Abstract:
In this paper, new sufficient optimality theorems for a solution of a differentiable bilevel multiobjective optimization problem (BMOP) are established. We start with a discussion on solution concepts in bilevel multiobjective programming; a theorem giving necessary and sufficient conditions for a decision vector to be called a solution of the BMOP and a proposition giving the relations between four types of solutions of a BMOP are presented and proved. Then, under the pseudoconvexity assumptions on the upper and lower level objective functions and the quasiconvexity assumptions on the constraints functions, we establish and prove two new sufficient optimality theorems for a solution of a general BMOP with coupled upper level constraints. Two corollary of these theorems, in the case where the upper and lower level objectives and constraints functions are convex are presented.

Abstract:
This short article shows that the functional equation on the equilibrium price function is more complicated than that considered by Lucas [1], and that modification is required to complete the proof. Furthermore, we shall provide a sufficient condition that guarantees the uniqueness of the equilibrium price function.

Abstract:
We study a linear delay differential equation with a single positive and a single negative term. We find a necessary condition for the oscillation of all solutions. We also find sufficient conditions for oscillation, which improve the known conditions.

This paper provides a solution to generalize the integrator and the
integral control action. It is achieved by defining two function sets to
generalize the integrator and the integral control action, respectively,
resorting to a stabilizing controller and adopting Lyapunov method to analyze
the stability of the closed-loop system. By originating a powerful Lyapunov
function, a universal theorem to ensure regionally as well as semi-globally
asymptotic stability is established by some bounded information. Consequently,
the justification of two propositions on the generalization of integrator and
integral control action is verified. Moreover, the conditions used to define
the function sets can be viewed as a class of sufficient conditions to design
the integrator and the integral control action, respectively.

Abstract:
This paper is concerned with the optimal distributed control problem governed by b-equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition. By contrasting with our previous result, the proof without considering viscous coefficient is a big improvement. Secondly, based on the well-posedness result, we find a unique optimal control for the controlled system with the quadratic cost functional. Moreover, by means of the optimal control theory, we obtain the sufficient and necessary optimality condition of an optimal control, which is another major novelty of this paper. Finally, we also present the optimality conditions corresponding to two physical meaningful distributive observation cases.

Abstract:
This paper isolates and studies a class of Markov chains with a special quasi-triangular form of the transition matrix [so-called ”m,n( ” ￠ € 2m,n)-matrix]. Many discrete stochastic processes encountered in applications (queues, inventories and dams) have transition matrices which are special cases of a ”m,n( ” ￠ € 2m,n)-matrix. Necessary and sufficient conditions for the ergodicity of a Markov chain with transition ”m,n( ” ￠ € 2m,n)-matrix are determined in the article in two equivalent versions. According to the first version, these conditions are expressed in terms of certain restrictions imposed on the generating functions Ai(x) of the elements of the i-th row of the transition matrix, i=0,1,2, ￠ € |; in the other version they are connected with the characterization of the roots of a certain associated function in the unit circle of the complex plane. Results obtained in the article generalize, complement, and refine similar results existing in the literature.

Abstract:
in this short note a sensitivity result for quadratic semidefinite programming is presented under a weak form of second order sufficient condition. based on this result, also the local convergence of a sequential quadratic semidefinite programming algorithm extends to this weak second order sufficient condition. mathematical subject classification: 90c22, 90c30, 90c31, 90c55.

Abstract:
This work is devoted to practical stability of a class of regime-switching diffusions. First, the notion of practical stability is introduced. Then, sufficient conditions for practical stability and practical instability in probability and in pth mean are provided using a Lyapunov function argument. In addition, easily verifiable conditions on drift and diffusion coefficients are also given. Moreover, examples are supplied for demonstration purposes.