On some extension of optimal control theory

2013

We consider control systems governed by nonlinear O.D.E.'s that are affine in the time-derivative du/dt of the control u. The latter is allowed to be an integrable, possibly of unbounded variation function, which gives the system an impulsive character. As is well-known, the corresponding Cauchy problem cannot be interpreted in terms of Schwartz distributions, even in the commutative case. A robust notion of solution already proposed in the literature is here adopted and slightly generalized to the case where an ordinary, bounded, control is present in the dynamics as well. For a problem in the Mayer form we then investigate the question whether this notion of solution provides a "proper extension" of the standard problem with absolutely continuous controls u. Furthermore, we show that this impulsive problem is a variational limit of problems corresponding to controls u with bounded variation.

2010

SIAM Journal on Control and Optimization, 2020

We consider a nonlinear system, affine with respect to an unbounded control u which is allowed to range in a closed cone. To this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to u. This lack of coercivity gives the problem an impulsive character, meaning that minimizing sequences of trajectories happen to converge towards discontinuous paths. As is known, a distributional approach does not make sense in such a nonlinear setting, where, instead, a suitable embedding in the graph-space is needed. We provide higher order necessary optimality conditions for properly defined impulsive minima, in the form of equalities and inequalities involving iterated Lie brackets of the dynamical vector fields. These conditions are derived under very weak regularity assumptions and without any constant rank conditions.

Journal of Mathematical Sciences, 2010

This paper considers constrained impulsive control problems for which the authors propose a new mathematical concept of control required for the impulsive framework. These controls can arise in engineering, in particular, in problems of space navigation. We derive necessary extremum conditions in the form of the Pontryagin maximum principle and also study conditions under which the constraint regularity clarifications become weaker. In the proof of the main result, Ekeland's variational principle is used.

fe.up.pt

2003

Nondegenerate second-order necessary conditions of optimality for general nonlinear optimization problems are presented and discussed in this article. Besides functional equality and inequality constraints, we also consider constraints in the form of an inclusion into a given closed set. Without assuming a priori normality, our conditions remain informative for abnormal points, and, under very general assumptions also take into account the second order effect of the curvature of the set in the inclusion constraints.

Journal of Dynamical and Control Systems, 2003

First and second order necessary conditions of optimality for an impulsive control problem are presented and derived. One of the main features of these results is that no a priori normality assumptions are required and they are informative for abnormal control processes as well. This feature follows from the fact that the conditions are derived from an extremal principle, which is proved for an abstract minimization problem with equality and inequality type constraints and constraints given by convex cone. Two simple examples illustrate the power of our result. 2000 Mathematics Subject Classification. 49K24, 49N25. Key words and phrases. Optimal control, impulsive control, extremal principle, second order conditions of optimality, abnormality.

52nd IEEE Conference on Decision and Control, 2013

We consider control systems governed by nonlinear O.D.E.'s that are affine in the time-derivative du/dt of the control u. The latter is allowed to be an integrable, possibly of unbounded variation function, which gives the system an impulsive character. As is well-known, the corresponding Cauchy problem cannot be interpreted in terms of Schwartz distributions, even in the commutative case. A robust notion of solution already proposed in the literature is here adopted and slightly generalized to the case where an ordinary, bounded, control is present in the dynamics as well. For a problem in the Mayer form we then investigate the question whether this notion of solution provides a "proper extension" of the standard problem with absolutely continuous controls u. Furthermore, we show that this impulsive problem is a variational limit of problems corresponding to controls u with bounded variation.

2000

The motivation to study dynamic optimization problems whose solutions might involve discontinuous trajectories have been documented in a large number of publications, [4], [5], [7], [9], and [10], addressing various application areas ranging from finance, and mechanics to resources management, and space navigation. Therefore, it is not surprising that impulsive dynamic optimization problems, either in the control or in the calculus of variations formulation, have been addressed by a vast body of literature. Closer to the current work and without the preoccupation of being exhaustive, we cite [2], [3], [6], [7], [8], [11], [13], [15], [16], [17], [19], [20] and [21]. In this article, we address impulsive control problems for which the dynamics are defined by a differential inclusion with a vector valued control measure.

Applied Mathematics & Optimization, 2013

The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption 1 .