In some preceding works we consider a class $\mathcal{O}\mathcal{P}$ of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line $l=l_{\gamma (\cdot )}$, regarded as determined by its (variable) curvature function $\gamma (\cdot )$ of domain $\left[s_0,s_1\right]$. Assume that the problem $\tilde{\mathcal{P}}\in \mathcal{O}\mathcal{P}$ is regular but has an impulsive monotone character in the sense that near each of some points ${\delta }_1$ to $\delta {}_{\nu }\gamma (\cdot )$ is monotone and $\left|\gamma {}^{\prime }\right(\cdot \left)\right|$ is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify $\tilde{\mathcal{P}}$ into a structurally...

In this Note (which will be followed by a second) we consider a Lagrangian system $\mathrm{\Sigma }$ (possibly without any Lagrangian function) referred to $N+1$ coordinates $q_1\mathrm{\cdots },q_N$, $u$, with $u$ to be used as a control, and precisely to add to $\mathrm{\Sigma }$ a frictionless constraint of the type $u=u\left(t\right)$. Let $\mathrm{\Sigma }$'s (frictionless) constraints be represented by the manifold $V_t$ generally moving in Hertz's space. We also consider an instant $d$ (to be used for certain limit discontinuity-properties), a point $\left(\overline{q},\overline{u}\right)$ of $V_d$, a value $\overline{p}$ for $\mathrm{\Sigma }$'s momentum...

Let $\left(A,B\right)$ be a linear time dependent control process, defined on an open interval $J=]a,\omega [$ with $a\ge -\mathrm{\infty }$ and $\omega \le \mathrm{\infty }$; in this paper we give a description of the function $\tau :I\to J$, $\tau \left(t\right)=inf\mathit{\{}t{}^{\prime }>t:(A,B)$ is $\left[t,t^{\prime }\right]$-globally controllable from $0\mathit{\}}$ where $I=\mathit{\{}t\in J:\mathrm{\exists }t{}^{\prime }\in J$ with $\left(A,B\right)\left[t,t^{\prime }\right]$-globally controllable from $0\mathit{\}}$.

Let $\mathrm{\Omega }$ be a bounded open convex set of class $C^2$. Let $a\left(x,H\left(u\right)\right)$ be a non linear operator satisfying the condition (A) (elliptic) with constants $\alpha$, $\gamma$, $\delta$. We prove that a number $\lambda \ge 0$ is an eigenvalue for the operator $a\left(x,H\left(u\right)\right)$ if and only if the number $\alpha \lambda$ is an eigen-value for the operator $\mathrm{\Delta }u$. If $\lambda \ge 0$ , the two systems $a\left(x,H\left(u\right)\right)=\lambda u$ and $\mathrm{\Delta }u=\alpha \lambda u$ have the same solutions. In particular, also the eventual eigen-values of the operator $a\left(x,H\left(u\right)\right)$ should all be negative. Finally, we obtain a sufficient condition for the existence of solutions $u\in H^2\cap H_0^1\left(\mathrm{\Omega }\right)$...