Let $\dot{x}=f(x,u)$ be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke’s generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov...

Let $\dot{x}=f(x,u)$ be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov...

We consider a finite-dimensional control system $\left(\Sigma \right)\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer $k$ that renders $\dot{x}=f(x,k\left(x\right))$ globally asymptotically stable. Moreover, for $(H,p,q)$ with $H$ an output map and $1\le p\le q\le \infty$, we assume that there exists a ${𝒦}_{\infty }$-function $\alpha$ such that $\parallel H\left(x_u\right){\parallel }_q\le {\alpha (\parallel u\parallel }_p)$, where $x_u$ is the maximal solution of ${\left(\Sigma \right)}_k\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, corresponding to $u$ and to the initial condition $x\left(0\right)=0$. Then, the gain function $G_{(H,p,q)}$ of $(H,p,q)$ given by$$G_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_p=X}{sup}{\parallel H\left(x_u\right)\parallel }_q,$$is well-defined. We call profile of $k$ for $(H,p,q)$ any ${𝒦}_{\infty }$-function which is of the same order of magnitude as $G_{(H,p,q)}$. For the double integrator...

We consider a finite-dimensional control system $\left(\Sigma \right)\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer k that renders $\dot{x}=f(x,k\left(x\right))$ globally asymptotically stable. Moreover, for (H,p,q) with H an output map and $1\le p\le q\le \infty$, we assume that there exists a ${𝒦}_{\infty }$-function α such that $\parallel H\left(x_u\right){\parallel }_q\le {\alpha (\parallel u\parallel }_p)$, where xu is the maximal solution of ${\left(\Sigma \right)}_k\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, 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 14.5cm $$G_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_p=X}{sup}{\parallel H\left(x_u\right)\parallel }_q,$$ is well-defined. We call profile of k for (H,p,q) any ${𝒦}_{\infty }$-function which is of the same order of...