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Costate Equations
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 differential equations : \dot^(t)=-\frac where the right-hand side is the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables. Interpretation The costate variables \lambda(t) can be interpreted as Lagrange multipliers associated with the state equations. The state equations represent constraints of the minimization problem, and the costate variables represent the marginal cost of violating those constraints; in economic terms the costate variables are the shadow prices. Solution The state equation is subject to an initial condition and is solved forwards in time. The costate equation must satisfy a transversality condition and is solved backwards in time, from the final time towards the beginning. For more details see Pontryagin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (geometry)
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for z=f(x, y, \ldots), the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: :f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 to be optimized over a certain time period. Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. Problem statement and definition of the Hamiltonian Consider a dynamical system of n first-order differential equations :\dot(t) = \mathbf(\mathbf(t),\mathbf(t),t) where \mathbf(t) = \left x_(t), x_(t), \ldots, x_(t) \right denotes a vector of state variables, and \mathbf(t) = \left u_(t), u_(t), \ldots, u_(t) \right a vector of control variables. Once initial conditions \mathbf(t_) = \mathbf_ and controls \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Multipliers
In mathematical optimization, the method of Lagrange multipliers 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). It is named after the mathematician Joseph-Louis Lagrange. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function. The method can be summarized as follows: in order to find the maximum or minimum of a function f(x) subjected to the equality constraint g(x) = 0, form the Lagrangian function :\mathcal(x, \lambda) = f(x) + \lambda g(x) and find the stationary points of \mathcal considered as a function of x and the Lagrange mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Cost
In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it refers to the rate of change of total cost as output is increased by an infinitesimal amount. As Figure 1 shows, the marginal cost is measured in dollars per unit, whereas total cost is in dollars, and the marginal cost is the slope of the total cost, the rate at which it increases with output. Marginal cost is different from average cost, which is the total cost divided by the number of units produced. At each level of production and time period being considered, marginal cost includes all costs that vary with the level of production, whereas costs that do not vary with production are fixed. For example, the marginal cost of producing an automobile will include the costs of labor and parts needed for the additional automobile but not the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shadow Price
A shadow price is the monetary value assigned to an abstract or intangible commodity which is not traded in the marketplace. This often takes the form of an externality. Shadow prices are also known as the recalculation of known market prices in order to account for the presence of distortionary market instruments (e.g. quotas, tariffs, taxes or subsidies). Shadow Prices are the real economic prices given to goods and services after they have been appropriately adjusted by removing distortionary market instruments and incorporating the societal impact of the respective good or service. A shadow price is often calculated based on a group of assumptions and estimates because it lacks reliable data, so it is subjective and somewhat inaccurate. The need for shadow prices arises as a result of “externalities” and the presence of distortionary market instruments. An externality is defined as a cost or benefit incurred by a third party as a result of production or consumption of a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transversality Condition
In optimal control theory, a transversality condition is a Boundary value problem, boundary condition for the terminal values of the costate equation, costate variables. They are one of the necessary conditions for optimality infinite-horizon optimal control problems without an endpoint Constraint (mathematics), constraint on the state variables. See also * Pontryagin's maximum principle Further reading * * Boundary conditions Optimal control {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for the state or input controls. It states that it is necessary for any optimal control along with the optimal state trajectory to solve the so-called Hamiltonian system, which is a two-point boundary value problem, plus a maximum condition of the control Hamiltonian. These necessary conditions become sufficient under certain convexity conditions on the objective and constraint functions. The maximum principle was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students, and its initial application was to the maximization of the terminal speed of a rocket. The result was derived using ideas from the classical calculus of variations. After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the pert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Equation
An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization, Flow control (fluid), fluid flow control and uncertainty quantification. For example dX_t = a(X_t)dt + b(X_t)dW this is an Itō calculus, Itō stochastic differential equation. Now by using Euler scheme, we integrate the parts of this equation and get another equation, X_ = X_n + a \Delta t + \zeta b \sqrt, here \zeta is a random variable, later one is an adjoint equation. Example: Advection-Diffusion PDE Consider the following linear, scalar Convection–diffusion_equation, advection-diffusion equation for the primal solution u(\vec), in the domain \Omega with Dirichlet boundary conditions: : \begin \nabla \cdot \left(\vec u - \mu \nabla u \righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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,Ross, I. M., “A Historical Introduction to the Covector Mapping Principle,” Proceedings of the 2005 AAS/AIAA Astrodynamics Specialist Conference, August 7–11, 2005 Lake Tahoe, CA. AAS 05-332.Ross, I. M. and Fahroo, F., “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, New York, 2003, pp 327–342.Ross, I. M. and Fahroo, F., “Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems,” Proceedings of the American Control Conference, June 2004, Boston, MARoss, I. M. and Fahroo, F., “A Pseudospectral Transformation of the Covectors of Optimal Control Systems,” Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
