In the present paper, we study the problem of small-time local attainability (STLA) of a closed set. For doing this, we introduce a new concept of variations of the reachable set well adapted to a given closed set and prove a new attainability result for a general dynamical system. This provide our main result for nonlinear control systems. Some applications to linear and polynomial systems are discussed and STLA necessary and sufficient conditions are obtained when the considered set...

The present paper studies the following constrained vector optimization problem: ${min}_{C}f\left(x\right)$, $g\left(x\right)\in -K$, $h\left(x\right)=0$, where $f:{ℝ}^n\to {ℝ}^m$, $g:{ℝ}^n\to {ℝ}^p$ are locally Lipschitz functions, $h:{ℝ}^n\to {ℝ}^q$ is $C^1$ function, and $C\subset {ℝ}^m$ and $K\subset {ℝ}^p$ are closed convex cones. Two types of solutions are important for the consideration, namely $w$-minimizers (weakly efficient points) and $i$-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point $x^0$ to be a $w$-minimizer and first-order sufficient conditions for $x^0$...

Given the probability measure $\nu$ over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set $\Sigma$ composed by $n$ points in $\Omega$ in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all $n$ points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

Given the probability measure ν over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

New $L^1$-lower semicontinuity and relaxation results for integral functionals defined in BV($\Omega$) are proved, under a very weak dependence of the integrand with respect to the spatial variable $x$. More precisely, only the lower semicontinuity in the sense of the $1$-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to $x$. Under this further BV dependence, a...

New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a...

A lower semicontinuity result in BV is obtained for quasiconvex integrals with subquadratic growth. The key steps in this proof involve obtaining boundedness properties for an extension operator, and a precise blow-up technique that uses fine properties of Sobolev maps. A similar result is obtained by Kristensen in [Calc. Var. Partial Differ. Equ. 7 (1998) 249–261], where there are weaker asssumptions on convergence but the integral needs to satisfy a stronger growth condition.

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where $\mu$ is a given positive measure on ${ℝ}^n$, and the vector-valued function $u$ belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with $\mu$. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to $\mu$. More precisely, for fully general $\mu$, a notion of quasiconvexity for $f$ along the tangent bundle to $\mu$, turns out to be necessary for lower...

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where μ is a given positive measure on ${ℝ}^n$, and the vector-valued function u belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower...

We prove a lower semicontinuity result for variational integrals associated with a given first order elliptic complex, extending, in this general setting, a well known result in the case ${}^{\text{'}}(ℝⁿ,ℝ){\to }^{\nabla }\text{'}(ℝⁿ,ℝⁿ){\to }^{curl}\text{'}(ℝⁿ,{ℝ}^{n\times n})$.

The lower semicontinuity of functionals of the type ${\int }_{\Omega }f(x,u,v,\nabla u)dx$ with respect to the $\left(W^{1,1}\times L^p\right)$-weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1,1}\times L^p$ for the lower semicontinuous envelope is also provided.

A self-balancing bicycle robot based on the concept of an inverted pendulum is an unstable and nonlinear system. To stabilize the system in this work, the following three main components are required, i. e., (1) an IMU sensor that detects the tilt angle of the bicycle robot, (2) a controller that is used to control motion of a reaction wheel, and (3) a reaction wheel that is employed to produce reactionary torque to balance the bicycle robot. In this paper, we propose three control strategies: linear...