Integral representation for bracket-generating multi-flows

If $f_1,f_2$ are smooth vector fields on an open subset of an Euclidean space and $[f_1,f_2]$ is their Lie bracket, the asymptotic formula $$\Psi_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 [f_1,f_2](x) +o(t_1t_2),$$ where we have set $\Psi_{[f_1,f_2]}(t_1,t_2)(x) := \exp(-t_2f_2)\circ\exp(-t_1f_1)\circ\exp(t_2f_2)\circ\exp(t_1f_1)(x)$, is valid for all $t_1,t_2$ small enough. In fact, the integral, exact formula \begin{equation}\label{abstractform} \Psi_{[f_1,f_2]}(t_1,t_2)(x) - x = \int_0^{t_1}\int_0^{t_2}[f_1,f_2]^{(s_2,s_1)} (\Psi(t_1,s_2)(x))ds_1\,ds_2 , \end{equation} where $[f_1,f_2]^{(s_2,s_1)}(y) := D\Big(\exp(s_1f_1)\circ \exp(s_2f_2{{)}}\Big)^{-1}\cdot [f_1,f_2](\exp(s_1f_1)\circ \exp(s_2f_2){(y)}),$ with ${{y = \Psi(t_1,s_2)(x)}}$ has also been proven. Of course the integral formula can be regarded as an improvement of the asymptotic formula. In this paper we show that an integral representation holds true for any iterated bracket made from elements of a family of vector fields ${f_1,\dots,f_{{k}}}$. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving nonsmooth vector fields.Comment: 1 figur

Asymptotic controllability and optimal control

We consider a control problem where the state must reach asymptotically a target while paying an integral payoff with a non-negative Lagrangian. The dynamics is just continuous, and no assumptions are made on the zero level set of the Lagrangian. Through an inequality involving a positive number $\bar p_0$ and a Minimum Restraint Function $U=U(x)$ --a special type of Control Lyapunov Function-- we provide a condition implying that (i) the control system is asymptotically controllable, and (ii) the value function is bounded above by $U/\bar p_0$

Necessary conditions involving Lie brackets for impulsive optimal control problems; the commutative case

In this article we study control problems with systems that are governed by ordinary differential equations whose vector fields depend linearly in the time derivatives of some components of the control. The remaining components are considered as classical controls. This kind of system is called `impulsive system'. We assume that the vector fields multiplying the derivatives of each component of the control are commutative. We derive new necessary conditions in terms of the adjoint state and the Lie brackets of the data functions

A note on systems with ordinary and impulsive controls

We investigate an everywhere defined notion of solution for control systems whose dynamics depend nonlinearly on the control $u$ and state $x,$ and are affine in the time derivative $\dot u.$ For this reason, the input $u,$ which is allowed to be Lebesgue integrable, is called impulsive, while a second, bounded measurable control $v$ is denominated ordinary. The proposed notion of solution is derived from a topological (non-metric) characterization of a former concept of solution which was given in the case when the drift is $v$-independent. Existence, uniqueness and representation of the solution are studied, and a close analysis of effects of (possibly infinitely many) discontinuities on a null set is performed as well.Comment: Article published in IMA J. Math. Control Infor

On the Control of Non Holonomic Systems by Active Constraints

The paper is concerned with mechanical systems which are controlled by implementing a number of time-dependent, frictionless holonomic constraints. The main novelty is due to the presence of additional non-holonomic constraints. We develop a general framework to analyze these problems, deriving the equations of motion and studying the continuity properties of the "control-to-trajectory" maps. Various geometric characterizations are provided, in order that the equations be affine w.r.t. the time derivative of the control. In this case the system is fit for jumps, and the evolution is well defined also in connection with discontinuous control functions. The classical Roller Racer provides an example where the non-affine dependence of the equations on the derivative of the control is due only to the non-holonomic constraint. This is a case where the presence of quadratic terms in the equations can be used for controllability purposes.Comment: 26 pages, 3 figures. The current version will appear on Discrete and Continuous Dynamical Systems, Series

Necessary conditions involving Lie brackets for impulsive optimal control problems

We obtain higher order necessary conditions for a minimum of a Mayer optimal control problem connected with a nonlinear, control-affine system, where the controls range on an m-dimensional Euclidean space. Since the allowed velocities are unbounded and the absence of coercivity assumptions makes big speeds quite likely, minimizing sequences happen to converge toward "impulsive", namely discontinuous, trajectories. 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 will illustrate how the chance of using impulse perturbations makes it possible to derive a Higher Order Maximum Principle which includes both the usual needle variations (in space-time) and conditions involving iterated Lie brackets. An example, where a third order necessary condition rules out the optimality of a given extremal, concludes the paper.Comment: Conference pape

A Higher-order Maximum Principle for Impulsive Optimal Control Problems

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 {\it 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

Integral representations for bracket-generating multi-flows

If f1, f2 are smooth vector fields on an open subset of an Euclidean space and [f1, f2] is their Lie bracket, the asymptotic formula (Equation presented), (1) where we have set \u3a6[f1 f2](t1,t2)(x)def= exp(-t2f2) oexp(-t1f1) o exp(t2f2)\ub0 exp(t1f1)(x), is valid for all t1,t2 small enough. In fact, the integral, exact formula (Equation presented), (2) where [f1, f2](s2,s1)(y)def= D exp(s1f1) oexp(s2f2))-1(y) \ub7 [f1, f2](exp(s1f1) o exp(s2f2)(y)), has also been proven. Of course (2) can be regarded as an improvement of (1). In this paper we show that an integral representation like (2) holds true for any iterated Lie bracket made of elements of a family f1,..., fm of vector fields. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving non-smooth vector fields

A geometrically based criterion to avoid infimum-gaps in Optimal Control

In optimal control theory the expression infimum gap means a strictly negative difference between the infimum value of a given minimum problem and the infimum value of a new problem obtained by the former by extending the original family V of controls to a larger family W. Now, for some classes of domain-extensions -- like convex relaxation or impulsive embedding of unbounded control problems -- the normality of an extended minimizer has been shown to be sufficient for the avoidance of an infimum gaps. A natural issue is then the search of a general hypothesis under which the criterium 'normality implies no gap' holds true. We prove that, far from being a peculiarity of those specific extensions and from requiring the convexity of the extended dynamics, this criterium is valid provided the original family V of controls is abundant in the extended family W. Abundance, which is stronger than the mere C^0-density of the original trajectories in the set of extended trajectories, is a dynamical-topological notion introduced by J. Warga, and is here utilized in a 'non-convex' version which, moreover, is adapted to differential manifolds. To get the main result, which is based on set separation arguments, we prove an open mapping result valid for Quasi-Differential-Quotient (QDQ) approximating cones, a notion of 'tangent cone' resulted as a peculiar specification of H. Sussmann's Approximate-Generalized-Differential-Quotients (AGDQ) approximating cone