Theorem 1). The optimal control law of the first controller is such that E ( x 1 )
Witsenhausen's_counterexample
French mathematical economist
Pintus, Patrick (2012). "On the optimal control of a linear neutral differential equation arising in economics". Optimal Control Applications and Methods. 33
Raouf_Boucekkine
In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The
Optimal_projection_equations
deterministic control theory such as optimal control theory. The basic idea of the MCAM is to approximate the original controlled process by a chosen controlled markov
Markov chain approximation method
Markov_chain_approximation_method
Model". Optimal Control Applications and Methods. 4 (2): 179–184. doi:10.1002/oca.4660040207. S2CID 123673289. Sethi, S.P. (2021). Optimal Control Theory:
Sethi_model
Optimal Processes, Wiley, New York, 1962. Ross, I. M., Gong, Q., Fahroo, F. and Kang, W., "Practical Stabilization Through Real-Time Optimal Control,"
Caratheodory-π_solution
Algorithm for trajectory optimization
Differential dynamic programming (DDP) is an optimal control algorithm of the trajectory optimization class. The algorithm was introduced in 1966 by Mayne
Differential dynamic programming
Differential_dynamic_programming
Subfield of machine learning, intelligent control, and control theory
Machine learning control (MLC) is a subfield of machine learning, intelligent control, and control theory which aims to solve optimal control problems with
Machine_learning_control
computational methods for optimal control and guidance and control of aerospace vehicles. He is the co-creator of the optimal control software GPOPS-II and
Anil_V._Rao
British engineer
including Linear Robust Control (with Michael Green), and Dynamics and Optimal Control of Road Vehicles (with Matteo Massaro). He is the recipient of two
David_Limebeer
NP-hard problem in combinatorial optimization
that, instead of seeking optimal solutions, would produce a solution whose length is provably bounded by a multiple of the optimal length, and in doing so
Travelling_salesman_problem
Mathematical model for sequential decision making under uncertainty
solve the equation to find the optimal value function V ∗ {\displaystyle V^{*}} , which in turns yield the optimal control at any time t {\displaystyle
Markov_decision_process
Double-setpoint control is quite similar to bang–bang control. It is an element of a feedback-loop and therefore evaluated by application of control theory. It
Double-setpoint_control
desirable to have an optimal navigation function with respect to a given cost functional J {\displaystyle J} . Formalized as an optimal control problem, we can
Navigation_function
certain optimal control problems with multiple optimal solutions Legendre–Clebsch condition — second-order condition for solution of optimal control problem
List of numerical analysis topics
List_of_numerical_analysis_topics
Topics referred to by the same term
electrons and nuclei in a molecule Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system Hamiltonian path
Hamiltonian
products. The problem can be modeled using mathematical techniques of optimal control, dynamic programming and network optimization. The study of such models
Inventory_theory
Optimization for dynamical systems
application to optimal control in queueing networks. Lyapunov optimization refers to the use of a Lyapunov function to optimally control a dynamic system
Lyapunov_optimization
Algorithm for solving boundary value problems of the Eikonal equation
{\displaystyle x} . The fast marching method takes advantage of this optimal control interpretation of the problem in order to build a solution outwards
Fast_marching_method
American mathematician (1920–1984)
equation which is central to optimal control theory. The solution of the HJB equation is the 'value function', which gives the optimal cost-to-go for a given
Richard_Bellman
natural and social phenomena, and has close ties to the theories of optimal control and set-valued analysis. Many systems, organizations, and networks
Viability_theory
Motivating example in mathematical study
filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics. The mathematical formulation of the problem
Obstacle_problem
equation Merton's portfolio problem Optimal stopping Malthusian growth model Mean field game theory Optimal rotation age Sovereign debt accumulation
List of named differential equations
List_of_named_differential_equations
Terminal command scheme used to transfer data
protocol innovates upon naive "data blaster" protocols through an optimal control-theoretic retransmission algorithm and implementation that achieves
Fast_and_Secure_Protocol
Canadian researcher
Shogi. His research focuses on machine learning and statistics for optimal control and decision making, as well as using these mathematical frameworks
Timothy_Lillicrap
Optimal control technique
pseudospectral methods are a broad collection of pseudospectral methods for optimal control. Examples of the Ross–Fahroo pseudospectral methods are the pseudospectral
Ross–Fahroo pseudospectral method
Ross–Fahroo_pseudospectral_method
Konidaris and Leslie Kaelbling and Tomas Lozano-Perez (2012). "LQR-RRT*: Optimal sampling-based motion planning with automatically derived extension heuristics"
Linear-quadratic regulator rapidly exploring random tree
Linear-quadratic_regulator_rapidly_exploring_random_tree
Topics referred to by the same term
Navier-Stokes equations Kalman filter, an approximating algorithm in optimal control applications and problems Filter (social media), an appearance-altering
Filter
Classification scheme for mathematics
92: Biology and other natural sciences 93: Systems theory; control (including optimal control) 94: Information and communication, circuits 97: Mathematics
Mathematics Subject Classification
Mathematics_Subject_Classification
Principle in control theory
computational optimal control. An application of Pontryagin's minimum principle to Problem B {\displaystyle B} , a given optimal control problem generates
Covector_mapping_principle
British physicist
atoms, electron spin resonance problems in structural biology, quantum optimal control theory, and gravitation sensors using atom interferometers. Kuprov
Ilya_Kuprov
Problem of finding the optimal shape under given conditions
optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost
Shape_optimization
travel, tourism, insurance
OPTIMAL CONTROL
OPTIMAL CONTROL
OPTIMAL CONTROL
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OPTIMAL CONTROL
OPTIMAL CONTROL
OPTIMAL CONTROL
OPTIMAL CONTROL
OPTIMAL CONTROL
travel, tourism, insurance