Determining the viability for hybrid control systems on a region with piecewise smooth boundary
Yanli Han Yan Gao
Numerical Algebra, Control & Optimization 2015, 5(1): 1-9 doi: 10.3934/naco.2015.5.1
This paper is devoted to determining the viability of hybrid control systems on a region which is expressed by inequalities of piecewise smooth functions. Firstly, the viability condition for the differential inclusion is discussed based on nonsmooth analysis. Secondly, the result is generalized to hybrid differential inclusion. Finally, the viability condition of differential inclusion on a region with the max-type function is given.
keywords: piecewise smooth function Hybrid control system differential inclusion. viability nonsmooth analysis
Yan Gao Zhiqiang Xu Lei Wang Honglei Xu
Numerical Algebra, Control & Optimization 2015, 5(1): i-ii doi: 10.3934/naco.2015.5.1i
This Special Issue of Numerical Algebra, Control and Optimization (NACO) is dedicated to Professor Enmin Feng for his important contributions in Applied Optimization, Optimal Control, System Identification and Large Scale Computing and their Engineering Applications and on the occasion of his 75th Birthday.

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Substitution secant/finite difference method to large sparse minimax problems
Junxiang Li Yan Gao Tao Dai Chunming Ye Qiang Su Jiazhen Huo
Journal of Industrial & Management Optimization 2014, 10(2): 637-663 doi: 10.3934/jimo.2014.10.637
We present a substitution secant/finite difference (SSFD) method to solve the finite minimax optimization problems with a number of functions whose Hessians are often sparse, i.e., these matrices are populated primarily with zeros. By combining of a substitution method, a secant method and a finite difference method, the gradient evaluations can be employed as efficiently as possible in forming quadratic approximations to the functions, which is more effective than that for large sparse unconstrained differentiable optimization. Without strict complementarity and linear independence, local and global convergence is proven and $q$-superlinear convergence result and $r$-convergence rate estimate show that the method has a good convergence property. A handling method of a nonpositive definitive Hessian is given to solve nonconvex problems. Our numerical tests show that the algorithm is robust and quite effective, and that its performance is comparable to or better than that of other algorithms available.
keywords: partition finite difference secant method. nondifferentiable optimization substitution sparsity Minimax problem
A characterization of Sierpiński carpet rational maps
Yan Gao Jinsong Zeng Suo Zhao
Discrete & Continuous Dynamical Systems - A 2017, 37(9): 5049-5063 doi: 10.3934/dcds.2017218

In this paper we prove that a postcritically finite rational map with non-empty Fatou set is Thurston equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpiński carpet.

keywords: Low dimensional dynamics rational maps Sierpiński carpet Julia sets thurston equivalent expanding Thurston maps
Second order sufficient optimality conditions for hybrid control problems with state jump
Lihua Li Yan Gao Hongjie Wang
Journal of Industrial & Management Optimization 2015, 11(1): 329-343 doi: 10.3934/jimo.2015.11.329
In this paper, an optimal control problem for a class of hybrid systems is considered. By introducing a new time variable and transforming the hybrid optimal control problem into an equivalent problem, second order sufficient optimality conditions for this hybrid problem are derived. It is shown that sufficient optimality conditions can be verified by checking the Legendre-Clebsch condition and solving some Riccati equations with certain boundary and jump conditions. An example is given to show the effectiveness of the main results.
keywords: Riccati equation. optimal control Hybrid system sufficient optimality condition necessary optimality condition

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