Non-Lipschitz points and the SBV regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function $T$ under controllability conditions which do not imply the Lipschitz continuity of $T$. We consider first the case of normal linear control systems with constant coefficients in $\mathbb{R}^N$. We characterize points around which $T$ is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call $\mathcal{S}$. Furthermore, we show that $\mathcal{S}$ is $\mathcal{H}^{N-1}$-rectifiable with positive $\mathcal{H}^{N-1}$-measure. Second, we consider a class of control-affine \textit{planar} nonlinear systems satisfying a second order controllability condition: we characterize the set $\mathcal{S}$ in a neighborhood of the origin in a similar way and prove the $\mathcal{H}^1$-rectifiability of $\mathcal{S}$ and that $\mathcal{H}^1(\mathcal{S})>0$. In both cases, $T$ is known to have epigraph with positive reach, hence to be a locally $BV$ function (see \cite{CMW,GK}). Since the Cantor part of $DT$ must be concentrated in $\mathcal{S}$, our analysis yields that $T$ is $SBV$, i.e., the Cantor part of $DT$ vanishes. Our results imply also that $T$ is locally of class $\mathcal{C}^{1,1}$ outside a $\mathcal{H}^{N-1}$-rectifiable set. With small changes, our results are valid also in the case of multiple control input.Comment: 23 page

Exterior sphere condition and time optimal control for differential inclusions

The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened set-up, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well-known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz

On the Burgers-Poisson Equation

In this paper, we prove the existence and uniqueness of weak entropy solutions to the Burgers-Poisson equation for initial data in L^1(R). Additional an Oleinik type estimate is established and some criteria on local smoothness and wave breaking for weak entropy solutions are provided.Comment: 22 page

The relationship between proactive personality and performance: Why and when?

In this dissertation, I developed two measures of the value congruence constructs: value congruence with organization and value congruence with supervisor. I also hypothesized that value congruence with one\u27s supervisor and one\u27s organization will be a function of the extent to which the subordinate and supervisor share similarity in proactive personality and that the two value congruence constructs would mediate the relationship between proactive personality and proactive behavior. I further hypothesized that the relationships between value congruence and proactive behavior would be the function of the extent to which employees perceived support from their supervisor and their organization. Finally, I hypothesized that the relationship between proactive behavior and performance would be moderated by supervisor perceptions of subordinate value congruence. All the hypotheses were fully or partially supported. By investigating the role of value congruence and by emphasizing the role supervisors\u27 play in fostering and rewarding proactive behavior, our study contributes to both the proactivity literature and the broader person-environment fit literature

Improved N-Type 4h-Sic Epitaxial Layer Radiation Detectors and Noise Analysis of Front-End Readout Electronics

Schottky barrier radiation detectors were fabricated on n-type 4H-SiC epitaxial layers (12 – 50 μm) grown by hot wall CVD process on highly nitrogen doped 4H-SiC (0001) substrates with 4-8º off-cut towards the ̅ direction. Ni/4H-SiC Schottky barrier radiation detectors, a very low leakage current of 0.18 nA at 250 V bias, revealing low thermal noise, was observed in current-voltage (I-V) measurements. Using a thermionic emission model, junction properties such as barrier height of ≥1.10 eV and an ideality factor of ≤1.29 were determined. An effective carrier concentration of 1.03×1015 cm-3 was calculated by capacitance-voltage (C-V) measurement. Deep level transient spectroscopy (DLTS) was used to investigate electrically active defects in epilayer. Defect parameters such as activation energy, capture cross-section, and density of defects were calculated from Arrhenius plots. DLTS revealed the presence of shallow level defects related to titanium impurities, electrically active lifetime killer Z1/2 defect, and deep level defects assigned as EH6/7 which are related to carbon and carbon-silicon vacancies. The density of Z1/2 defect, the most detrimental to detector performance, was 1.6×1012 cm-3, orders of magnitude lower compared to other 4H-SiC detectors. Detector performances were evaluated in terms of the energy resolution at full-width at half-maximum (FWHM) using pulse height spectroscopy (PHS) measurements with 0.1 μCi 241Am source. Charge collection efficiency was investigated using a drift-diffusion charge transport model. The energy resolution for 5.486 MeV alpha particles was 166 keV with charge collection efficiency of 22.6%. Electronic noise analysis of front-end readout system was carried out in terms of equivalent noise charge (ENC) in order to study the contribution of white series noise, pink noise ( parallel and ⁄ series) and white parallel noise to the total electronic noise in the detection system. New edge termination was developed using surface passivating layers of silicon dioxide (SiO2) and silicon nitride (Si3N4) in order to improve detector performance. With edge termination, reverse leakage current of Ni/4H-SiC epilayer detector was improved significantly (nA to pA) leading to an increased signal-to-noise ratio. Improved Schottky properties such as barrier height of ~1.7 eV and diode ideality factor of ~1.07 were observed indicating a better surface uniformity that enhanced charge collection efficiency. C-V measurement confirmed a doping concentration of 2.4 x 1014 cm-3 ensuring a fully depleted (~20 μm) detector at bias voltages as low as ~70 V. DLTS analysis showed a decreased concentration of performance limiting Z1/2 defect level and absence of EH6/7 deep-levels with edge termination, ensuing a more complete charge collection. Alpha spectroscopy measurements revealed an improved detector energy resolution from ~0.7% to ~0.4% for 5.48 MeV alpha particles with edge termination. 4H-SiC epitaxial detector with ruthenium (Ru) Schottky barrier contact (in addition to Ni being used in above studies) was investigated for operation in harsh environments with high temperature and high radiation. Ru/4H-SiC Schottky detectors exhibited excellent rectification and improved junction properties, even without edge termination. However, inhomogeneity of the Schottky barrier heights was observed due to interfacial defects resulting from a solid-state reaction involving Ru, Si, and C. As a result, pulse-height spectra with 241Am source were broad, and the three characteristic alpha peaks were not resolved. The energy resolution was calculated to be ~ 0.75% at 180 V reverse bias at room temperature