P. Zezza

Strong Optimality for a Bang-Bang Trajectory

In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C0 topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.

The conjugate point condition for smooth control sets

Abstract In this paper we present, for optimal control problems with smooth control constraints, second-order necessary conditions stated in terms of conjugate points and a Riccati-type equation. We show explicitly how to construct a set of feasible directions and how to derive the corresponding accessory problem.

Optimality Conditions for a Constrained Control Problem

This paper is devoted to the study of necessary or sufficient second-order conditions for a weak local minimum in an optimal control problem. The problem is stated in the Mayer form and includes equality constraints both on the endpoints and on the state-control trajectory. The second-order conditions are stated through an associated linear--quadratic problem.

Necessary conditions for optimal control problems: conjugate points

In this paper we introduce a definition of “normality” and of “conjugate points” for a general optimal control problem. Using these concepts we obtain new second-order necessary conditions for optimality. In the special case when the control set U is the whole space or in the classical setting of calculus of variations, our conditions reduce to known results, namely, the Jacobi condition and the existence of a solution to a certain Riccati equation.

Coupled points in the calculus of variations and applications to periodic problems

The aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions, and to develop second order necessary conditions for optimality. When one of the end points is fixed, our necessary conditions reduce to the known ones involving conjugate points. We also apply our results to the periodic problems of the calculus of variations.

Nonlinear boundary value problems in Banach spaces for multivalue differential equations on a non-compact interval

where A and L are linear operators and M is a continuous, generally nonlinear operator. We want to show that, under suitable hypotheses, the problem (1.1) has a solution defined on an interval J = [a, b), x < a < b d + xc;. The analogous problem for ordinary differential equations on a compact interval has been treated by Scrucca [ 11. The linear boundary value problem has been studied by Cecchi et al. [Z] in the univoque case and by Anichini and Zecca [3,4], in the multivoque one. For a wide bibliography and exposition on boundary value problems for differential equations see Conti [5]. 2. NOTATIONS AND HYPOTHESES

On the asymptotic behavior of the solutions of a class of second-order linear differential equations

Abstract We give necessary and sufficient conditions for the solutions of the differential equation ( p ( t ) x ′( t ))′ = q ( t ) x ( t ) to be bounded together with their first derivatives. We also study the asymptotic behavior of the solutions.

Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.

Coupled points in optimal control theory

The concept of coupled points is introduced for an optimal control problem where both state endpoints are allowed to vary. This definition leads to the extension of the theory of conjugate points to the optimal-control setting. Under suitable controllability assumptions, weaker than those previously considered, it is shown that the nonexistence of coupled points in the open interval (a, b) is a necessary condition for weak local optimality. This result generalizes the ones of the same kind known from the calculus of variations. In the special case when one or both state endpoints are fixed, the notion of coupled points is more general than those of focal or conjugate points. >

An invariant second variation in optimal control

For an optimal control problem we define a second variation which is invariant under change of coordinates, and realize it as a linear-quadratic problem. When the strong Legendre condition is satisfied we give a complete Hamiltonian characterization of the index and of the nullity of the second variation.