We consider nonlinear optimization programs with matrix inequality constraints, also known as nonlinear semidefinite programs. We prove local convergence for an augmented Lagrangian method which uses smooth spectral penalty functions. The sufficient second-order no-gap optimality condition and a suitable implicit function theorem are used to prove local linear convergence without the need to dr...

A version of the nonlinear Hodge equations is introduced in which the irrotationality condition is weakened. An elliptic estimate for solutions is derived. 1991 MSC: 58E15 (Classical field theory)

This paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps. By the Itô formula, stochastic inequality and semi-martingale convergence theorem, we study the asymptotic stability in the pth moment and almost sure exponential stability of solutions under the local Lipschitz condition and nonlinear growth condition....

This paper presents a predictor-corrector algorithm for solving the strongly nonlinear general noncovex variational inequality, which is a class of nonconvex variational inequalities involving three nonlinear operators. We establish the equivalence between the strongly nonlinear general nonconvex variational inequalities and the fixed point problem, and show that the convergence of the predicto...

This paper addresses robust H∞ control problems for nonlinear systems with parameter uncertainty. By using the technique of partition of unity, necessary and sufficient conditions for existence of strong robust H∞ dynamic compensators and static state feedback controllers are established in terms of nonlinear matrix inequality (NLMI) approach, respectively. These conditions reduce the robust H∞...

These notes contain a short exposition of selected results about parabolic equations: Schauder estimates for linear parabolic equations with Hölder coefficients, some existence, uniqueness and regularity results for viscosity solutions of fully nonlinear parabolic equations (including degenerate ones), the Harnack inequality for fully nonlinear uniformly parabolic equations. MSC. 35K55, 35D40, ...

We study the initial-boundary value problem for a system of nonlinear wave equations with nonlinear damping and source terms, in a bounded domain. The decay estimates of the energy function are established by using Nakao’s inequality. The nonexistence of global solutions is discussed under some conditions on the given parameters.

Assume that g(t) ≥ 0, and ġ(t) ≤ −γ(t)g(t) + α(t, g(t)) + β(t), t ≥ 0; g(0) = g0; ġ := dg dt , on any interval [0, T ) on which g exists and has bounded derivative from the right, ġ(t) := lims→+0 g(t+s)−g(t) s . It is assumed that γ(t), and β(t) are nonnegative continuous functions of t defined on R+ := [0,∞), the function α(t, g) is defined for all t ∈ R+, locally Lipschitz with respect to g u...