In this paper the diierential equations describing the minimal length curves satisfying the integral constraining relations of a general type are obtained. Moreover, an additional necessary condition supplementing Pontryagin maximum principle for the generalized isoperimetric problem is established. All results are illustrated by the analysis of generalized Dido's problem.

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin Maximum Principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.

In this paper we will study three hypotheses proposed by L. I. Rozonoer [1] in optimal control theory in order to derive conditions for the existence of an optimal control under all initial conditions, and the relationships between Pontryagin maximum principle and the dynamic programming method.

The purpose of this paper is to provide an a]ternate statement of the Pontryagin maximum principle as applied to systems which are most conveniently and naturally described by matrix, rather than vector, differential or difference equations. The use of gradient matrices facilitates the manipulation of the resultant equations. The ~heory is applied to the solution of a simple optimization problem.

In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow-Rashevski theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite dimensional setting.

We review some recent results on the theory of Lagrangian systems on Lie algebroids. In particular we consider the symplectic and variational formalism and we study reduction. Finally we also consider optimal control systems on Lie algebroids and we show how to reduce Pontryagin maximum principle.

An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler–Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical met...

We obtain a discrete time analog of E. Noether’s theorem in Optimal Control, asserting that integrals of motion associated to the discrete time Pontryagin Maximum Principle can be computed from the quasiinvariance properties of the discrete time Lagrangian and discrete time control system. As corollaries, results for first-order and higher-order discrete problems of the calculus of variations a...

The purpose of this paper is to show that modern control theory, both in the form of the “classical” ideas developed in the 1950s and 1960s, and in that of later, more recent methods such as the “nonsmooth,” “very nonsmooth” and “differentialgeometric” approaches, provides the best and mathematically most natural setting to do justice to Johann Bernoulli’s famous 1696 “brachistochrone problem.”...