Diameter-girth sufficient conditions for optimal extraconnectivity in graphs

For a connected graph G, the rth extraconnectivity κ_r(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ₀(G) and κ₁(G), respectively. The minimum r-tree degree of G, denoted by ξ_r(G), is the minimum cardinality of N(T) taken over all trees T⊆G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neighbors of some vertex of T. When r=1, any such considered tree is just an edge of G. Then, ξ₁(G) is equal to the so-called minimum edge-degree of G, defined as ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short κ_r-optimal, if κ_r(G)?ξ_r(G). In this paper, we present some sufficient conditions that guarantee κ_r(G)?ξ_r(G) for r?2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r=1.     

Necessary and sufficient conditions for the existence of periodic solutions of Riccati's equation with periodic coefficients

In this paper, the concept of generalized-periodic solution is given for Riccati's equationy'=a(t)y ²+b(t)y+c(t) with perriodic coefficients, the relation between generalized-periodic solutions and the characteristic numbers of systemx'₁=c(t)x ₂,x'₂=–a(t)x ₁-b(t)x ₂ is indicated, and several necessary and sufficient conditions are given using the coefficients. Moreover, in the case ofa(t) without zero, the relation between the number of continuous-periodic solutions ofy'=a(t)y ²+b(t)y+c(t)+ and the parameter is given; thus the problem on the existence of continuous-periodie solutions is basically solved.     

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 $$H^\circ (x,\lambda ,t) = \max \{ H(x,u,\lambda ,t):u\varepsilon K\} $$ .     

Arrow-type sufficient optimality conditions for nondifferentiable optimal control problems with state constraints

In this paper a class of nondifferentiable optimal control problems governed by differential inclusions and subject to state variable inequality constraints is considered. Sufficient conditions using the concavity of the maximized Hamiltonian are given. Furthermore, a counterexample is presented that shows that in the nondifferentiable case the maximum principle does not form sufficient optimality conditions if the adjoint relation is formulated in terms of the ordinary Hamiltonian rather than the maximized one. Finally, it is shown that the sufficient conditions correspond to Clarke's necessary conditions with some additional assumptions such as concavity.     

Integral formulation for a steady-state transport process with general boundary conditions

A linear equation for a particle steady-state transport process in a homogeneous slab of finite thickness with boundary conditions of general type is derived. This equation differs from the well-known integral equation for no-reentry boundary conditions because of the presence of a per-turbance linear operator which describes the effect of the re-emission of the particles incident at the wall. The properties of the resulting operator are investigated. The dependence of the first positive eigenvalue on physical parameters is studies in detail. The results obtained are enough to discuss the existence of a critical strictly positive solution of the physical problems which motivated this research.     

First and second order sufficient conditions for optimal control and the calculus of variations

In this paper we develop first and second order sufficient conditions for optimal control and the calculus of variations problems. Our conditions are derived from the Hamilton-Jacobi approach [15, Thm. 2], which was obtained for the generalized problem of Bolza. We do not require any convexity on the data [7] and [11], or that the control setU is polyhedral [14], or that the control function is in the interior ofU [8]. Instead, we assume a certain inequality which is satisfied in each of the above mentioned cases.The publication of this report has been made possible due to a grant of the Fonds FCAC for the help and support of research.     

The necessary and sufficient conditions for the existence of periodic orbits in a Lotka-Volterra system

By extending Darboux method to three dimension, we present necessary and sufficient conditions for the existence of periodic orbits in three species Lotka-Volterra systems with the same intrinsic growth rates. Therefore, all the published sufficient or necessary conditions for the existence of periodic orbits of the system are included in our results. Furthermore, we prove the stability of periodic orbits. Hopf bifurcation is shown for the emergence of periodic orbits and new phenomenon is presented: at critical values, each equilibrium are surrounded by either equilibria or periodic orbits.     

Necessary and sufficient conditions for the simultaneous diagonability of two quadratic forms

A new necessary and sufficient condition is obtained for the simultaneous diagonability of two quadratic forms in n variables. The condition is such that several known sufficient conditions can easily be derived from it.     

