The Lojasiewicz Inequality And Gradient Flows In Infinite Demensional Space

The gradient system is widely used in differential equations,optimization and engineering,and the finite length property of the orbits of gradient flows is important in theoretical analysis and practical application.In this thesis,the finite length property of gradient flows,which is generated by real analytic functions and convex functions in infinite dimensional Hilbert space,is studied.The Lojasiewicz inequality is an important tool for studying the asymptotic behavior of gradient flows of real analytic functions in finite dimensional space,but fails in infinite dimensional space.In the past decades,people have been trying to find the conditions under which the Lojasiewicz inequality is effective in infinite dimensional Hilbert space.This thesis analyzes the reason why the Lojasiewicz inequality fails in infinite dimensional Hilbert space,and an example is given to illustrate that the Lojasiewicz inequality is invalid in the intersection of a compact set and the neighbourhood of the critical point.Finally,we propose the notion of Cofinal Set and prove that the Lojasiewicz inequality is valid in this kind of sets.Based on this we can prove that the trajectory of gradient flows of real analytic functions which converge to the Cofinal Set has finite length.Self-contracted curves were proposed to deal with the gradient flows of convex functions in finite dimensional space.In this thesis,the definition of self-contracted curves is extended to infinite dimensional Hilbert space,and the gradient flows of convex functions are proved to be bounded self-contracted curves.Moreover,the orbits of gradient flows of convex functions are convergent if they are in a compact set.