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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 nonlinear partial differential equation in the value function, which means its solution the value function itself. Once this solution is known, it can be used to obtain the optimal control by taking the maximizer (or minimizer) of the Hamiltonian involved in the HJB equation. The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The connection to the Hamilton–Jacobi equation from classical physics was first drawn by Rudolf Kálmán. In discrete-time problems, the corresponding difference equation is usually referred to as the Bellman equation. While classical variational problems, such as the brachistochrone problem, can be solved using the Hamilton–Jacobi–Bellman equation, the method can be applied to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control Theory
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Problem
The calculus of variations (or Variational Calculus) 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. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael G
Michael may refer to: People * Michael (given name), a given name * Michael (surname), including a list of people with the surname Michael Given name "Michael" * Michael (archangel), ''first'' of God's archangels in the Jewish, Christian and Islamic religions * Michael (bishop elect), English 13th-century Bishop of Hereford elect * Michael (Khoroshy) (1885–1977), cleric of the Ukrainian Orthodox Church of Canada * Michael Donnellan (1915–1985), Irish-born London fashion designer, often referred to simply as "Michael" * Michael (footballer, born 1982), Brazilian footballer * Michael (footballer, born 1983), Brazilian footballer * Michael (footballer, born 1993), Brazilian footballer * Michael (footballer, born February 1996), Brazilian footballer * Michael (footballer, born March 1996), Brazilian footballer * Michael (footballer, born 1999), Brazilian footballer Rulers =Byzantine emperors= *Michael I Rangabe (d. 844), married the daughter of Emperor Nikephoros I * Mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre-Louis Lions
Pierre-Louis Lions (; born 11 August 1956) is a French people, French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal and the 1991 Prize of the Altria, Philip Morris tobacco and cigarette company. Biography Lions graduated from the École Normale Supérieure, École normale supérieure in 1977, and received his doctorate from the University of Pierre and Marie Curie in 1979. He holds the position of Professor of ''Partial differential equations and their applications'' at the Collège de France in Paris as well as a position at École Polytechnique. Since 2014, he has also been a visiting professor at the University of Chicago. In 1979, Lions married Lila Laurenti, with whom he has one son. Lions' parents were Andrée Olivier and the renowned mathematician Jacques-Louis Lions, at the time a professor at the University of Nancy, and from 1991 through ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backward Induction
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by examining the last point at which a decision is to be made and then identifying what action would be most optimal at that moment. Using this information, one can then determine what to do at the second-to-last time of decision. This process continues backwards until one has determined the best action for every possible situation (i.e. for every possible information set) at every point in time. Backward induction was first used in 1875 by Arthur Cayley, who uncovered the method while trying to solve the infamous Secretary problem. In the mathematical optimization method of dynamic programming, backward induction is one of the main methods for solving the Bellman equation. In game theory, backward induction is a method used to compute subgame perfect equilibria in sequential games. The only difference is that optimizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Little-o Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Optimality
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices. This breaks a dynamic optimization problem into a sequence of simpler subproblems, as Bellman's “principle of optimality" prescribes. The equation applies to algebraic structures with a total ordering; for algebraic structures with a partial ordering, the generic Bellman's equation can be used. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory; though the basic concepts of dynamic programming are prefigured in John von Neumann and Oskar Morgenstern's ''Theory of Games and Econo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bequest Value
Bequest value, in economics, is the value of satisfaction from preserving a natural environment or a historic environment, in other words natural heritage or cultural heritage for future generations. It is often used when estimating the value of an environmental service or good. Together with the existence value, it makes up the non-use value Non-use value is the value that people assign to economic goods (including public goods) even if they never have and never will use it. It is distinguished from use value, which people derive from direct use of the good. The concept is most common ... of such an environmental service or good. References Environmental economics {{econ-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subderivative
In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f:I \to \mathbb be a real-valued convex function defined on an open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ... of the real line. Such a function need not be differentiable at all points: For example, the absolute value function ''f''(''x'')=, ''x'', is nondifferentiable when ''x''=0. However, as seen in the graph on the right (where ''f(x)'' in blue has non-differentiable kinks similar to the absolute value function), for any ''x''0 in the domain of the function one can draw a line which goes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viscosity Solution
In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems, as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games. The classical concept was that a PDE : F(x,u,Du,D^2 u) = 0 over a domain x\in\Omega has a solution if we can find a function ''u''(''x'') continuous and differentiable over the entire domain such that x, u, Du, D^2 u satisfy the above equation at every point. If a scalar equation is degenerate el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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