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# Optimal control

Optimal control theory deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. [1]

53 relations: APMonitor, ASTOS, Bellman equation, Bellman pseudospectral method, Boundary value problem, Brachistochrone curve, Calculus of variations, Collocation method, Constraint (mathematics), Control theory, Controllability, DIDO (software), Differential equation, Digital data, Discrete time and continuous time, DNSS point, Dynamic programming, Dynamical system, Edward J. McShane, Fortran, Free energy principle, Function (mathematics), Gauss pseudospectral method, Generalized filtering, GPOPS-II, Hamilton–Jacobi–Bellman equation, Hamiltonian (control theory), Hamiltonian system, Initial condition, JModelica.org, Kalman filter, Lagrange multiplier, Lev Pontryagin, Linear–quadratic regulator, Mathematical optimization, MATLAB, Matrix (mathematics), Model predictive control, Necessity and sufficiency, Optimality criterion, Pontryagin's maximum principle, PROPT, Pseudospectral optimal control, Pursuit-evasion, Riccati equation, Richard E. Bellman, Rudolf E. Kálmán, Shadow price, Sliding mode control, SNOPT, ... Expand index (3 more) »

## APMonitor

Advanced process monitor (APMonitor), is a modeling language for differential algebraic (DAE) equations.

## ASTOS

ASTOS is a tool dedicated to mission analysis, Trajectory optimization, vehicle design and simulation for space scenarios, i.e. launch, re-entry missions, orbit transfers, Earth observation, navigation, coverage and re-entry safety assessments.

## Bellman equation

A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming.

## Bellman pseudospectral method

The Bellman pseudospectral method is a pseudospectral method for optimal control based on Bellman's principle of optimality.

## Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.

## Brachistochrone curve

In mathematics and physics, a brachistochrone curve, or curve of fastest descent, is the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.

## Calculus of variations

Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.

## Collocation method

In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.

## Constraint (mathematics)

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

## Control theory

Control theory in control systems engineering deals with the control of continuously operating dynamical systems in engineered processes and machines.

## Controllability

Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.

## DIDO (software)

DIDO is a software product for solving general-purpose optimal control problems.

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Digital data

Digital data, in information theory and information systems, is the discrete, discontinuous representation of information or works.

## Discrete time and continuous time

In mathematics and in particular mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.

## DNSS point

DNSS points, also known as Skiba points, arise in optimal control problems that exhibit multiple optimal solutions.

## Dynamic programming

Dynamic programming is both a mathematical optimization method and a computer programming method.

## Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

## Edward J. McShane

Edward James McShane (May 10, 1904 – June 1, 1989) was an American mathematician noted for his advancements of the calculus of variations, integration theory, stochastic calculus, and exterior ballistics.

## Fortran

Fortran (formerly FORTRAN, derived from Formula Translation) is a general-purpose, compiled imperative programming language that is especially suited to numeric computation and scientific computing.

## Free energy principle

The free energy principle tries to explain how (biological) systems maintain their order (non-equilibrium steady-state) by restricting themselves to a limited number of states.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## 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 program (NLP).

## Generalized filtering

Generalized filtering is a generic Bayesian filtering scheme for nonlinear state-space models.

## 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.

## Hamilton–Jacobi–Bellman equation

The Hamilton–Jacobi–Bellman (HJB) equation is a partial differential equation which is central to optimal control theory.

## Hamiltonian (control theory)

The Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle.

## Hamiltonian system

A Hamiltonian system is a dynamical system governed by Hamilton's equations.

## Initial condition

In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t.

## JModelica.org

JModelica.org is a free and open source software platform based on the Modelica modeling language for modeling, simulating, optimizing and analyzing complex dynamic systems.

## Kalman filter

Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe.

## Lagrange multiplier

In mathematical optimization, the method of Lagrange multipliers (named after Joseph-Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables).

## Lev Pontryagin

Lev Semyonovich Pontryagin (Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician.

The theory of optimal control is concerned with operating a dynamic system at minimum cost.

## Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.

## MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

## Model predictive control

Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints.

## Necessity and sufficiency

In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements.

## Optimality criterion

In statistics, an optimality criterion provides a measure of the fit of the data to a given hypothesis, to aid in model selection.

## Pontryagin's maximum principle

Pontryagin's maximum (or minimum) 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 for the state or input controls.

## 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.

## Pseudospectral optimal control

Pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems.

## Pursuit-evasion

Pursuit-evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment.

## Riccati equation

In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function.

## Richard E. Bellman

Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and important contributions in other fields of mathematics.

## Rudolf E. Kálmán

Rudolf Emil Kálmán (Kálmán Rudolf Emil; May 19, 1930 – July 2, 2016) was a Hungarian-born American electrical engineer, mathematician, and inventor.

A shadow price is commonly referred to as a monetary value assigned to currently unknowable or difficult-to-calculate costs.

## Sliding mode control

In control systems, sliding mode control, or SMC, is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal (or more rigorously, a set-valued control signal) that forces the system to "slide" along a cross-section of the system's normal behavior.

## SNOPT

SNOPT, for Sparse Nonlinear OPTimizer, is a software package for solving large-scale nonlinear optimization problems written by Philip Gill, Walter Murray and Michael Saunders.

## Stochastic control

Stochastic control or stochastic optimal control is a subfield of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system.

## TOMLAB

The TOMLAB Optimization Environment is a modeling platform for solving applied optimization problems in MATLAB.

## Trajectory optimization

Trajectory optimization is the process of designing a trajectory that minimizes (or maximizes) some measure of performance while satisfying a set of constraints.

## References

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