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

Almost periodic solutions of periodic linear partial functional differential equations

We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.Comment: 17 page

On regular and singular points of the minimum time function

In this thesis, we study the regularity of the minimum time function Τ for both linear and nonlinear control systems in Euclidean space. We first consider nonlinear problems satisfying Petrov condition. In this case, Τ is locally Lipschitz and then is differentiable almost everywhere. In general, Τ fails to be differentiable at points where there are multiple time optimal trajectories and its differentiability at a point does not guarantee continuous differentiability around this point. We show that, under some regularity assumptions, the non-emptiness of proximal subdifferential of the minimum time function at a point x implies its continuous differentiability on a neighborhood of Υ. The technique consists of deriving sensitivity relations for the proximal subdifferential of the minimum time function and excluding the presence of conjugate points when the proximal subdifferential is nonempty. We then study the regularity the minimum time function Τ to reach the origin under controllability conditions which do not imply the Lipschitz continuity of Τ. Basing on the analysis of zeros of the switching function, we find out singular sets (e.g., non - Lipschitz set, non - differentiable set) and establish rectifiability properties for them. The results imply further regularity properties of Τ such as the SBV regularity, the differentiability and the analyticity. The results are mainly for linear control problems

Single-valued core selection and aggregate-monotonic solutions to equitable cost allocation

Single-valued core selection, equity and aggregate monotonicity are desirable properties for cost allocation but offer challenges to develop suitable allocation methods to satisfy. This paper attempts to develop a new solution approach to meeting these properties. The idea is first to identify the critical value of the grand total cost for non-empty core, using an LP to maximize the grand cost subject to individual and group rationality conditions. The critical grand cost is then allocated by any single-valued core allocation methods available such as the nucleolus and its variants. The non-critical grand cost values are allocated simply by scaling up or down from the critical case so as to ensure aggregate-monotonicity. In addition, a new core allocation method is proposed for the critical case, based on prorating the best and worse costs that are feasible bounds for each player in the grand cooperation. Existence of such proration fractions is proved along with lower and upper bounds identified explicitly, which indicates some sense of equity. The new method is found to be desirable for the cases with two or three players, and more research is being conducted for general cases of more than three players

In vitro susceptibility of zoliflodacin and solithromycin in Neisseria gonorrhoeae isolated in Republic of Korea from 2016 to 2018

Background: Zoliflodacin (topoisomerase II inhibitor) and solithromycin (first fluoroketolide) are new drugs which are in clinical development for treatment of uncomplicated gonorrhea. In this study, we determined the in vitro activity of zoliflodacin and solithromycin in gonococcal isolates from Republic of Korea. Methods: A total of 250 isolates of N. gonorrhoeae collected throughout Republic of Korea from 2016 to 2018 were tested to determine MIC of therapeutic antimicrobials using CLSI agar dilution method. Results: Most isolates (86.4%, 234/250) were non-susceptible to penicillin G, tetracycline and ciprofloxacin, but all isolates were susceptible to ceftriaxone and spectinomycin. The MIC range for zoliflodacin and solithromycin were ≤0.015–0.12 mg/L and ≤0.015–0.5 mg/L, respectively. MIC50 and MIC90 were 0.03 and 0.06 mg/L for zoliflodacin and 0.06 and 0.12 mg/L for solithromycin. All isolates belonged to the wild-type MIC distribution for zoliflodacin and there were no cross-resistance between zoliflodacin and ciprofloxacin (also a topoisomerase II inhibitor). The azithromycin-resistant isolate with the highest MIC of azithromycin (32 mg/L) had the highest MIC of solithromycin (0.5 mg/L), illustrating cross-resistance between azithromycin and solithromycin at higher MICs. Conclusion: Zoliflodacin and solithromycin showed potent antimicrobial activity against contemporary multidrug-resistant N. gonorrhoeae isolates in Republic of Korea. However, the recently finished phase III clinical trial for solithromycin identified relatively many treatment failures. Zoliflodacin remains more promising and a multi-continental phase III clinical trial will be initiated in 2019open석