# Category: Optimal control

Unscented optimal control
In mathematics, unscented optimal control combines the notion of the unscented transform with deterministic optimal control to address a class of uncertain optimal control problems. It is a specific a
Sethi model
The Sethi model was developed by Suresh P. Sethi and describes the process of how sales evolve over time in response to advertising. The model assumes that the rate of change in sales depend on three
Pseudospectral knotting method
In applied mathematics, the pseudospectral knotting method is a generalization and enhancement of a standard pseudospectral method for optimal control. The concept was introduced by I. Michael Ross an
Legendre pseudospectral method
The Legendre 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 by Ross. A basic vers
Ross' π lemma
Ross' π lemma, named after I. Michael Ross, is a result in computational optimal control. Based on generating Carathéodory-π solutions for feedback control, Ross' π-lemma states that there is fundamen
Hydrological optimization
Hydrological optimization applies mathematical optimization techniques (such as dynamic programming, linear programming, integer programming, or quadratic programming) to water-related problems. These
Microgrid
A microgrid is a local electrical grid with defined electrical boundaries, acting as a single and controllable entity. It is able to operate in grid-connected and in island mode. A 'Stand-alone microg
Singular control
In optimal control, problems of singular control are problems that are difficult to solve because a straightforward application of Pontryagin's minimum principle fails to yield a complete solution. On
In mathematical optimization, Zermelo's 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 fr
Linear-quadratic regulator rapidly-exploring random tree (LQR-RRT) is a sampling based algorithm for kinodynamic planning. A solver is producing random actions which are forming a funnel in the state
Caratheodory-π solution
A Carathéodory-π solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory. Its practicality was demo
PROPT
The PROPT MATLAB Optimal Control Software is a new generation platform for solving applied optimal control (with ODE or DAE formulation) and parameters estimation problems. The platform was developed
Transversality condition
In optimal control theory, a transversality condition is a boundary condition for the terminal values of the costate variables. They are one of the necessary conditions for optimality infinite-horizon
Dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields
DIDO (software)
DIDO (/ˈdaɪdoʊ/ DY-doh) is a MATLAB optimal control toolbox for solving general-purpose optimal control problems. It is widely used in academia, industry, and NASA. Hailed as a breakthrough software,
Bang–bang control
In control theory, a bang–bang controller (2 step or on–off controller), is a feedback controller that switches abruptly between two states. These controllers may be realized in terms of any element t
Hamilton–Jacobi–Bellman equation
In optimal control theory, the Hamilton-Jacobi-Bellman (HJB) equation gives a necessary and sufficient condition for optimality of a control with respect to a loss function. It is, in general, a nonli
GPOPS-II
GPOPS-II (pronounced "GPOPS 2") is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse nonlinear programming.
Beltrami identity
The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange equation serves to extremize action functionals
Ross–Fahroo pseudospectral method
Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control. Examples of the Ross–Fahroo pseudospectral met
PDE-constrained optimization
PDE-constrained optimization is a subset of mathematical optimization where at least one of the constraints may be expressed as a partial differential equation. Typical domains where these problems ar
Optimal rotation age
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 specific to each stand and to the economic and sus
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 repeatedly for model predictive control. It
Bellman pseudospectral method
The Bellman pseudospectral method is a pseudospectral method for optimal control based on Bellman's principle of optimality. It is part of the larger theory of pseudospectral optimal control, a term c
Covector mapping principle
The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by Ross and co-workers, It provides condit
Ross–Fahroo lemma
Named after I. Michael 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, non-commutative opera
Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has nume
Pontryagin's maximum principle
Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints fo
Gauss pseudospectral method
The Gauss pseudospectral method (GPM), one of many topics named after Carl Friedrich Gauss, is a direct transcription method for discretizing a continuous optimal control problem into a nonlinear prog
Pseudospectral optimal control
Pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems. It combines pseudospectral (PS) theory with optimal control theory to produce PS optima
Flat pseudospectral method
The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo. The method combines the concept of differential flatness with pseudospectr
Value function
The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters of the problem. In a controlled dynamical system,
Stochastic dynamic programming
Originally introduced by Richard E. Bellman in, stochastic dynamic programming is a technique for modelling and solving problems of decision making under uncertainty. Closely related to stochastic pro
Chebyshev pseudospectral method
The Chebyshev pseudospectral method 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 term coined b
Optimal projection equations
In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The linear-quadratic-Gaussian (LQG) control problem i
DNSS point
DNSS points, also known as Sethi-Skiba points, arise in optimal control problems that exhibit multiple optimal solutions. A DNSS pointnamed alphabetically after Deckert and Nishimura, Sethi, and Skiba
Hamiltonian (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 the Lagrangian expression of the problem that is
Costate equation
The costate equation is related to the state equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order
Double-setpoint control
A 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 has two setpoints on which it switches
Legendre–Clebsch condition
In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum. For the problem of minim