arXiv subjects

Yacine Chitour

Publications and source records attributed to Yacine Chitour.

93 records · Page 6Linked to original sources

arXiv2006

Singular trajectories of control-affine systems

When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -- with respect to the Whitney topology --, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend previous results. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the LARC, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues.

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arXiv2005

Time Optimal Synthesis for Left--Invariant Control Systems on SO(3)

Consider the control system given by $\dot x=x(f+ug)$, where $x\in SO(3)$, $|u|\leq 1$ and $f,g\in so(3)$ define two perpendicular left-invariant vector fields normalized so that $\|f\|=\cos(\al)$ and $\|g\|=\sin(\al)$, $\al\in ]0,π/4[$. In this paper, we provide an upper bound and a lower bound for $N(α)$, the maximum number of switchings for time-optimal trajectories. More precisely, we show that $N_S(\al)\leq N(\al)\leq N_S(\al)+4$, where $N_S(\al)$ is a suitable integer function of $\al$ which for $\al\to 0$ is of order $π/(4α).$ The result is obtained by studying the time optimal synthesis of a projected control problem on $R P^2$, where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere $S^2$. It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.

math.OC↗

arXiv2004

Common Polynomial Lyapunov Functions for Linear Switched Systems

In this paper, we consider linear switched systems $\dot x(t)=A_{u(t)} x(t)$, $x\in\R^n$, $u\in U$, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching ({\bf UAS} for short). We first prove that, given a {\bf UAS} system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the {\bf UAS} systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.

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