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On the L p -stabilization of the double integrator subject to input saturation

Yacine Chitour — 2001

ESAIM: Control, Optimisation and Calculus of Variations

We consider a finite-dimensional control system ( Σ ) x ˙ ( t ) = f ( x ( t ) , u ( t ) ) , such that there exists a feedback stabilizer k that renders x ˙ = f ( x , k ( x ) ) globally asymptotically stable. Moreover, for ( H , p , q ) with H an output map and 1 p q , we assume that there exists a 𝒦 -function α such that H ( x u ) q α ( u p ) , where x u is the maximal solution of ( Σ ) k x ˙ ( t ) = f ( x ( t ) , k ( x ( t ) ) + u ( t ) ) , corresponding to u and to the initial condition x ( 0 ) = 0 . Then, the gain function G ( H , p , q ) of ( H , p , q ) given by G ( H , p , q ) ( X ) = def sup u p = X H ( x u ) q , is well-defined. We call profile of k for ( H , p , q ) any 𝒦 -function which is of the same order of magnitude as...

A continuation method for motion-planning problems

Yacine Chitour — 2006

ESAIM: Control, Optimisation and Calculus of Variations

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

A continuation method for motion-planning problems

Yacine Chitour — 2005

ESAIM: Control, Optimisation and Calculus of Variations

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

On the -stabilization of the double integrator subject to input saturation

Yacine Chitour — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We consider a finite-dimensional control system ( Σ ) x ˙ ( t ) = f ( x ( t ) , u ( t ) ) , such that there exists a feedback stabilizer that renders x ˙ = f ( x , k ( x ) ) globally asymptotically stable. Moreover, for with an output map and 1 p q , we assume that there exists a 𝒦 -function such that H ( x u ) q α ( u p ) , where is the maximal solution of ( Σ ) k x ˙ ( t ) = f ( x ( t ) , k ( x ( t ) ) + u ( t ) ) , corresponding to and to the initial condition . Then, the gain function G ( H , p , q ) of given by 14.5cm G ( H , p , q ) ( X ) = def sup u p = X H ( x u ) q , is well-defined. We call profile of for any 𝒦 -function which is of the same order of magnitude as G ( H , p , q ) ....

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