VARIATIONAL ANALYSIS FOR THE MAXIMAL TIME FUNCTION IN NORMED SPACES

For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.

A Characterization of Paratingent Cone and P-subderivative with Applications in Nonsmooth Analysis

We give a new characterization of the paratingent cone in terms of contingent cones,i. e., the paratingent cone to any open set at a boundary point is the upper limit of the contingentcones at the neighboring points. We use this result to characterize the strict differentiability in termsof the contingent directional derivatives. We also define a P-subderivative for continuous functionsand develop a subdifferential calculus with applications to optimality conditions in mathematicalprograming.

Stable Flocking Motion of Mobile Agents Following a Leader in Fixed and Switching Networks

Multiple mobile agents with double integrator dynamics, following a leader to achieve a flocking motion formation, are studied in this paper. A class of local control laws for a group of mobile agents is proposed. From a theoretical proof, the following conclusions are reached： （i） agents globally align their velocity vectors with a leader, （ii） they converge their velocities to the leaders velocity, （iii） collisions among interconnected agents are avoided, and （iv） agent＇s artificial potential functions are minimized. We model the interaction and/or communication relationship between agents by algebraic graph theory. Stability analysis is achieved by using classical Lyapunov theory in a fixed network topology, and differential inclusions and nonsmooth analysis in a switching network topology respectively. Simulation examples are provided.

MULTIPLICITY OF SOLUTIONS FOR A QUASILINEAR ELLIPTIC EQUATION

We study a quasilinear elliptic equation with polynomial growth coefficients. The existence of infinitely many solutions is obtained by a dual method and a nonsmooth critical point theory.

Nonsmooth critical point theory and applications to the spectral graph theory

Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critical sets could be either very vague or too large.To overcome these difficulties,we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons.This yields natural extensions of classical results in the critical point theory,such as the Liusternik-Schnirelmann multiplicity theorem.More importantly,eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes,which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.

VIABILITY CRITERIA FOR DIFFERENTIAL INCLUSIONS

A method of verifying the viability criterion at a given point for a region with nonsmooth boundary, which is expressed by a quasidifferentiabl function, under a differential inclusion which is a convex hull of finitely many functions, is proposed. By this method, determining the viability is transformed into solving a number of systems of linear inequalities, or equivalently solving a number of linear programming problems. For the other differential inclusion, called the generalized convex process, it is shown that viability condition holds for a polytope if and only if it holds at all of its vertices. This result is an extension of corresponding one for a linear control system.

STABILITY FOR THE MIX-DELAYED COHEN-GROSSBERG NEURAL NETWORKS WITH NONLINEAR IMPULSE

In this paper,the authors are concerned with the stability of the mix-delayed Cohen-Grossbergneural networks with nonlinear impulse by the nonsmooth analysis.Some novel sufficientconditions are obtained for the existence and the globally asymptotic stability of the unique equilibriumpoint,which include the well-known results on some impulsive systems and non-impulsive systems asits particular cases.The authores also analyze the globally exponential stability of the equilibriumpoint.Two examples are exploited to illustrate the feasibility and effectiveness of our results.

DELAY DIFFERENTIAL EQUATIONS UNDER NONLINEAR IMPULSIVE CONTROL AND APPLICATIONS TO NEURAL NETWORK MODELS

In this paper, a class of delay differential equations with nonlinear impulsive control is discussed. Based on the nonsmooth analysis, criteria of stability are obtained for delay differential equations with nonlinear impulses control under certain conditions. These criteria can be applied to some neural network models. At the end of the paper, two examples are provided to illustrate the feasibility and effectiveness of the proposed results.

Finite-time consensus protocols for multi-dimensional multi-agent systems

A fnite.-time consensus protocol is proposed for multi -dimensional multi- agent systems, using direction peserving signumcontrols. Flipp solutions and nonsmooh analysis tehniques are adopted to handle discontinuities. Suficient and ncessaryconditions are provided to guarantee infinte time convergence and boundedness of the solutions. It turns out that the numberof agents which have cotinuous contol law plays an ssenan role in fnite-tine conerence In adidio it is shown thatthe unit bals itoduced bylp, norms. where p ∈[1,∞] , are inariat for the closed lop.