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LaSalle's Invariance Principle
LaSalle's invariance principle (also known as the invariance principle, Barbashin-Krasovskii-LaSalle principle, or Krasovskii-LaSalle principle) is a criterion for the asymptotic stability of an autonomous (possibly nonlinear) dynamical system. Global version Suppose a system is represented as : \dot = f \left(\mathbf x \right) where \mathbf x is the vector of variables, with : f \left( \mathbf 0 \right) = \mathbf 0. If a C^1(see Smoothness) function V(\mathbf x) can be found such that : \dot(\mathbf x) \le 0 for all \mathbf x (negative semidefinite), then the set of accumulation points of any trajectory is contained in where is the union of complete trajectories contained entirely in the set \. If we additionally have that the function V is positive definite, i.e. : V( \mathbf x) > 0 , for all \mathbf x \neq \mathbf 0 : V( \mathbf 0) = 0 and if contains no trajectory of the system except the trivial trajectory \mathbf x(t) = \mathbf 0 for t \geq 0, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x_e stay near x_e forever, then x_e is Lyapunov stable. More strongly, if x_e is Lyapunov stable and all solutions that start out near x_e converge to x_e, then x_e is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolai Nikolaevich Krasovsky
Nikolay Nikolayevich Krasovsky (russian: Никола́й Никола́евич Красо́вский; 7 September 1924 – 4 April 2012) was a Russian mathematician who worked in the mathematical theory of control, the theory of dynamical systems, and the theory of differential games. He was the author of Krasovskii-LaSalle principle and the chief of the Ural scientific school in mathematical theory of control and the theory of differential games. Biography Nikolay Krasovsky was born in Yekaterinburg, Soviet Union (renamed later to Sverdlovsk) in the family of a doctor. In 1949, he graduated summa cum laude from the department of metallurgical science at the Ural State Technical University. In 1954, he presented his first thesis and received his ''kandidat nauk'' degree in mathematics. In 1957, he defended his second thesis for the degree of ''doktor nauk'' and became a professor of mathematics. From 1949 to 1959, he worked at the Ural State Technical University. Since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purdue University
Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and money to establish a college of science, technology, and agriculture in his name. The first classes were held on September 16, 1874, with six instructors and 39 students. It has been ranked as among the best public universities in the United States by major institutional rankings, and is renowned for its engineering program. The main campus in West Lafayette offers more than 200 majors for undergraduates, over 70 masters and doctoral programs, and professional degrees in pharmacy, veterinary medicine, and doctor of nursing practice. In addition, Purdue has 18 intercollegiate sports teams and more than 900 student organizations. Purdue is the founding member of the Big Ten Conference and enrolls the largest student body of any individual univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caltech
The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasionally referred to as "CIT", most notably in its alma mater, but this is uncommon. is a private university, private research university in Pasadena, California. Caltech is ranked among the best and most selective academic institutions in the world, and with an enrollment of approximately 2400 students (acceptance rate of only 5.7%), it is one of the world's most selective universities. The university is known for its strength in science and engineering, and is among a small group of Institute of Technology (United States), institutes of technology in the United States which is primarily devoted to the instruction of pure and applied sciences. The institution was founded as a preparatory and vocational school by Amos G. Throop in 1891 and began ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
North Carolina State University
North Carolina State University (NC State) is a public land-grant research university in Raleigh, North Carolina. Founded in 1887 and part of the University of North Carolina system, it is the largest university in the Carolinas. The university forms one of the corners of the Research Triangle together with Duke University in Durham and the University of North Carolina at Chapel Hill. It is classified among "R1: Doctoral Universities – Very high research activity". The North Carolina General Assembly established the North Carolina College of Agriculture and Mechanic Arts, now NC State, on March 7, 1887, originally as a land-grant college. The college underwent several name changes and officially became North Carolina State University at Raleigh in 1965. However, by longstanding convention, the "at Raleigh" portion is usually omitted. Today, NC State has an enrollment of more than 35,000 students, making it among the largest in the country. NC State has historical strengths ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Texas A&M University
Texas A&M University (Texas A&M, A&M, or TAMU) is a public, land-grant, research university in College Station, Texas. It was founded in 1876 and became the flagship institution of the Texas A&M University System in 1948. As of late 2021, Texas A&M has the largest student body in the United States, and is the only university in Texas to hold simultaneous designations as a land, sea, and space grant institution. In 2001, it was inducted into the Association of American Universities. The university's students, alumni, and sports teams are known as Aggies, and its athletes compete in eighteen varsity sports as a member of the Southeastern Conference. The university was the first public higher-education institution in Texas; it opened for classes on October 4, 1876, as the Agricultural and Mechanical College of Texas (A.M.C.) under the provisions of the 1862 Morrill Land-Grant Act. In the following decades, the college grew in size and scope, expanding to its largest enrol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New York City
New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the List of United States cities by population density, most densely populated major city in the United States, and is more than twice as populous as second-place Los Angeles. New York City lies at the southern tip of New York (state), New York State, and constitutes the geographical and demographic center of both the Northeast megalopolis and the New York metropolitan area, the largest metropolitan area in the world by urban area, urban landmass. With over 20.1 million people in its metropolitan statistical area and 23.5 million in its combined statistical area as of 2020, New York is one of the world's most populous Megacity, megacities, and over 58 million people live within of the city. New York City is a global city, global Culture of New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Providence, Rhode Island
Providence is the capital and most populous city of the U.S. state of Rhode Island. One of the oldest cities in New England, it was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony. He named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city developed as a busy port as it is situated at the mouth of the Providence River in Providence County, at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool, jewelry, and silverware industries. Today, the city of Providence is home to eight hospitals and List of colleges and universities in Rhode Island#Institutions, eight institutions of higher learning which have shifted the city's economy into service industries, though it still retains some manufacturin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doklady Akademii Nauk SSSR
The ''Proceedings of the USSR Academy of Sciences'' (russian: Доклады Академии Наук СССР, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), french: Comptes Rendus de l'Académie des Sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (russian: Доклады Академии Наук), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance. In dynamical systems, an orbit is called ''Lyapunov stable'' if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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