This paper deals with the regularity of solutions of the Hamilton-Jacobi Inequality which arises in H∞ control. It shows by explicit counterexamples that there are gaps between existence of continuous and locally Lipschitz (positive definite and proper) solutions, and between Lipschitz and continuously differentiable ones. On the other hand, it is shown that it is always possible to smooth-out ...

An optimal control problem with quadratic cost functional for the steady-state Navier-Stokes equations with no-slip boundary condition is considered. Lipschitz stability of locally optimal controls with respect to certain perturbations of both the cost functional and the equation is proved provided a second-order sufficient optimality condition holds. For a sufficiently small Reynolds number, e...

We show that local minimizers of functionals of the form Z Ω [f(Du(x)) + g(x , u(x))] dx, u ∈ u0 + W 1,p 0 (Ω), are locally Lipschitz continuous provided f is a convex function with p − q growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

The relationship between a minimization problem on the in nite horizon and local stabilizability is studied for a ne control systems. A su cient condition for a system to be locally stabilizable in the sense of Filippov solutions is given in the case that the value function associated to the minimization problem is locally Lipschitz continuous.

In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...

It is shown that local epi-sub-Lipschitz continuity of the function-valued mapping associated with a perturbed optimization problem yields the local Lipschitz continuity of the inf-projections (= marginal functions, = infimal functions). The use of the theorem is illustrated by considering perturbed nonlinear optimization problems with linear constraints.