On the L p -stabilization of the double integrator subject to input saturation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 291-331.

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 1pq, 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 G (H,p,q) . For the double integrator subject to input saturation and stabilized by k L (x)=-(11) T x, we determine the profiles corresponding to the main output maps. In particular, if σ 0 is used to denote the standard saturation function, we show that the L 2 -gain from the output of the saturation nonlinearity to u of the system x ¨=σ 0 (-x-x ˙+u) with x(0)=x ˙(0)=0, is finite. We also provide a class of feedback stabilizers k F that have a linear profile for (x,p,p), 1p. For instance, we show that the L 2 -gains from x and x ˙ to u of the system x ¨=σ 0 (-x-x ˙-(x ˙) 3 +u) with x(0)=x ˙(0)=0, are finite.

Classification : 93D15,  93D21,  93D30
Mots clés : nonlinear control systems, L p -stabilization, input-to-state stability, finite-gain stability, input saturation, Lyapunov function
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     author = {Chitour, Yacine},
     title = {On the $L^p$-stabilization of the double integrator subject to input saturation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {291--331},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
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     mrnumber = {1824105},
     language = {en},
     url = {http://archive.numdam.org/item/COCV_2001__6__291_0/}
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Chitour, Yacine. On the $L^p$-stabilization of the double integrator subject to input saturation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 291-331. http://archive.numdam.org/item/COCV_2001__6__291_0/

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