Necessary and sufficient conditions for stabilization of discrete-time planar switched systems

This paper discusses the stabilization problem for single-input discrete-time planar switched systems. We establish necessary and sufficient conditions for the stabilization of planar switched systems. A series of linear inequalities are presented for describing the set of all common quadratic Lyapunov functions. Our results are not only easily testable, but also constructive.     

Necessary and sufficient conditions for robust global asymptotic stabilization of discrete-time systems

We give Lyapunov-like conditions for non-uniform in time output stabilization of discrete-time systems. Particularly, it is proved that for a discrete-time control system there exists a (continuous) output stabilizing feedback if and only if there exists a (strong) output control Lyapunov function (OCLF). Moreover, strategies for the construction of continuous robust feedback stabilizers are presented.     

Necessary and sufficient conditions for transversals of countable set systems

This paper proves a conjecture of C. St. J. A. Nash-Williams giving necessary and sufficient conditions for an arbitrary countable system of sets to have a transversal.     

Necessary and sufficient conditions for distinguishability of linear control systems

Distinguishability plays a crucial rule in studying observability of hybrid system such as switched system. Recently, for two linear systems, Lou and Si gave a condition not only necessary but also sufficient to the distinguishability of linear systems. However, the condition is not easy enough to verify. This paper will give a new equivalent condition which is relatively easy to verify.     

Consensus of second-order and high-order discrete-time multi-agent systems with random networks

This paper studies the convergence and convergence speed for the second-order and the high-order discrete-time multi-agent systems with random networks and arbitrary weights. Random networks mean that the existence of any edge is probabilistic and independent of any other edge. By introducing the agreement set, velocity control gain and high-order state control gain, some consensus protocols are provided for the discrete-time random networks. Moreover, the per-step and asymptotic convergence factors are proposed to measure the convergence and convergence speed. Some examples and simulation results are given to illustrate the effectiveness of the obtained theoretical results.     

Necessary and sufficient conditions for optimal control of quasilinear partial differential systems

First-order necessary and sufficient conditions are obtained for the following quasilinear distributed-parameter optimal control problem: $$max\left\{ {J(u) = \int_\Omega {F(x,u,t) d\omega + } \int_{\partial \Omega } {G(x,t) \cdot d\sigma } } \right\},$$ subject to the partial differential equation $$A(t)x = f(x,u,t),$$ wheret,u,G are vectors andx,F are scalars. Use is made of then-dimensional Green's theorem and the adjoint problem of the equation. The second integral in the objective function is a generalized surface integral. Use of then-dimensional Green's theorem allows simple generalization of single-parameter methods. Sufficiency is proved under a concavity assumption for the maximized Hamiltonian $$H^\circ (x,\lambda ,t) = \max \{ H(x,u,\lambda ,t):u\varepsilon K\} $$ .     

Necessary and sufficient conditions for hyperplane transversals

We prove that a finite family ={B ₁,B ₂, ...,B _n } of connected compact sets in ^d has a hyperplane transversal if and only if for somek there exists a set of pointsP={p ₁,p ₂, ...,p _n } (i.e., ak-dimensional labeling of the family) which spans ^k and everyk+2 sets of are met by ak-flat consistent with the order type ofP. This is a common generalization of theorems of Hadwiger, Katchalski, Goodman-Pollack and Wenger.Supported in part by NSF grant DMS-8501947 and CCR-8901484, NSA grant MDA904-89-H-2030, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648.Supported by the National Science and Engineering Research Council of Canada and DIMACS.     

Necessary and sufficient conditions for bang-bang control

Necessary and sufficient conditions for the optimal control to be bang-bang are presented for a nonlinear system. The payoff, which is not necessarily quadratic, is assumed to be described by a Hilbert-space norm and to be differentiable and convex. The results are extensions of Ref. 1 to the case of nonlinear systems.     

Necessary and sufficient conditions for multivariate majorization

Let X and Y be m × n matrices whose elements are in K, a real or complex field. We obtain necessary and sufficient conditions for the existence of a matrix A belonging to the convex hull of a certain subgroup of the general linear group GL_n(K) such that X = YA, which unite and generalize several known results concerning majorization.     

Necessary and sufficient conditions for quadratic minimality

We provide necessary and sufficient conditions for a (non-convex) quadratic function to take a local minimum over a convex set. Various limiting examples are given.     

Necessary and sufficient conditions for the existence of stabilizing feedback for control systems

We prove that the existence of a non-smooth control Lyapunovfunction is a necessary and sufficient condition for the existenceof an ordinary smooth time-varying feedback that stabilizesan affine time-varying control system. Results concerning thenon-affine case are also provided.     

Necessary and sufficient conditions for interval-valued differentiability

This paper presents necessary and sufficient conditions for generalized Hukuhara differentiability of interval-valued functions and counterexamples of some equivalences previously presented in the literature, for which important results are based on. Moreover, applications of interval generalized Hukuhara differentiability are presented.     

Necessary and sufficient conditions for finiteness of lifetime

We give a geometric characterization for the finiteness of conditioned Brownian motion for a general class of simply connected domains, extending previous results and exhibit some new examples of domains with infinite area and finite lifetime.I would like to thank Professor Rodrigo Bañuelos, my academic advisor, for his help and guidance on this paper which is part of my Ph.D. thesis.     

Necessary and sufficient conditions for the Robbins-Monro method

This paper examines the relation between convergence of the Robbins-Monro iterates $\text{X}{}_{\mathrm{n+1}}\text{= X}{}_{\mathrm{n}}\text{?a}{}_{\mathrm{n}}\text{?(X}{}_{\mathrm{n}}\text{)+a}{}_{\mathrm{n}}\text{ξ}{}_{\mathrm{n}}\text{, ?(θ)=0}$, and the laws of large numbers S_n=a_nΣ^n?1_j=0ξ_j→0 as n→+∞. If a_n is decreasing at least as rapidly as c/n, then X_n→θw.p. 1 (resp. in L_p, p?1) implies S_n→0 w.p. 1 (resp. in L_p, p?1) as n→+∞. If a_n is decreasing at least as slowly as c?n and lim_n→+∞a_n=0, then S_n→0 w.p. 1 (resp. in L_p, p?2) implies X_n→θw.p. 1 (resp. in L_p, p?2) as n →+∞. Thus, there is equivalence in the frequently examined case $\text{a}{}_{\mathrm{n}}\text{?c?n}$. Counter examples show that the LLN must have the form of S_n, that the rate of decrease conditions are sharp, that the weak LLN is neither necessary nor sufficient for the convergence in probability of X_n to θ when $\text{a}{}_{\mathrm{n}}\text{?c?n}$.     

Necessary and sufficient conditions in constrained optimization

Additional conditions are attached to the Kuhn-Tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization, under appropriate constraint qualifications. Necessary and sufficient conditions are also given for optimality of the dual problem. Duality and converse duality are treated accordingly.