Patchy vector fields and asymptotic stabilization
Patchy Vector Fields and Asymptotic Stabilization
This paper is concerned with the structure of asymptotically stabilizing feedbacks for a nonlinear control system on . We first introduce a family of discontinuous, piecewise smooth vector fields and derive a number of properties enjoyed by solutions of the corresponding O.D.E's. We then define a class of “patchy feedbacks” which are obtained by patching together a locally finite family of smooth controls. Our main result shows that, if a system is asymptotically controllable at the origin,...
Pointwise minimization of supplemented variational problems
We describe an approach to variational problems, where the solutions appear as pointwise (finite-dimensional) minima for fixed t of the supplemented Lagrangian. The minimization is performed simultaneously with respect to the state variable x and ẋ, as opposed to Pontryagin's maximum principle, where optimization is done only with respect to the ẋ-variable. We use the idea of the equivalent problems of Carathéodory employing suitable (and simple) supplements to the original minimization problem....
Pontryagin Maximum Principle for coupled slow and fast systems
P-order necessary and sufficient conditions for optimality in singular calculus of variations
This paper is devoted to singular calculus of variations problems with constraint functional not regular at the solution point in the sense that the first derivative is not surjective. In the first part of the paper we pursue an approach based on the constructions of the p-regularity theory. For p-regular calculus of variations problem we formulate and prove necessary and sufficient conditions for optimality in singular case and illustrate our results by classical example of calculus of variations...