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Barrier Certificate
A barrier certificate or barrier function is used to prove that a given region is forward invariant for a given ordinary differential equation or hybrid dynamical system. That is, a barrier function can be used to show that if a solution starts in a given set, then it cannot leave that set. Showing that a set is forward invariant is an aspect of ''safety'', which is the property where a system is guaranteed to avoid obstacles specified as an ''unsafe set''. Barrier certificates play the analogical role for safety to the role of Lyapunov functions for stability. For every ordinary differential equation that robustly fulfills a safety property of a certain type there is a corresponding barrier certificate. History The first result in the field of barrier certificates was the Nagumo theorem by Mitio Nagumo in 1942. The term "barrier certificate" was introduced later based on similar concept in convex optimization called barrier functions. Barrier certificates were gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forward Invariant
Forward is a relative direction, the opposite of backward. Forward may also refer to: People *Forward (surname) Sports * Forward (association football) * Forward (basketball), including: ** Point forward ** Power forward (basketball) ** Small forward * Forward (ice hockey) ** Power forward (ice hockey) * In rugby football: ** Forwards (rugby league), in rugby league football ** Forwards (rugby union), in rugby union football * Forward Sports, a Pakistan sportswear brand * BK Forward, a Swedish club for association football and bandy Politics * Avante (political party) (Portuguese for ''forward''), a political party in Brazil * Forward (Belgium), a political party in Belgium * Forward (Denmark), a political party in Denmark * Forward (Greenland), a political party in Greenland * Forward Party (United States), a centrist American political party * Kadima (Hebrew for ''forward''), a political party in Israel * La République En Marche! (sometimes translated as ''Forw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hybrid System
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine or automaton). Often, the term "hybrid dynamical system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena. In general, the ''state'' of a hybrid system is defined by the values of the ''continuous variables'' and a discrete ''mode''. The state changes either continuously, according to a flow condition, or discretely according to a ''control graph''. Continuous flow is permitted as long as so-called ''invariants'' hold, while discrete transitions can occur as soon as given ''jump conditions'' are satisfied. Discrete tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Set
A set is the mathematical model for a collection of different things; a set contains ''elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work '' Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Functions
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bony–Brezis Theorem
In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant under the flow defined by a vector field, namely at each point of the closed set the vector field must have non-positive inner product with any exterior normal vector to the set. A vector is an ''exterior normal'' at a point of the closed set if there is a real-valued continuously differentiable function maximized locally at the point with that vector as its derivative at the point. If the closed subset is a smooth submanifold with boundary, the condition states that the vector field should not point outside the subset at boundary points. The generalization to non-smooth subsets is important in the theory of partial differential equations. The theorem had in fact been previously discovered by Mitio Nagumo in 1942 and is also known as the Nagumo theorem. Statement Let ''F'' be cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mitio Nagumo
Mitio (Michio) Nagumo ( ja, 南雲 道夫; May 7, 1905 – February 6, 1995) was a Japanese mathematician, who specialized in the theory of differential equations. He gave the first necessary and sufficient condition for positive invariance of closed sets under the flow induced by ordinary differential equations ( Nagumo/Bony-Brezis theorem). Biography Mitio Nagumo graduated from the Department of Mathematics at the Imperial University of Tokyo in March 1928. In March 1931 he was appointed Lecturer in the Faculty of Technology at the Imperial University of Kyushu. In February 1932 he left Japan for an academic visit to Göttingen, where he remained for two years. Upon his return from Göttingen in March 1934, he was appointed Lecturer in the Department of Mathematics at the Imperial University of Osaka, and was promoted to Associate Professor in September that year, becoming Professor in the Faculty of Science in March 1936. In March 1937 Nagumo received a Doctor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, finance, statistics (optimal experimental design), and structural optimization, where the approximation concept has proven to be efficient. With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming. Definition A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrier Function
In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible region of an optimization problem. Such functions are used to replace inequality constraints by a penalizing term in the objective function that is easier to handle. The two most common types of barrier functions are inverse barrier functions and logarithmic barrier functions. Resumption of interest in logarithmic barrier functions was motivated by their connection with primal-dual interior point methods. Motivation Consider the following constrained optimization problem: :minimize :subject to where is some constant. If one wishes to remove the inequality constraint, the problem can be re-formulated as :minimize , :where if , and zero otherwise. This problem is equivalent to the first. It gets rid of the inequality, but introduces the issue that the penalty function , and theref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hybrid Systems
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both ''flow'' (described by a differential equation) and ''jump'' (described by a state machine or automaton). Often, the term "hybrid dynamical system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena. In general, the ''state'' of a hybrid system is defined by the values of the ''continuous variables'' and a discrete ''mode''. The state changes either continuously, according to a flow condition, or discretely according to a ''control graph''. Continuous flow is permitted as long as so-called ''invariants'' hold, while discrete transitions can occur as soon as given ''jump conditions'' are satisfied. Discrete tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stephen Prajna
Stephen or Steven is a common English first name. It is particularly significant to Christians, as it belonged to Saint Stephen ( grc-gre, Στέφανος ), an early disciple and deacon who, according to the Book of Acts, was stoned to death; he is widely regarded as the first martyr (or " protomartyr") of the Christian Church. In English, Stephen is most commonly pronounced as ' (). The name, in both the forms Stephen and Steven, is often shortened to Steve or Stevie. The spelling as Stephen can also be pronounced which is from the Greek original version, Stephanos. In English, the female version of the name is Stephanie. Many surnames are derived from the first name, including Stephens, Stevens, Stephenson, and Stevenson, all of which mean "Stephen's (son)". In modern times the name has sometimes been given with intentionally non-standard spelling, such as Stevan or Stevon. A common variant of the name used in English is Stephan ; related names that have found some cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ali Jadbabaie
Ali Jadbabaie is an Iranian-American systems scientist and decision theorist and the JR East Professor of Engineering at Massachusetts Institute of Technology. Prior to joining MIT, he was the Alfred Fitler Moore Professor of Network Science in the Department of Electrical and Systems Engineering at the University of Pennsylvania and a postdoc at the department of Electrical and Computer Engineering at Yale University under A. Stephen Morse (2001–2002). Jadbabaie is an internationally renowned expert in the control and coordination of multi-robot formations, distributed optimization, network economics, and network science. He is currently the head of the Civil and Environmental Engineering Department at MIT. Previously he served as the Associate director of the Institute for Data, Systems and Society (IDSS) at MIT and was the program Head for the Social and Engineering Systems PhD program. He was a cofounder and director of the Singh Program in Networked & Social Systems Engine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
