This paper presents a novel design method for discrete-time repetitive control systems （RCS） based on two-dimensional （2D） discrete-time model. Firstly, the 2D model of an RCS is established by considering both th...This paper presents a novel design method for discrete-time repetitive control systems （RCS） based on two-dimensional （2D） discrete-time model. Firstly, the 2D model of an RCS is established by considering both the control action and the learning action in RCS. Then, through constructing a 2D state feedback controller, the design problem of the RCS is converted to the design problem of a 2D system. Then, using 2D system theory and linear matrix inequality （LMI） method, stability criterion is derived for the system without and with uncertainties, respectively. Parameters of the system can be determined by solving the LMI of the stability criterion. Finally, numerical simulations validate the effectiveness of the proposed method.展开更多
Suppose that G is a planar cubic graph withχi(G)>5.We show that ifχi(H)<χi(G)for each planar cubic graph H of order less thanG,thenG is either a 3-connected simple planar cubic graph,or a planar graph obtaine...Suppose that G is a planar cubic graph withχi(G)>5.We show that ifχi(H)<χi(G)for each planar cubic graph H of order less thanG,thenG is either a 3-connected simple planar cubic graph,or a planar graph obtained from a simple cubic 3-connected planar graph by adding some earrings.This shows that a minimum non-5-injectively colorable simple planar cubic graph must be 3-connected.展开更多
In 2003, Borodin and Raspaud proved that if G is a plane graph without 5-circuits and without triangles of distance less than four, then G is 3-colorable. In this paper, we prove that if G is a plane graph without 5- ...In 2003, Borodin and Raspaud proved that if G is a plane graph without 5-circuits and without triangles of distance less than four, then G is 3-colorable. In this paper, we prove that if G is a plane graph without 5- and 6-circuits and without triangles of distance less than 2, then G is 3-colorable.展开更多
Let G be a graph.We say that G is 2-divisible if for each induced subgraph H of G,either V(H)is a stable set,or V(H)can be partitioned into two sets A and B such thatω(H[A])<ω(H)andω(H[B])<ω(H).A hole is an ...Let G be a graph.We say that G is 2-divisible if for each induced subgraph H of G,either V(H)is a stable set,or V(H)can be partitioned into two sets A and B such thatω(H[A])<ω(H)andω(H[B])<ω(H).A hole is an induced cycle of length at least 4,a bull is a graph consisting of a triangle with two disjoint pendant edges,a diamond is the graph obtained from K4 by removing an edge,a dart denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 3,and a racket denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 2.In this paper,we prove that every{odd hole,H}-free graph is 2-divisible,where H is a dart,or a racket,or a bull.As corollaries,X(G)≤min{2ω^(G)-1,(ω^(G)/2+1)}if G is{odd hole,dart}-free,or{odd hole,racket}-free,or{odd hole,bull}-free.展开更多
基金supported by National Natural Science Foundation of China (Nos. 60974045 and 60674016)the Research Foundation of Education Bureau of Hunan Province, China (No. 08C090)
文摘This paper presents a novel design method for discrete-time repetitive control systems （RCS） based on two-dimensional （2D） discrete-time model. Firstly, the 2D model of an RCS is established by considering both the control action and the learning action in RCS. Then, through constructing a 2D state feedback controller, the design problem of the RCS is converted to the design problem of a 2D system. Then, using 2D system theory and linear matrix inequality （LMI） method, stability criterion is derived for the system without and with uncertainties, respectively. Parameters of the system can be determined by solving the LMI of the stability criterion. Finally, numerical simulations validate the effectiveness of the proposed method.
基金This research was supported by the National Natural Science Foundation of China(Nos.11571180 and 11331003)the Natural Science Foundation of Jiangsu Higher Education Institutions of China(No.17KJB110010).
文摘Suppose that G is a planar cubic graph withχi(G)>5.We show that ifχi(H)<χi(G)for each planar cubic graph H of order less thanG,thenG is either a 3-connected simple planar cubic graph,or a planar graph obtained from a simple cubic 3-connected planar graph by adding some earrings.This shows that a minimum non-5-injectively colorable simple planar cubic graph must be 3-connected.
基金Supported by National Natural Science Foundation of China(No.10931003 and 11171160)the Doctoral Fund of Ministry of Education of China
文摘In 2003, Borodin and Raspaud proved that if G is a plane graph without 5-circuits and without triangles of distance less than four, then G is 3-colorable. In this paper, we prove that if G is a plane graph without 5- and 6-circuits and without triangles of distance less than 2, then G is 3-colorable.
基金supported by the National Natural Science Foundation of China(No.11931006)。
文摘Let G be a graph.We say that G is 2-divisible if for each induced subgraph H of G,either V(H)is a stable set,or V(H)can be partitioned into two sets A and B such thatω(H[A])<ω(H)andω(H[B])<ω(H).A hole is an induced cycle of length at least 4,a bull is a graph consisting of a triangle with two disjoint pendant edges,a diamond is the graph obtained from K4 by removing an edge,a dart denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 3,and a racket denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 2.In this paper,we prove that every{odd hole,H}-free graph is 2-divisible,where H is a dart,or a racket,or a bull.As corollaries,X(G)≤min{2ω^(G)-1,(ω^(G)/2+1)}if G is{odd hole,dart}-free,or{odd hole,racket}-free,or{odd hole,bull}-free.
