In this article, a general Lyapunov stability theory of nonlinear systems is put forward and it contains asymptotic/finite-time/fast finite-time/fixed-time stability. Especially, a more accurate estimate of the settli...In this article, a general Lyapunov stability theory of nonlinear systems is put forward and it contains asymptotic/finite-time/fast finite-time/fixed-time stability. Especially, a more accurate estimate of the settling-time function is exhibited for fixedtime stability, and it is still extraneous to the initial conditions.This can be applied to obtain less conservative convergence time of the practical systems without the information of the initial conditions. As an application, the given fixed-time stability theorem is used to resolve time-varying(TV) convex optimization problem.By the Newton's method, two classes of new dynamical systems are constructed to guarantee that the solution of the dynamic system can track to the optimal trajectory of the unconstrained and equality constrained TV convex optimization problems in fixed time, respectively. Without the exact knowledge of the time derivative of the cost function gradient, a fixed-time dynamical non-smooth system is established to overcome the issue of robust TV convex optimization. Two examples are provided to illustrate the effectiveness of the proposed TV convex optimization algorithms. Subsequently, the fixed-time stability theory is extended to the theories of predefined-time/practical predefined-time stability whose bound of convergence time can be arbitrarily given in advance, without tuning the system parameters. Under which, TV convex optimization problem is solved. The previous two examples are used to demonstrate the validity of the predefined-time TV convex optimization algorithms.展开更多
In this paper, the authors give the boundedness of the commutator [b,μΩ,γ] from the homogeneous Sobolev space LP（R^n） to the Lebesgue space L^p（R^n） for 1 〈 p 〈 ∞, where μΩ,γ denotes the Marcinkiewicz int...In this paper, the authors give the boundedness of the commutator [b,μΩ,γ] from the homogeneous Sobolev space LP（R^n） to the Lebesgue space L^p（R^n） for 1 〈 p 〈 ∞, where μΩ,γ denotes the Marcinkiewicz integral with rough hypersingular kernel defined by μΩ,γf（x）=（∫0^∞｜∫｜x-y｜≤tΩ（x-y）/｜x-y｜^n-1f（y）dy｜^2dt/t^3＋2γ）^1/2,with Ω∈L^1（S^n-1）for 0〈γ〈min（n/2,n/p）or Ω∈L（log＋L）^β（S^n-1）for｜1-2/p｜〈β〈1（0〈γ〈n/2）,respectively.展开更多
基金supported in part by the National Natural Science Foundation of China(62203281)。
文摘In this article, a general Lyapunov stability theory of nonlinear systems is put forward and it contains asymptotic/finite-time/fast finite-time/fixed-time stability. Especially, a more accurate estimate of the settling-time function is exhibited for fixedtime stability, and it is still extraneous to the initial conditions.This can be applied to obtain less conservative convergence time of the practical systems without the information of the initial conditions. As an application, the given fixed-time stability theorem is used to resolve time-varying(TV) convex optimization problem.By the Newton's method, two classes of new dynamical systems are constructed to guarantee that the solution of the dynamic system can track to the optimal trajectory of the unconstrained and equality constrained TV convex optimization problems in fixed time, respectively. Without the exact knowledge of the time derivative of the cost function gradient, a fixed-time dynamical non-smooth system is established to overcome the issue of robust TV convex optimization. Two examples are provided to illustrate the effectiveness of the proposed TV convex optimization algorithms. Subsequently, the fixed-time stability theory is extended to the theories of predefined-time/practical predefined-time stability whose bound of convergence time can be arbitrarily given in advance, without tuning the system parameters. Under which, TV convex optimization problem is solved. The previous two examples are used to demonstrate the validity of the predefined-time TV convex optimization algorithms.
基金Supported by National Natural Science Foundation of China （Grant Nos. 10931001, 10901017） and Specialized Research Fund for the Doctoral Program of China （Grant No. 20090003110018）Acknowledgements The authors would like to express their gratitude to the referee for his/her very careful reading and many important valuable comments.
文摘In this paper, the authors give the boundedness of the commutator [b,μΩ,γ] from the homogeneous Sobolev space LP（R^n） to the Lebesgue space L^p（R^n） for 1 〈 p 〈 ∞, where μΩ,γ denotes the Marcinkiewicz integral with rough hypersingular kernel defined by μΩ,γf（x）=（∫0^∞｜∫｜x-y｜≤tΩ（x-y）/｜x-y｜^n-1f（y）dy｜^2dt/t^3＋2γ）^1/2,with Ω∈L^1（S^n-1）for 0〈γ〈min（n/2,n/p）or Ω∈L（log＋L）^β（S^n-1）for｜1-2/p｜〈β〈1（0〈γ〈n/2）,respectively.