Necessary and sufficient conditions for quadratic stabilizability of an uncertain system

Consider an uncertain system (Σ) described by the equationx(t)=A(r(t))x(t)+B(s(t))u(t), wherex(t) ∈R ⁿ is the state,u(t) ∈R ^m is the control,r(t) ∈ ? ?R ^p represents the model parameter uncertainty, ands(t) ∈L ?R ^l represents the input connection parameter uncertainty. The matrix functionsA(·),B(·) are assumed to be continuous and the restraint sets ?,L are assumed to be compact. Within this framework, a notion of quadratic stabilizability is defined. It is important to note that this type of stabilization is robust in the following sense: The Lyapunov function and the control are constructed using only the bounds ?,L. Much of the previous literature has concentrated on a fundamental question: Under what conditions onA(·),B(·), ?,L can quadratic stabilizability be assured? In dealing with this question, previous authors have shown that, if (Σ) satisfies certain matching conditions, then quadratic stabilizability is indeed assured (e.g., Refs. 1–2). Given the fact that matching is only a sufficient condition for quadratic stabilizability, the objective here is to characterize the class of systems for which quadratic stabilizability can be guaranteed.     

A note on the use of direct sufficient conditions in optimal control problems

It is always possible to transform a nonautonomous optimal control problem into an autonomous one. However, the direct sufficient conditions may yield no information when applied to this autonomous problem, even though they do allow one to conclude sufficiency when applied to the original nonautonomous problem.This research was supported by the Air Force Office of Scientific Research, under Grant No. AFOSR-76-2923.     

Necessary and sufficient conditions of some strong optimal solutions to the interval linear programming

This paper considers optimal solutions of general interval linear programming problems. Necessary and sufficient conditions of (A,b)$(\mathbf{A},\mathbf{b})$-strong and (A,b,c)$(\mathbf{A},\mathbf{b},\mathbf{c})$-strong optimal solutions to the interval linear programming with inequality constraints are proposed. The features of the proposed methods are illustrated by some examples.     

Necessary conditions for an infinite time optimal control problem

In this paper, we deal with an optimal control problem with infinite transfer time. Using methods which involve local conditional stability, we obtain necessary conditions for optimality under the assumption that the right-hand side of the state equation is Fréchet-differentiable at every point of the optimal solution, and under some weak assumptions about the asymptotical behaviour of the set of perturbations of the solution. The results are illustrated in a specific case considered by Pontryagain et al.     

The necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional differential equations

In this paper, using Fourier series, we study the problem of the existence of periodic solutions of a type of periodic neutral differential difference system. Some necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional equation system are obtained, and at the same time, we present a method with formula shows how to find the periodic solutions.     

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

The paper deals with the existence of positive periodic solutions to a system of degenerate parabolic equations with delayed nonlocal terms and Dirichlet boundary conditions. Taking in each equation a meaningful function as a control parameter, we show that for a suitable choice of a class of such controls we have, for each of them, a time-periodic response of the system under different assumptions on the kernels of the nonlocal terms. Finally, we consider the problem of the minimization of a cost functional on the set of pairs: control-periodic response. The considered system may be regarded as a possible model for the coexistence problem of two biological populations, which dislike crowding and live in a common territory, under different kind of intra- and inter-specific interferences.     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

Second order sufficient conditions and sensitivity analysis for the optimal control of a container crane under state constraints

Stability and sensitivity analysis of parametric control problems has recently been elaborated for optimal control problems subject to pure state constraints. This paper illustrates the numerical aspects of sensitivity analysis for a complex practical example: the optimal control of a container crane with a state constraint on the vertical velocity. The multiple shooting method is used to determine a nominal solution satisfying first order necessary conditions. Second order sufficient conditions are checked by showing that an associated Riccati equation has a bounded solution. Sensitivity differentials of optimal solutions an computed with respect to variations in the swing angle