1,721,077 research outputs found
A geometrically based criterion to avoid infimum gaps in optimal control
Full text linkIn optimal control theory, infimum gap means a non-zero difference between the infimum values of a given minimum problem and an extended problem obtained by embedding the original family V of controls in a larger family W. For some embeddings – like standard convex relaxations or impulsive extensions – the normality of an extended minimizer has been shown to be sufficient for the avoidance of 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 this criterium is actually valid as soon as V is abundant in W, without any convexity assumption on the extended dynamics. Abundance, which was introduced by J. Warga in a convex context and was later generalized by B. Kaskosz, strengthens density, the latter being not sufficient for the mentioned criterium to hold true
High order Lyapunov-like functions for optimal control
Full text linkWe consider an optimal control problem where the state has to approach asymp -totically a closed target, while paying an integral cost with a non-negative Lagrangian £. We generalize the dissipative relation that usually defines a Control Lvapunov Function bv introducing a weaker differential inequality, which involves both the Lagrangian £ and higher order dynamics' directions expressed in form of iterated Lie brackets up to a certain degree k. The existence of a solution U of the resulting extended relation turns out to be sufficient for a twofold goal: on the one hand, it ensures that the system is globally asymptotically controllable to the target, and, 011 the other hand, it implies that the value function associated to the minimization problem is bounded above by a [/-dependent function. We call such a solution U a degree-k Minimum Restraint Function (k > 1). A11 example is provided where a smooth degree-1 Minimum Restraint Function fails to exist, while the distance from the target happens to be a C°° degree-2 Minimum Restraint Function
UNBOUNDED CONTROL, INFIMUM GAPS, AND HIGHER ORDER NORMALITY
No full textIn optimal control theory one sometimes extends the minimization domain of a given problem, with the aim of achieving the existence of an optimal control. However, this issue is naturally confronted with the possibility of a gap between the original infimum value and the extended one. Avoiding this phenomenon is not a trivial issue, especially when the trajectories are subject to endpoint constraints. However, since the seminal works by Warga, some authors have recognized "normality"of an extended minimizer as a condition guaranteeing the absence of an infimum gap. Yet, normality is far from being necessary for this goal, a fact that makes the search for weaker assumptions a reasonable aim. In relation to a control-affine system with unbounded controls, in this paper we prove a sufficient no-gap condition based on a notion of higher order normality, which is less demanding than the standard normality and involves iterated Lie brackets of the vector fields defining the dynamics
A Lie-bracket-based notion of stabilizing feedback in optimal control
Full text linkFor a control system two major objectives can be considered: the stabilizability with respect to a given target, and the minimization of an integral functional (while the trajectories reach this target). Here we consider a problem where stabilizability or controllability are investigated together with the further aim of a 'cost regulation', namely a state-dependent upper bound of the functional. This paper is devoted to a crucial step in the programme of establishing a chain of equivalences among degree-k stabilizability with regulated cost, asymptotic controllability with regulated cost and the existence of a degree-k Minimum Restraint Function (which is a special kind of Control Lyapunov Function). Besides the presence of a cost we allow the stabilizing 'feedback' to give rise to directions that range in the union of original directions and the family of iterated Lie bracket of length <= k. In the main result asymptotic controllability [resp. with regulated cost] is proved to be necessary for degree-k stabilizability [resp. with regulated cost]. Further steps of the above-mentioned logical chain as, for instance, a Lyapunov-type inverse theorem - i.e. the possibility of deriving existence of a Minimum Restraint Function from stabilizability - are proved in companion papers
Calibration results of a new generation capillary type thermal mass flowmeter for natural gas
No full textHJ Inequalities Involving Lie Brackets and Feedback Stabilizability with Cost Regulation
Full text linkWith reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree k = 1, and we call any of its (suitably defined) solutions a degree -k Minimum Restraint Function. We prove that the existence of a degree -k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost
On Quasi Differential Quotients
No full textWe discuss basic properties and some applications of a generalized notion of differential for set-valued maps, called Quasi Differential Quotient. The latter was proved to be extremely useful to deal with several open questions in Optimal Control, such as the so called "Infimum Gap" problem. Furthermore, it recently revealed to be a valuable tool to prove second order necessary conditions expressed in terms of non-smooth Lie-brackets when the vector field is merely Lipschitz continuous. In this paper, we present a first step towards a more systematic theory of Quasi Differential Quotients and we study the relation between Quasi Differential Quotients and other generalized differentials such as the Sussmann's Generalized Differential Quotient and Sussman's Approximate Generalized Differential Quotient, the Clarke's Generalized Jacobian and the Warga's Derivative Container
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