Functional inequalities on path space of sub-Riemannian manifolds and applications

in Nonlinear Analysis: Theory, Methods and Applications (2021), 210(112387), 1-30

We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians ... [more ▼]

We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians of a sub-Riemannian structure with a chosen complement. We introduce a concept of gradient for cylindrical functionals on path space in such a way that the gradient operators are closable in L^2. With this structure in place, we show that a bound on horizontal Ricci curvature is equivalent to several inequalities for functions on path space, such as a gradient inequality, log-Sobolev inequality and Poincaré inequality. As a consequence, we also obtain a bound for the spectral gap of the Ornstein-Uhlenbeck operator. [less ▲]

Exponential Stabilization of a Class of Nonlinear Systems : A Generalized Gronwall-Bellman Lemma Approach

in Nonlinear Analysis: Theory, Methods and Applications (2011), 74

In this paper, stabilizing control design for a class of nonlinear affine systems is presented by using a new generalized Gronwall-Bellman lemma approach. The nonlinear systems under consideration can be ... [more ▼]

In this paper, stabilizing control design for a class of nonlinear affine systems is presented by using a new generalized Gronwall-Bellman lemma approach. The nonlinear systems under consideration can be non Lipschitz. Two cases are treated for the exponential stabilization: the static state feedback and the static output feedback. The robustness of the proposed control laws with regards to parameter uncertainties is also studied. A numerical example is given to show the effectiveness of the proposed method. [less ▲]

Initial-irregular oblique derivative problems for nonlinear parabolic complex equations of second order with measurable coefficients

in Nonlinear Analysis: Theory, Methods and Applications (2000)

In this paper, initial-irregular oblique derivative boundary value problems for nonlinear and non-divergence parabolic complex equations of second order in multiply connected domains are discussed, where ... [more ▼]

In this paper, initial-irregular oblique derivative boundary value problems for nonlinear and non-divergence parabolic complex equations of second order in multiply connected domains are discussed, where coefficients of equations are measurable. Firstly, the uniqueness of solutions for the above problems is verified, and then a priori estimates of solutions for the problems are given. Finally, by using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. [less ▲]